bench_coq_unzip_run_meta_hammer_lemma.txt

========
Experiment configuration: python experiments.py --experiment_name run_meta.py --problem_name problem_unzip --language Coq --use_hammer True --extract_lemma_depth 1 --n_trials 10 --mins_timeout 10 --seed 42 


[2024-01-28 00:19:00,440] [INFO] [real_accelerator.py:158:get_accelerator] Setting ds_accelerator to cuda (auto detect)
Trial  1
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
SCORE
1.0
['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  (map fst l, map snd l).

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n\nDefinition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  (map fst l, map snd l).\n\nLemma unzip_length : forall l, length (fst (unzip l)) = length l /\\ length (snd (unzip l)) = length l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  (map fst l, map snd l).

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\nProof.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  (map fst l, map snd l).

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  intros l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  (map fst l, map snd l).

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  split.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  split.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l

### given assumptions

l as follows.

l : list (nat * nat).
### Need to show

Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  (map fst l, map snd l).

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  split.
  - rewrite <- (map_length fst l).
    hammer.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  - rewrite <- (map_length fst l).\n    hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip l)) =
Datatypes.length (map fst l)

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  (map fst l, map snd l).

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  split.
  - rewrite <- (map_length fst l).
    hammer.
  - rewrite <- (map_length snd l).
    hammer.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  - rewrite <- (map_length snd l).\n    hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (snd (unzip l)) =
Datatypes.length (map snd l)

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  (map fst l, map snd l).

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  split.
  - rewrite <- (map_length fst l).
    hammer.
  - rewrite <- (map_length snd l).
    hammer.
Qed.
SCORE
1.0
['\nQed.']
CHOSEN SOLUTION

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (snd (unzip l)) =
Datatypes.length (map snd l)

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  (map fst l, map snd l).

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  split.
  - rewrite <- (map_length fst l).
    hammer.
  - rewrite <- (map_length snd l).
    hammer.
Qed.```

STATS

number of nodes: 17
number of gen nodes: 9 (including leaves: 1)
number of back nodes: 8 (including leaves: 8)

expansion count: 8 (including failed: 0)

cache stats {'hit': 0, 'miss': 8}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20]

Trial calls: [8]


Time Statistics:

  Min: 20.00 sec, Max: 20.00 sec, Average: 20.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 8 calls, Max: 8 calls, Average: 8.00 calls, Standard Deviation: 0.00 calls

=====SUMMARY STATISTICS FOR NON_TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20]

Trial calls: [8]


Time Statistics:

  Min: 20.00 sec, Max: 20.00 sec, Average: 20.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 8 calls, Max: 8 calls, Average: 8.00 calls, Standard Deviation: 0.00 calls

Trial  2
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Section Main.
Error: The section Main needs to be closed.


SCORE
-1.0
['Section Main.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import ListNotations.
File "./ex.v", line 8, characters 7-20:
Error: Cannot find module ListNotations


SCORE
-1.0
['Import ListNotations.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.
File "./ex.v", line 8, characters 8-9:
Error:
Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).


SCORE
None
['From Coq.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.Lists.
File "./ex.v", line 8, characters 14-15:
Error:
Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).


SCORE
None
['Lists.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.Lists.List Import List.
File "./ex.v", line 8, characters 20-26:
Error:
Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).


SCORE
-1.0
['List Import List.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > From Coq>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
SCORE
1.0
['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
SCORE
1.0
['\nImport ListNotations.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define unzip function *)
Fixpoint unzip (l : (nat * nat) list) : (nat list) * (nat list) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.
File "./ex.v", line 12, characters 20-36:
Error: Illegal application (Non-functional construction): 
The expression "(nat * nat)%type" of type "Type"
cannot be applied to the term
 "list" : "Type -> Type"


SCORE
-1.0
["\n\n(* (1) Define unzip function *)\nFixpoint unzip (l : (nat * nat) list) : (nat list) * (nat list) :=\n  match l with\n  | [] => ([], [])\n  | (x, y) :: l' =>\n    let (xs, ys) := unzip l' in\n    (x :: xs, y :: ys)\n  end."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
File "./ex.v", line 11, characters 0-251:
Error: Syntax Error: Lexer: Unterminated comment


SCORE
None
['\n\n(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)
Fixpoint unzip (l : (nat * nat) list) : (nat list * nat list) :=
  match l with
  | [] => ([], [])
  | (x, y) :: tl => (x :: (fst (unzip tl)), y :: (snd (unzip tl)))
  end.
File "./ex.v", line 12, characters 20-36:
Error: Illegal application (Non-functional construction): 
The expression "(nat * nat)%type" of type "Type"
cannot be applied to the term
 "list" : "Type -> Type"


SCORE
-1.0
[' *)\nFixpoint unzip (l : (nat * nat) list) : (nat list * nat list) :=\n  match l with\n  | [] => ([], [])\n  | (x, y) :: tl => (x :: (fst (unzip tl)), y :: (snd (unzip tl)))\n  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > Import List.
     > Import ListNotations. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  (map fst l, map snd l).

Lemma unzip_length : forall l : list (nat * nat),
  length (map fst (unzip l)) = length l /\ length (map snd (unzip l)) = length l.
File "./ex.v", line 14, characters 19-26:
Error:
In environment
l : list (nat * nat)
The term "unzip l" has type "(list nat * list nat)%type"
while it is expected to have type "list (?A1 * ?B0)".


SCORE
-1.0
['\n\nDefinition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  (map fst l, map snd l).\n\nLemma unzip_length : forall l : list (nat * nat),\n  length (map fst (unzip l)) = length l /\\ length (map snd (unzip l)) = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

['Lists.']
['List Import List.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > From Coq>>>.<<< simpl.
   Results past this point may be unreliable.

['Import ListNotations.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.Lists.List Require Import List.
File "./ex.v", line 8, characters 0-40:
Error: Cannot find a physical path bound to logical path
List with prefix Coq.Lists.List.


SCORE
None
['List Require Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.Lists.List Require Import List.

Definition unzip {A B} (l : (A * B) list) : (A list * B list) :=
  match l with
  | [] => ([], [])
  | (a, b) :: t =>
    let (la, lb) := unzip t in
    (a :: la, b :: lb)
  end.
File "./ex.v", line 8, characters 0-40:
Error: Cannot find a physical path bound to logical path
List with prefix Coq.Lists.List.


SCORE
-1.0
['\n\nDefinition unzip {A B} (l : (A * B) list) : (A list * B list) :=\n  match l with\n  | [] => ([], [])\n  | (a, b) :: t =>\n    let (la, lb) := unzip t in\n    (a :: la, b :: lb)\n  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.Lists.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > From Coq.Lists>>>.<<< simpl.
   Results past this point may be unreliable.

['List Import List.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > From Coq>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* Function unzip *)
Fixpoint unzip (l: (string * nat) list) : (string list * nat list) :=
  match l with
  | [] => ([], [])
  | (s, n) :: l' =>
    let (sl, nl) := unzip l' in
    (s :: sl, n :: nl)
  end.
File "./ex.v", line 12, characters 19-38:
Error: Illegal application (Non-functional construction): 
The expression "(string * nat)%type" of type "Type"
cannot be applied to the term
 "list" : "Type -> Type"


SCORE
-1.0
["\n\n(* Function unzip *)\nFixpoint unzip (l: (string * nat) list) : (string list * nat list) :=\n  match l with\n  | [] => ([], [])\n  | (s, n) :: l' =>\n    let (sl, nl) := unzip l' in\n    (s :: sl, n :: nl)\n  end."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

['Lists.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.Lists.List Import Length.
File "./ex.v", line 8, characters 20-26:
Error:
Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).


SCORE
-1.0
['List Import Length.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > From Coq>>>.<<< simpl.
   Results past this point may be unreliable.

['List Require Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.Lists.List Require Import List.

Definition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=
  let rec unzip_rec (l : list (A * B)) (l1 : list A) (l2 : list B) :=
    match l with
    | nil => (l1, l2)
    | (a, b) :: l' => unzip_rec l' (a
File "./ex.v", line 8, characters 0-40:
Error: Cannot find a physical path bound to logical path
List with prefix Coq.Lists.List.


SCORE
None
["\n\nDefinition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=\n  let rec unzip_rec (l : list (A * B)) (l1 : list A) (l2 : list B) :=\n    match l with\n    | nil => (l1, l2)\n    | (a, b) :: l' => unzip_rec l' (a"]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.Lists.List Require Import List.

Definition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=
  let rec unzip_rec (l : list (A * B)) (l1 : list A) (l2 : list B) :=
    match l with
    | nil => (l1, l2)
    | (a, b) :: l' => unzip_rec l' (a :: l1) (b :: l2)
    end
  in
  unzip_rec l nil nil.
File "./ex.v", line 8, characters 0-40:
Error: Cannot find a physical path bound to logical path
List with prefix Coq.Lists.List.


SCORE
-1.0
[' :: l1) (b :: l2)\n    end\n  in\n  unzip_rec l nil nil.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.Lists.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > From Coq.Lists>>>.<<< simpl.
   Results past this point may be unreliable.

['List Import List.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > From Coq>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)
Fixpoint unzip (l : (nat * nat) list) : (nat list * nat list) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: l' =>
    let (l1, l2) := unzip l' in
    (a :: l1, b :: l2)
  end.
File "./ex.v", line 12, characters 20-36:
Error: Illegal application (Non-functional construction): 
The expression "(nat * nat)%type" of type "Type"
cannot be applied to the term
 "list" : "Type -> Type"


SCORE
-1.0
[" *)\nFixpoint unzip (l : (nat * nat) list) : (nat list * nat list) :=\n  match l with\n  | nil => (nil, nil)\n  | (a, b) :: l' =>\n    let (l1, l2) := unzip l' in\n    (a :: l1, b :: l2)\n  end."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > Import List.
     > Import ListNotations. >>>simpl<<<.
   Results past this point may be unreliable.

['Lists.']
['List Import Length.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > From Coq>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.Lists.List Require Import List. Import ListNotations.
File "./ex.v", line 8, characters 0-40:
Error: Cannot find a physical path bound to logical path
List with prefix Coq.Lists.List.


SCORE
-1.0
[' Import ListNotations.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.Lists.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > From Coq.Lists>>>.<<< simpl.
   Results past this point may be unreliable.

['List Import List.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > From Coq>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.
SCORE
1.0
[" *)\n\nFixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n  | [] => ([], [])\n  | (x, y) :: l' =>\n    let (xs, ys) := unzip l' in\n    (x :: xs, y :: ys)\n  end."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
File "./ex.v", line 21, characters 0-147:
Error: Syntax Error: Lexer: Unterminated comment


SCORE
None
['\n\n(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
[' *)\n\nLemma unzip_length : forall l, length (fst (unzip l)) = length l /\\ length (snd (unzip l)) = length l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\nProof.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  intros l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| hd tl].
  - split.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  induction l as [| hd tl].\n  - split.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip [])) =
Datatypes.length []

### given assumptions

(none).
### Need to show

Datatypes.length (snd (unzip [])) =
Datatypes.length []

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| hd tl].
  - split.
    + reflexivity.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n    + reflexivity.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip [])) =
Datatypes.length []

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| hd tl].
  - split.
    + reflexivity.
    + reflexivity.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n    + reflexivity.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (snd (unzip [])) =
Datatypes.length []

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| hd tl].
  - split.
    + reflexivity.
    + reflexivity.
  - destruct hd as [x y].
    + simpl.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  - destruct hd as [x y].\n    + simpl.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl) /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl)

### given assumptions

x, y, tl, IHtl as follows.

x, y : nat
tl : list (nat * nat)
IHtl : Datatypes.length (fst (unzip tl)) =
Datatypes.length tl /\
Datatypes.length (snd (unzip tl)) =
Datatypes.length tl.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| hd tl].
  - split.
    + reflexivity.
    + reflexivity.
  - destruct hd as [x y].
    + simpl.
      rewrite IHtl.
File "./ex.v", line 32, characters 6-18:
Error:
Found no subterm matching "Datatypes.length (fst (unzip tl)) =
                           Datatypes.length tl" in the current goal.


SCORE
-1.0
['\n      rewrite IHtl.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| hd tl].
  - split.
    + reflexivity.
    + reflexivity.
  - destruct hd as [x y].
    + simpl.
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall (x y : nat) (tl : list (nat * nat)),
Datatypes.length (fst (unzip tl)) =
Datatypes.length tl /\
Datatypes.length (snd (unzip tl)) =
Datatypes.length tl ->
Datatypes.length
  (fst
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl) /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl)

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)


Lemma helper0: forall x y : nat, forall tl : list (nat * nat), forall IHtl : Datatypes.length (fst (unzip tl)) =
Datatypes.length tl /\
Datatypes.length (snd (unzip tl)) =
Datatypes.length tl, Datatypes.length
  (fst
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl) /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl).
Proof.
(* do not use induction, try using hammer *)
  intros.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  intros.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl) /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl)

### given assumptions

x, y, tl, IHtl as follows.

x, y : nat
tl : list (nat * nat)
IHtl : Datatypes.length (fst (unzip tl)) =
Datatypes.length tl /\
Datatypes.length (snd (unzip tl)) =
Datatypes.length tl.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)


Lemma helper0: forall x y : nat, forall tl : list (nat * nat), forall IHtl : Datatypes.length (fst (unzip tl)) =
Datatypes.length tl /\
Datatypes.length (snd (unzip tl)) =
Datatypes.length tl, Datatypes.length
  (fst
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl) /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl).
Proof.
(* do not use induction, try using hammer *)
  intros.
  hammer.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['\n  hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: hauto 


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl) /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl)

### given assumptions

x, y, tl, IHtl as follows.

x, y : nat
tl : list (nat * nat)
IHtl : Datatypes.length (fst (unzip tl)) =
Datatypes.length tl /\
Datatypes.length (snd (unzip tl)) =
Datatypes.length tl.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)


Lemma helper0: forall x y : nat, forall tl : list (nat * nat), forall IHtl : Datatypes.length (fst (unzip tl)) =
Datatypes.length tl /\
Datatypes.length (snd (unzip tl)) =
Datatypes.length tl, Datatypes.length
  (fst
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl) /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl).
Proof.
(* do not use induction, try using hammer *)
  intros.
  hammer.
Qed.
SCORE
1.0
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sauto 


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl) /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl)

### given assumptions

x, y, tl, IHtl as follows.

x, y : nat
tl : list (nat * nat)
IHtl : Datatypes.length (fst (unzip tl)) =
Datatypes.length tl /\
Datatypes.length (snd (unzip tl)) =
Datatypes.length tl.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)


Lemma helper0: forall x y : nat, forall tl : list (nat * nat), forall IHtl : Datatypes.length (fst (unzip tl)) =
Datatypes.length tl /\
Datatypes.length (snd (unzip tl)) =
Datatypes.length tl, Datatypes.length
  (fst
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl) /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl).
Proof.
(* do not use induction, try using hammer *)
  intros.
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| hd tl].
  - split.
    + reflexivity.
    + reflexivity.
  - destruct hd as [x y].
    + simpl. apply (@helper0 x y tl IHtl).
  Qed.
SCORE
1.0
['  Qed.']
CHOSEN SOLUTION

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sauto 


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl) /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl)

### given assumptions

x, y, tl, IHtl as follows.

x, y : nat
tl : list (nat * nat)
IHtl : Datatypes.length (fst (unzip tl)) =
Datatypes.length tl /\
Datatypes.length (snd (unzip tl)) =
Datatypes.length tl.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list. *)

Fixpoint unzip (l: list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list. *)


Lemma helper0: forall x y : nat, forall tl : list (nat * nat), forall IHtl : Datatypes.length (fst (unzip tl)) =
Datatypes.length tl /\
Datatypes.length (snd (unzip tl)) =
Datatypes.length tl, Datatypes.length
  (fst
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl) /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip tl in (x :: xs, y :: ys))) =
S (Datatypes.length tl).
Proof.
(* do not use induction, try using hammer *)
  intros.
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| hd tl].
  - split.
    + reflexivity.
    + reflexivity.
  - destruct hd as [x y].
    + simpl. apply (@helper0 x y tl IHtl).
  Qed.```

STATS

number of nodes: 51
number of gen nodes: 26 (including leaves: 5)
number of back nodes: 25 (including leaves: 21)

expansion count: 43 (including failed: 18)

cache stats {'hit': 10, 'miss': 33}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116]

Trial calls: [8, 43]


Time Statistics:

  Min: 20.00 sec, Max: 116.00 sec, Average: 68.00 sec, Standard Deviation: 48.00 sec

Call Statistics:

  Min: 8 calls, Max: 43 calls, Average: 25.50 calls, Standard Deviation: 17.50 calls

=====SUMMARY STATISTICS FOR NON_TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116]

Trial calls: [8, 43]


Time Statistics:

  Min: 20.00 sec, Max: 116.00 sec, Average: 68.00 sec, Standard Deviation: 48.00 sec

Call Statistics:

  Min: 8 calls, Max: 43 calls, Average: 25.50 calls, Standard Deviation: 17.50 calls

Trial  3
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
SCORE
1.0
['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
SCORE
1.0
['\nImport ListNotations.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.
File "./ex.v", line 13, characters 20-21:
Error:
Syntax error: 'in' expected after [term level 200] (in [binder_constr]).


SCORE
None
['\n\nDefinition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=\n  let f (x : A * B) : A * B := (fst x, snd x) in\n  let (a, b) := List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.
File "./ex.v", line 14, characters 7-8:
Error: Syntax error: ',' or ')' expected after [term level 200] (in [term]).


SCORE
None
['map f l in\n  (List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.
File "./ex.v", line 14, characters 23-24:
Error: Syntax error: ',' or ')' expected after [term level 200] (in [term]).


SCORE
None
['map fst a, List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.map snd b).

Lemma unzip_length {A B} (l : list (A * B)) : 
  let (a, b) := unzip l in
  length a = length b = length l.
File "./ex.v", line 13, characters 2-66:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) : \n  let (a, b) := unzip l in\n  length a = length b = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in
     >   (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  (length (map fst l)) = (length (map snd l)) = (length l).
Proof.
File "./ex.v", line 15, characters 2-58:
Error: Unknown interpretation for notation "_ = _ = _".


SCORE
-1.0
['\n\nDefinition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=\n  (map fst l, map snd l).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  (length (map fst l)) = (length (map snd l)) = (length l).\nProof.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   (map fst l, map snd l). >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.
File "./ex.v", line 13, characters 37-38:
Error: Syntax error: ',' or ')' expected after [term level 200] (in [term]).


SCORE
None
['map f l in (List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
File "./ex.v", line 13, characters 53-54:
Error: Syntax error: ',' or ')' expected after [term level 200] (in [term]).


SCORE
None
['map fst a, List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.map snd b).

Lemma unzip_length {A B} (l : list (A * B)) :
  let (a, b) := unzip l in
  length a = length b = length l.
File "./ex.v", line 13, characters 2-64:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in\n  length a = length b = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

['map fst a, List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.map snd b).

Lemma unzip_length_lemma {A B} (l : list (A * B)) : 
  let (a, b) := unzip l in
  length a = length b.
File "./ex.v", line 13, characters 2-66:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n\nLemma unzip_length_lemma {A B} (l : list (A * B)) : \n  let (a, b) := unzip l in\n  length a = length b.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in
     >   (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

['\nImport ListNotations.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Section Main.
Error: The section Main needs to be closed.


SCORE
-1.0
['\n\nSection Main.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > Import List.
     > Import ListNotations. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define unzip function *)
Fixpoint unzip (l: (nat * nat) list) : (nat list * nat list) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.
File "./ex.v", line 12, characters 19-35:
Error: Illegal application (Non-functional construction): 
The expression "(nat * nat)%type" of type "Type"
cannot be applied to the term
 "list" : "Type -> Type"


SCORE
-1.0
["\n\n(* (1) Define unzip function *)\nFixpoint unzip (l: (nat * nat) list) : (nat list * nat list) :=\n  match l with\n  | [] => ([], [])\n  | (x, y) :: l' =>\n    let (xs, ys) := unzip l' in\n    (x :: xs, y :: ys)\n  end."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=
  let rec unzip_helper (l : list (A * B)) (accA : list A) (accB : list B) :=
    match l with
      | nil => (accA, accB)
      | (a, b) :: l' => unzip_helper l' (a
File "./ex.v", line 14, characters 42-43:
Error: Syntax error: ',' or ')' expected after [term level 200] (in [term]).


SCORE
None
["\n\nDefinition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=\n  let rec unzip_helper (l : list (A * B)) (accA : list A) (accB : list B) :=\n    match l with\n      | nil => (accA, accB)\n      | (a, b) :: l' => unzip_helper l' (a"]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=
  let rec unzip_helper (l : list (A * B)) (accA : list A) (accB : list B) :=
    match l with
      | nil => (accA, accB)
      | (a, b) :: l' => unzip_helper l' (a :: accA) (b :: accB)
    end
  in
    unzip_helper l [] [].

Lemma unzip_length {A B : Type} (l : list (A * B)) :
  let (l1, l2) := unzip l in
  length l1 = length l2.
File "./ex.v", line 17, characters 19-21:
Error: Syntax error: '.' expected after [gallina] (in [vernac_aux]).


SCORE
-1.0
[' :: accA) (b :: accB)\n    end\n  in\n    unzip_helper l [] [].\n\nLemma unzip_length {A B : Type} (l : list (A * B)) :\n  let (l1, l2) := unzip l in\n  length l1 = length l2.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n\nDefinition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=\n  (map fst l, map snd l).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  length (fst (unzip l)) = length l /\\ length (snd (unzip l)) = length l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\nProof.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  intros.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros.
  split.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  split.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).
### Need to show

Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros.
  split.
  - induction l as [| x l'].
    + reflexivity.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
["\n  - induction l as [| x l'].\n    + reflexivity."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip nil)) =
Datatypes.length nil

### given assumptions

A, B as follows.

A : Type
B : Type.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros.
  split.
  - induction l as [| x l'].
    + reflexivity.
    + simpl.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n    + simpl.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

S (Datatypes.length (map fst l')) =
S (Datatypes.length l')

### given assumptions

A, B, x, l', IHl' as follows.

A : Type
B : Type
x : (A * B)%type
l' : list (A * B)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros.
  split.
  - induction l as [| x l'].
    + reflexivity.
    + simpl.
      rewrite IHl'.
      reflexivity.
File "./ex.v", line 21, characters 6-18:
Error:
Found no subterm matching "Datatypes.length (fst (unzip l'))" in the current goal.


SCORE
-1.0
["\n      rewrite IHl'.\n      reflexivity."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros.
  split.
  - induction l as [| x l'].
    + reflexivity.
    + simpl.
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall A B : Type,
A * B ->
forall l' : list (A * B),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' ->
S (Datatypes.length (map fst l')) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

forall A B : Type,
A * B ->
forall l' : list (A * B),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' ->
S (Datatypes.length (map fst l')) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.
SCORE
1.0
['\nDefined.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.
File "./ex.v", line 22, characters 123-124:
Error: Syntax error: ',' expected after [open_binders] (in [binder_constr]).


SCORE
None
["\n\nLemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

forall A B : Type,
A * B ->
forall l' : list (A * B),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' ->
S (Datatypes.length (map fst l')) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.
File "./ex.v", line 23, characters 9-10:
Error: Syntax error: ',' expected after [open_binders] (in [binder_constr]).


SCORE
None
["length (snd (unzip l')) =\nDatatypes."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

forall A B : Type,
A * B ->
forall l' : list (A * B),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' ->
S (Datatypes.length (map fst l')) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.
File "./ex.v", line 23, characters 33-34:
Error: Syntax error: ',' or ')' expected after [term level 200] (in [term]).


SCORE
None
["length l', S (Datatypes."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

forall A B : Type,
A * B ->
forall l' : list (A * B),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' ->
S (Datatypes.length (map fst l')) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.
File "./ex.v", line 24, characters 12-13:
Error: Syntax error: ',' or ')' expected after [term level 200] (in [term]).


SCORE
None
["length (map snd l')) =\nS (Datatypes."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

forall A B : Type,
A * B ->
forall l' : list (A * B),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' ->
S (Datatypes.length (map fst l')) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
Error: There are pending proofs in file ./ex.v: helper1.

SCORE
1.0
["length l').\nProof."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

forall A B : Type,
A * B ->
forall l' : list (A * B),
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
S (Datatypes.length (map snd l')) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Error: There are pending proofs in file ./ex.v: helper1.

SCORE
1.0
['\n(* do not use induction, try using hammer *)\nhammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

forall A B : Type,
A * B ->
forall l' : list (A * B),
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
S (Datatypes.length (map snd l')) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.
SCORE
1.0
['\nDefined.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.
File "./ex.v", line 31, characters 2-11:
Error: The reference Datatypes was not found in the current environment.


SCORE
None
['\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  Datatypes.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

forall A B : Type,
A * B ->
forall l' : list (A * B),
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
S (Datatypes.length (map snd l')) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.
File "./ex.v", line 31, characters 37-46:
Error: The reference Datatypes was not found in the current environment.


SCORE
None
['length (fst (unzip l)) = Datatypes.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

forall A B : Type,
A * B ->
forall l' : list (A * B),
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
S (Datatypes.length (map snd l')) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.
File "./ex.v", line 31, characters 59-68:
Error: The reference Datatypes was not found in the current environment.


SCORE
None
['length l /\\ Datatypes.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

forall A B : Type,
A * B ->
forall l' : list (A * B),
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
S (Datatypes.length (map snd l')) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.
File "./ex.v", line 31, characters 94-103:
Error: The reference Datatypes was not found in the current environment.


SCORE
None
['length (snd (unzip l)) = Datatypes.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

forall A B : Type,
A * B ->
forall l' : list (A * B),
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
S (Datatypes.length (map snd l')) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['length l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\nProof.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  induction l as [| x l' IHl'].
  - simpl.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
["\n  induction l as [| x l' IHl'].\n  - simpl."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

0 = 0 /\ 0 = 0

### given assumptions

A, B as follows.

A : Type
B : Type.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  induction l as [| x l' IHl'].
  - simpl. reflexivity.
File "./ex.v", line 34, characters 11-22:
Error:
 The relation and is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
[' reflexivity.']
[' reflexivity.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  intros.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  split.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  split.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).
### Need to show

Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  split.
  (* use the helper lemma for the first part *)
  apply helper0.
File "./ex.v", line 36, characters 8-15:
Error: In environment
A : Type
B : Type
l : list (A * B)
Unable to unify
 "S (Datatypes.length (map fst ?M1422)) = S (Datatypes.length ?M1422)" with
 "Datatypes.length (fst (unzip l)) = Datatypes.length l".


SCORE
-1.0
['\n  (* use the helper lemma for the first part *)\n  apply helper0.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).
### Need to show

Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  split.
  - apply (helper0 A B l).
  - apply (helper1 A B l).
Defined.
File "./ex.v", line 35, characters 23-24:
Error:
In environment
A : Type
B : Type
l : list (A * B)
The term "l" has type "list (A * B)" while it is expected to have type
 "(A * B)%type".


SCORE
-1.0
['\n  - apply (helper0 A B l).\n  - apply (helper1 A B l).\nDefined.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  (* do not use induction, try using hammer *)\n  hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...
Tactic srun eauto succeeded.
Replace the hammer tactic with:
	srun eauto use: map_length unfold: snd, unzip, fst.


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
SCORE
1.0
['\nDefined.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...
Tactic srun eauto succeeded.
Replace the hammer tactic with:
	srun eauto use: map_length unfold: snd, unzip, fst.


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

## Explanation

In this Coq code, we define a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elements is a list of all the second elements of the original list.
SCORE
1.0
['\n```\n\n## Explanation\n\nIn this Coq code, we define a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elements is a list of all the second elements of the original list.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...
Tactic srun eauto succeeded.
Replace the hammer tactic with:
	srun eauto use: map_length unfold: snd, unzip, fst.


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

## Proof

The proof of `unzip_length` relies on the helpers `helper0` and `helper1` that are proved using the `hammer` tactic.
SCORE
1.0
['```\n\n## Proof\n\nThe proof of `unzip_length` relies on the helpers `helper0` and `helper1` that are proved using the `hammer` tactic.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...
Tactic srun eauto succeeded.
Replace the hammer tactic with:
	srun eauto use: map_length unfold: snd, unzip, fst.


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

## Proof

In this code, we first define a `unzip` function that takes a list of pairs and returns a pair of lists.
SCORE
1.0
['```\n\n## Proof\n\nIn this code, we first define a `unzip` function that takes a list of pairs and returns a pair of lists.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...
Tactic srun eauto succeeded.
Replace the hammer tactic with:
	srun eauto use: map_length unfold: snd, unzip, fst.


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

## Proof
The `unzip` function is defined by mapping the `fst` and `snd` functions over the input list.
SCORE
1.0
['```\n\n## Proof\nThe `unzip` function is defined by mapping the `fst` and `snd` functions over the input list.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...
Tactic srun eauto succeeded.
Replace the hammer tactic with:
	srun eauto use: map_length unfold: snd, unzip, fst.


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

## Proof

The `unzip` function is defined by applying the `map` function to the first and second components of each pair in the list `l`. 

The lemma `helper0` is proved by applying the `map` function to the first component of each pair in the list `l'`. Similarly, the lemma `helper1` is proved by applying the `map` function to the second component of each pair in the list `l'`.
SCORE
1.0
["```\n\n## Proof\n\nThe `unzip` function is defined by applying the `map` function to the first and second components of each pair in the list `l`. \n\nThe lemma `helper0` is proved by applying the `map` function to the first component of each pair in the list `l'`. Similarly, the lemma `helper1` is proved by applying the `map` function to the second component of each pair in the list `l'`."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...
Tactic srun eauto succeeded.
Replace the hammer tactic with:
	srun eauto use: map_length unfold: snd, unzip, fst.


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

## Proof

This proof demonstrates how to use the `hammer` tactic in Coq to prove a lemma about the `unzip` function.
SCORE
1.0
['```\n\n## Proof\n\nThis proof demonstrates how to use the `hammer` tactic in Coq to prove a lemma about the `unzip` function.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

## Explanation

In this code, we define a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
SCORE
1.0
['```\n\n## Explanation\n\nIn this code, we define a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...
Tactic srun eauto succeeded.
Replace the hammer tactic with:
	srun eauto use: map_length unfold: snd, unzip, fst.


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

## Explanation

In this script, we first import the necessary libraries and modules.
SCORE
1.0
['```\n\n## Explanation\n\nIn this script, we first import the necessary libraries and modules.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...
Tactic srun eauto succeeded.
Replace the hammer tactic with:
	srun eauto use: map_length unfold: snd, unzip, fst.


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

## Proof

We first define helper lemmas `helper0` and `helper1` to prove that the length of the first and second projections of `unzip` is the same as the length of the original list.
SCORE
1.0
['```\n\n## Proof\n\nWe first define helper lemmas `helper0` and `helper1` to prove that the length of the first and second projections of `unzip` is the same as the length of the original list.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...
Tactic srun eauto succeeded.
Replace the hammer tactic with:
	srun eauto use: map_length unfold: snd, unzip, fst.


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

## Proof

We define a helper lemma `helper0` and `helper1` that prove that the length of the list obtained by mapping `fst` and `snd` respectively, is one more than the length of the original list.

SCORE
-1.0
['```\n\n## Proof\n\nWe define a helper lemma `helper0` and `helper1` that prove that the length of the list obtained by mapping `fst` and `snd` respectively, is one more than the length of the original list.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...
Tactic srun eauto succeeded.
Replace the hammer tactic with:
	srun eauto use: map_length unfold: snd, unzip, fst.


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

## Proof of `unzip_length`

To prove `unzip_length`, we use the `hammer` tactic, which tries to apply a set of pre-defined lemmas.
SCORE
1.0
['```\n\n## Proof of `unzip_length`\n\nTo prove `unzip_length`, we use the `hammer` tactic, which tries to apply a set of pre-defined lemmas.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

This code defines a function `unzip` that takes a list of pairs and returns a pair of lists.
SCORE
1.0
['```\n\nThis code defines a function `unzip` that takes a list of pairs and returns a pair of lists.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...
Tactic srun eauto succeeded.
Replace the hammer tactic with:
	srun eauto use: map_length unfold: snd, unzip, fst.


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

## Proof

The lemma `unzip_length` is proved using the helpers `helper0` and `helper1`. The helpers are proved using the `hammer` tactic, which automatically finds and applies the appropriate lemmas to discharge the goal.
SCORE
1.0
['```\n\n## Proof\n\nThe lemma `unzip_length` is proved using the helpers `helper0` and `helper1`. The helpers are proved using the `hammer` tactic, which automatically finds and applies the appropriate lemmas to discharge the goal.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: sfirstorder 
Detecting provers...
Eprover found
Vampire found
Z3 found
CVC4 found
Extracting features...
Running provers (8 threads)...
CVC4 (knn-64) succeeded
- dependencies: Lists.List.map_length
- definitions: Init.Datatypes.snd, ex.unzip, Init.Datatypes.fst
Reconstructing the proof...
Tactic srun eauto succeeded.
Replace the hammer tactic with:
	srun eauto use: map_length unfold: snd, unzip, fst.


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', S (Datatypes.length (map fst l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma helper1: forall A : Type, forall B : Type, forall x : (A * B)%type, forall l' : list (A * B), forall IHl' : Datatypes.length (snd (unzip l')) =
Datatypes.length l', S (Datatypes.length (map snd l')) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Defined.

Lemma unzip_length {A B} (l : list (A * B)) :
  Datatypes.length (fst (unzip l)) = Datatypes.length l /\ Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
  intros.
  (* do not use induction, try using hammer *)
  hammer.
Defined.
```

## Proof

In this Coq code, we first import the necessary libraries and define the `unzip` function.
SCORE
1.0
['```\n\n## Proof\n\nIn this Coq code, we first import the necessary libraries and define the `unzip` function.']
['map f l in (List.']
['map fst a, List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.map snd b).

Lemma unzip_length {A B} (l : list (A * B)) :
  (fst (unzip l)) <> 0 -> (snd (unzip l)) <> 0 -> (List.
File "./ex.v", line 13, characters 2-64:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  (fst (unzip l)) <> 0 -> (snd (unzip l)) <> 0 -> (List.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

['map fst a, List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.map snd b).
```

## Lemma

```Coq
Lemma unzip_length {A B} (l : list (A * B)) : 
  (fun (x : (list A) * (list B)) => length (fst x) = length (snd x)) (unzip l).
Proof.
File "./ex.v", line 13, characters 2-64:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n```\n\n## Lemma\n\n```Coq\nLemma unzip_length {A B} (l : list (A * B)) : \n  (fun (x : (list A) * (list B)) => length (fst x) = length (snd x)) (unzip l).\nProof.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.map snd b).
```

## Lemma

```Coq
Lemma unzip_length_lemma {A B} (l : list (A * B)) :
  let (a, b) := unzip l in
  length a = length l /\ length b = length l.
File "./ex.v", line 13, characters 2-66:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n```\n\n## Lemma\n\n```Coq\nLemma unzip_length_lemma {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in\n  length a = length l /\\ length b = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in
     >   (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.fold_right (fun x (a, b) => let (a1, b1) := f x in (a1 :: a, b1 :: b)) l ([], []).
Defined.
File "./ex.v", line 13, characters 41-42:
Error:
Syntax error: ':=' or ':' or [name] expected after [name] (in [closed_binder]).


SCORE
-1.0
['fold_right (fun x (a, b) => let (a1, b1) := f x in (a1 :: a, b1 :: b)) l ([], []).\nDefined.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > Import List.
     > Import ListNotations. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.map snd b).

Lemma unzip_length {A B} (l : list (A * B)) :
  let (a, b) := unzip l in
  length a = length l /\ length b = length l.
File "./ex.v", line 13, characters 2-64:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in\n  length a = length l /\\ length b = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

['map fst a, List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.map snd b).

Lemma unzip_length {A B} (l : list (A * B)) :
  let (a, b) := unzip l in
  length a = length b.
File "./ex.v", line 13, characters 2-66:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in\n  length a = length b.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in
     >   (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.map snd b).

Lemma unzip_length {A B} (l : list (A * B)) : 
  (List.
File "./ex.v", line 13, characters 2-66:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) : \n  (List.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in
     >   (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

['map fst a, List.']
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in\n  length a = length b = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in\n  length a = length b = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

['map fst a, List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.map snd b).
```

## Lemma

```Coq
Lemma unzip_length_lemma {A B} (l : list (A * B)) :
  let (a, b) := unzip l in
  length a = length b = length l.
File "./ex.v", line 13, characters 2-66:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n```\n\n## Lemma\n\n```Coq\nLemma unzip_length_lemma {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in\n  length a = length b = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in
     >   (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.map snd b).

Lemma unzip_length {A B} (l : list (A * B)) :
  let (a, b) := unzip l in
  length a = length b = length l.
File "./ex.v", line 13, characters 2-66:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in\n  length a = length b = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in
     >   (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

['map fst a, List.']
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in\n  length a = length b = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.map snd b).

Lemma unzip_length {A B} (l : list (A * B)) :
  let (a, b) := unzip l in length a = length b.
File "./ex.v", line 13, characters 2-64:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in length a = length b.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: l' =>
    let (la, lb) := unzip l' in
    (a :: la, b :: lb)
  end.
File "./ex.v", line 14, characters 20-25:
Error: The reference unzip was not found in the current environment.


SCORE
-1.0
["\n\nDefinition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=\n  match l with\n  | nil => (nil, nil)\n  | (a, b) :: l' =>\n    let (la, lb) := unzip l' in\n    (a :: la, b :: lb)\n  end."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

['map fst a, List.']
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in\n  length a = length l /\\ length b = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.map snd b).

Lemma unzip_length {A B} (l : list (A * B)) :
  (unzip l).1 = List.
File "./ex.v", line 13, characters 2-64:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  (unzip l).1 = List.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.map snd b).
```

## Lemma

```Coq
Lemma unzip_length {A B} (l : list (A * B)) :
  let (a, b) := unzip l in
  length a = length b = length l.
File "./ex.v", line 13, characters 2-66:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n```\n\n## Lemma\n\n```Coq\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in\n  length a = length b = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in
     >   (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

['map fst a, List.']
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in length a = length b.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in length a = length b.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.map snd b).

Lemma unzip_length {A B} (l : list (A * B)) :
  let (a, b) := unzip l in
  length a = length b.
File "./ex.v", line 13, characters 2-64:
Error: Destructing let is only for inductive types with one constructor.


SCORE
-1.0
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in\n  length a = length b.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

['map fst a, List.']
['map snd b).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  let (a, b) := unzip l in\n  length a = length b = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let f (x : A * B) : A * B := (fst x, snd x) in
  let (a, b) := List.map f l in
  (List.map fst a, List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: ',' or ')' expected after [term level 200] (in [term]).
   The offending chunk is delimited by >>>…<<< below:
     >   let f (x : A * B) : A * B := (fst x, snd x) in
     >   let (a, b) := List.map f l in
     >   (List.map fst a, List>>>.<<< simpl.
   Results past this point may be unreliable.

reached timelimit, stopping expansion on current node
CHOSEN SOLUTION
None

STATS

number of nodes: 123
number of gen nodes: 62 (including leaves: 17)
number of back nodes: 61 (including leaves: 45)

expansion count: 95 (including failed: 34)

cache stats {'hit': 21, 'miss': 73}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 602]

Trial calls: [8, 43, 94]


Time Statistics:

  Min: 20.00 sec, Max: 602.00 sec, Average: 246.00 sec, Standard Deviation: 254.76 sec

Call Statistics:

  Min: 8 calls, Max: 94 calls, Average: 48.33 calls, Standard Deviation: 35.31 calls

=====SUMMARY STATISTICS FOR TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [602]

Trial calls: [94]


Time Statistics:

  Min: 602.00 sec, Max: 602.00 sec, Average: 602.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 94 calls, Max: 94 calls, Average: 94.00 calls, Standard Deviation: 0.00 calls

=====SUMMARY STATISTICS FOR NON_TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116]

Trial calls: [8, 43]


Time Statistics:

  Min: 20.00 sec, Max: 116.00 sec, Average: 68.00 sec, Standard Deviation: 48.00 sec

Call Statistics:

  Min: 8 calls, Max: 43 calls, Average: 25.50 calls, Standard Deviation: 17.50 calls

Trial  4
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import ListNotations.
File "./ex.v", line 8, characters 7-20:
Error: Cannot find module ListNotations


SCORE
-1.0
['Import ListNotations.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

['Import ListNotations.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
SCORE
1.0
['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let rec unzip_helper (l : list (A * B)) (acc1 : list A) (acc2 : list B) : (list A) * (list B) :=
    match l with
    | [] => (acc1, acc2)
    | (a, b) :: tl =>
File "./ex.v", line 13, characters 6-7:
Error: Syntax error: 'end' expected after [branches] (in [term_match]).


SCORE
None
['\n\nDefinition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=\n  let rec unzip_helper (l : list (A * B)) (acc1 : list A) (acc2 : list B) : (list A) * (list B) :=\n    match l with\n    | [] => (acc1, acc2)\n    | (a, b) :: tl =>']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let rec unzip_helper (l : list (A * B)) (acc1 : list A) (acc2 : list B) : (list A) * (list B) :=
    match l with
    | [] => (acc1, acc2)
    | (a, b) :: tl => unzip_helper tl (a :: acc1) (b :: acc2)
    end
  in unzip_helper (List.
File "./ex.v", line 13, characters 6-7:
Error: Syntax error: 'end' expected after [branches] (in [term_match]).


SCORE
-1.0
[' unzip_helper tl (a :: acc1) (b :: acc2)\n    end\n  in unzip_helper (List.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(** * unzip function *)
Fixpoint unzip (l : (nat * nat) list) : (nat list) * (nat list) :=
  match l with
  | [] => ([], [])
  | (x1, x2) :: t =>
    let (l1, l2) := unzip t in
    (x1 :: l1, x2 :: l2)
  end.
File "./ex.v", line 13, characters 4-5:
Error: Syntax error: 'end' expected after [branches] (in [term_match]).


SCORE
-1.0
['\n\n(** * unzip function *)\nFixpoint unzip (l : (nat * nat) list) : (nat list) * (nat list) :=\n  match l with\n  | [] => ([], [])\n  | (x1, x2) :: t =>\n    let (l1, l2) := unzip t in\n    (x1 :: l1, x2 :: l2)\n  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import Pair.
File "./ex.v", line 9, characters 7-11:
Error: Cannot find module Pair


SCORE
-1.0
['\nImport Pair.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

[' unzip_helper tl (a :: acc1) (b :: acc2)\n    end\n  in unzip_helper (List.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
SCORE
1.0
['\nImport ListNotations.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.
SCORE
1.0
['\nImport Nat.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.

Lemma unzip_length : forall l : list (nat * nat),
  let (l1, l2) := unzip l in
  length l1 = length l2.
File "./ex.v", line 13, characters 18-23:
Error: The reference unzip was not found in the current environment.


SCORE
-1.0
['\n\nLemma unzip_length : forall l : list (nat * nat),\n  let (l1, l2) := unzip l in\n  length l1 = length l2.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > Import List.
     > Import ListNotations. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.
Import Pair.
File "./ex.v", line 11, characters 7-11:
Error: Cannot find module Pair


SCORE
-1.0
['\nImport Pair.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Import List.
     > Import ListNotations.
     > Import Nat. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n\nDefinition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=\n  (map fst l, map snd l).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  length (fst (unzip l)) = length l /\\ length (snd (unzip l)) = length l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\nProof.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

A, B, l as follows.

A : Type
B : Type
l : list (A * B).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  induction l as [| a l'].
  - split; reflexivity.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
["\n  induction l as [| a l'].\n  - split; reflexivity."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip [])) =
Datatypes.length [] /\
Datatypes.length (snd (unzip [])) =
Datatypes.length []

### given assumptions

A, B as follows.

A : Type
B : Type.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  induction l as [| a l'].
  - split; reflexivity.
  - destruct a as [a1 a2]; split; reflexivity.
File "./ex.v", line 19, characters 34-45:
Error:
In environment
A : Type
B : Type
a1 : A
a2 : B
l' : list (A * B)
IHl' : Datatypes.length (fst (unzip l')) = Datatypes.length l' /\
       Datatypes.length (snd (unzip l')) = Datatypes.length l'
Unable to unify "Datatypes.length ((a1, a2) :: l')" with
 "Datatypes.length (fst (unzip ((a1, a2) :: l')))".


SCORE
-1.0
['\n  - destruct a as [a1 a2]; split; reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  induction l as [| a l'].
  - split; reflexivity.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal. Focus next goal with bullet -.
   The offending chunk is delimited by >>>…<<< below:
     > Proof.
     >   induction l as [| a l'].
     >   - split; reflexivity. >>>simpl.<<<
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip [])) =
Datatypes.length [] /\
Datatypes.length (snd (unzip [])) =
Datatypes.length []

### given assumptions

A, B as follows.

A : Type
B : Type.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  induction l as [| a l'].
  - split; reflexivity.
  - destruct a as (a1, a2); split; [| reflexivity].
Qed.
File "./ex.v", line 19, characters 38-49:
Error:
In environment
A : Type
B : Type
a1 : A
a2 : B
l' : list (A * B)
IHl' : Datatypes.length (fst (unzip l')) = Datatypes.length l' /\
       Datatypes.length (snd (unzip l')) = Datatypes.length l'
Unable to unify "Datatypes.length ((a1, a2) :: l')" with
 "Datatypes.length (snd (unzip ((a1, a2) :: l')))".


SCORE
-1.0
['\n  - destruct a as (a1, a2); split; [| reflexivity].\nQed.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  induction l as [| a l'].
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Type -> Type -> 0 = 0 /\ 0 = 0

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Type -> Type -> 0 = 0 /\ 0 = 0

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.
SCORE
1.0
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip (a :: l'))) =
Datatypes.length (a :: l') /\
Datatypes.length (snd (unzip (a :: l'))) =
Datatypes.length (a :: l')

### given assumptions

A, B, a, l', IHl' as follows.

A : Type
B : Type
a : (A * B)%type
l' : list (A * B)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  induction l as [| a l']. apply (@helper0 A B).
  destruct (unzip l') as [lf ls]; simpl.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
["  destruct (unzip l') as [lf ls]; simpl."]
CHOSEN SOLUTION

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip (a :: l'))) =
Datatypes.length (a :: l') /\
Datatypes.length (snd (unzip (a :: l'))) =
Datatypes.length (a :: l')

### given assumptions

A, B, a, l', IHl' as follows.

A : Type
B : Type
a : (A * B)%type
l' : list (A * B)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).


Lemma helper0: forall A : Type, forall B : Type, 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.

Lemma unzip_length {A B} (l : list (A * B)) :
  length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  induction l as [| a l']. apply (@helper0 A B).
  destruct (unzip l') as [lf ls]; simpl.```

STATS

number of nodes: 25
number of gen nodes: 13 (including leaves: 3)
number of back nodes: 12 (including leaves: 10)

expansion count: 21 (including failed: 9)

cache stats {'hit': 3, 'miss': 18}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 602, 57]

Trial calls: [8, 43, 94, 21]


Time Statistics:

  Min: 20.00 sec, Max: 602.00 sec, Average: 198.75 sec, Standard Deviation: 235.32 sec

Call Statistics:

  Min: 8 calls, Max: 94 calls, Average: 41.50 calls, Standard Deviation: 32.79 calls

=====SUMMARY STATISTICS FOR TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [602]

Trial calls: [94]


Time Statistics:

  Min: 602.00 sec, Max: 602.00 sec, Average: 602.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 94 calls, Max: 94 calls, Average: 94.00 calls, Standard Deviation: 0.00 calls

=====SUMMARY STATISTICS FOR NON_TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 57]

Trial calls: [8, 43, 21]


Time Statistics:

  Min: 20.00 sec, Max: 116.00 sec, Average: 64.33 sec, Standard Deviation: 39.53 sec

Call Statistics:

  Min: 8 calls, Max: 43 calls, Average: 24.00 calls, Standard Deviation: 14.45 calls

Trial  5
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.
File "./ex.v", line 8, characters 8-9:
Error:
Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).


SCORE
None
['From Coq.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.Lists.
File "./ex.v", line 8, characters 14-15:
Error:
Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).


SCORE
None
['Lists.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.Lists.List Import List.
File "./ex.v", line 8, characters 20-26:
Error:
Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).


SCORE
-1.0
['List Import List.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: 'Extra' 'Dependency' or 'Require' expected after [global] (in [gallina_ext]).
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > From Coq>>>.<<< simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.
File "./ex.v", line 8, characters 0-30:
Error: Cannot find a physical path bound to logical path
Lists with prefix Coq.


SCORE
None
['From Coq Require Import Lists.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.
SCORE
1.0
['List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Definition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x1, y1) :: l' =>
    let (xs, ys) := unzip l' in
    (x1 :: xs, y1 :: ys)
  end.
File "./ex.v", line 12, characters 4-5:
Error: Syntax error: 'end' expected after [branches] (in [term_match]).


SCORE
-1.0
["\n\nDefinition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n  | [] => ([], [])\n  | (x1, y1) :: l' =>\n    let (xs, ys) := unzip l' in\n    (x1 :: xs, y1 :: ys)\n  end."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Cannot find a physical path bound to logical path Lists with prefix Coq.
   The offending chunk is delimited by >>>…<<< below:
     > >>>
     > From Hammer Require Import Tactics.
     > From Hammer Require Import Hammer.
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String.
     >
     > From Coq Require Import Lists. simpl.<<<
   Results past this point may be unreliable.
!! (WARNING/2) Orphaned message for sid b'6':
    >  Cannot find a physical path bound to logical path
    >  Lists with prefix Coq.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
SCORE
1.0
['\n\nImport ListNotations.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import List.
SCORE
1.0
['\nImport List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import List.

Definition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: l' =>
    let (l1, l2) := unzip l' in
    (a :: l1, b :: l2)
  end.
File "./ex.v", line 17, characters 20-25:
Error: The reference unzip was not found in the current environment.


SCORE
-1.0
["\n\nDefinition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=\n  match l with\n  | nil => (nil, nil)\n  | (a, b) :: l' =>\n    let (l1, l2) := unzip l' in\n    (a :: l1, b :: l2)\n  end."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > From Coq Require Import Lists.List.
     >
     > Import ListNotations. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import List.List.
File "./ex.v", line 11, characters 7-16:
Error: Cannot find module List.List


SCORE
-1.0
['List.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > Import ListNotations.
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
SCORE
1.0
['\nImport Nat.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.
SCORE
1.0
['\nImport List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.
Import Pair.
File "./ex.v", line 13, characters 7-11:
Error: Cannot find module Pair


SCORE
-1.0
['\nImport Pair.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Import ListNotations.
     > Import Nat.
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.

(* Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (la, lb) := unzip tl in
    (a :: la, b :: lb)
  end.
SCORE
1.0
['\n\n(* Part 1: Define unzip function *)\n\nFixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n  | [] => ([], [])\n  | (a, b) :: tl =>\n    let (la, lb) := unzip tl in\n    (a :: la, b :: lb)\n  end.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.

(* Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (la, lb) := unzip tl in
    (a :: la, b :: lb)
  end.

(* Part 2: Define a lemma *)

Lemma unzip_length_lemma : forall l,
  let (la, lb) := unzip l in
  length la = length lb = length l.
File "./ex.v", line 28, characters 2-34:
Error: Unknown interpretation for notation "_ = _ = _".


SCORE
-1.0
['\n\n(* Part 2: Define a lemma *)\n\nLemma unzip_length_lemma : forall l,\n  let (la, lb) := unzip l in\n  length la = length lb = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.

(* Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (la, lb) := unzip tl in
    (a :: la, b :: lb)
  end.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >     let (la, lb) := unzip tl in
     >     (a :: la, b :: lb)
     >   end. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.

(* Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (la, lb) := unzip tl in
    (a :: la, b :: lb)
  end.

(* Part 2: Prove that the lengths of the lists in the pair returned by unzip are the same as the length of the original list *)

Lemma unzip_length : forall l : list (nat * nat),
  let (la, lb) := unzip l in
  length la = length l /\ length lb = length l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n\n(* Part 2: Prove that the lengths of the lists in the pair returned by unzip are the same as the length of the original list *)\n\nLemma unzip_length : forall l : list (nat * nat),\n  let (la, lb) := unzip l in\n  length la = length l /\\ length lb = length l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
let (la, lb) := unzip l in
Datatypes.length la = Datatypes.length l /\
Datatypes.length lb = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.

(* Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (la, lb) := unzip tl in
    (a :: la, b :: lb)
  end.

(* Part 2: Prove that the lengths of the lists in the pair returned by unzip are the same as the length of the original list *)

Lemma unzip_length : forall l : list (nat * nat),
  let (la, lb) := unzip l in
  length la = length l /\ length lb = length l.
Proof.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\nProof.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
let (la, lb) := unzip l in
Datatypes.length la = Datatypes.length l /\
Datatypes.length lb = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.

(* Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (la, lb) := unzip tl in
    (a :: la, b :: lb)
  end.

(* Part 2: Prove that the lengths of the lists in the pair returned by unzip are the same as the length of the original list *)

Lemma unzip_length : forall l : list (nat * nat),
  let (la, lb) := unzip l in
  length la = length l /\ length lb = length l.
Proof.
  intros l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  intros l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

let (la, lb) := unzip l in
Datatypes.length la = Datatypes.length l /\
Datatypes.length lb = Datatypes.length l

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.

(* Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (la, lb) := unzip tl in
    (a :: la, b :: lb)
  end.

(* Part 2: Prove that the lengths of the lists in the pair returned by unzip are the same as the length of the original list *)

Lemma unzip_length : forall l : list (nat * nat),
  let (la, lb) := unzip l in
  length la = length l /\ length lb = length l.
Proof.
  intros l.
  induction l as [| hd tl l].
  - simpl.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  induction l as [| hd tl l].\n  - simpl.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

0 = 0 /\ 0 = 0

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.

(* Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (la, lb) := unzip tl in
    (a :: la, b :: lb)
  end.

(* Part 2: Prove that the lengths of the lists in the pair returned by unzip are the same as the length of the original list *)

Lemma unzip_length : forall l : list (nat * nat),
  let (la, lb) := unzip l in
  length la = length l /\ length lb = length l.
Proof.
  intros l.
  induction l as [| hd tl l].
  - simpl. reflexivity.
File "./ex.v", line 32, characters 11-22:
Error:
 The relation and is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
[' reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.

(* Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (la, lb) := unzip tl in
    (a :: la, b :: lb)
  end.

(* Part 2: Prove that the lengths of the lists in the pair returned by unzip are the same as the length of the original list *)

Lemma unzip_length : forall l : list (nat * nat),
  let (la, lb) := unzip l in
  length la = length l /\ length lb = length l.
Proof.
  intros l.
  induction l as [| hd tl l].
  - simpl.
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

0 = 0 /\ 0 = 0

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.

(* Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (la, lb) := unzip tl in
    (a :: la, b :: lb)
  end.

(* Part 2: Prove that the lengths of the lists in the pair returned by unzip are the same as the length of the original list *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
hammer.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

0 = 0 /\ 0 = 0

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.

(* Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (la, lb) := unzip tl in
    (a :: la, b :: lb)
  end.

(* Part 2: Prove that the lengths of the lists in the pair returned by unzip are the same as the length of the original list *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.
SCORE
1.0
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

0 = 0 /\ 0 = 0

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.

(* Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (la, lb) := unzip tl in
    (a :: la, b :: lb)
  end.

(* Part 2: Prove that the lengths of the lists in the pair returned by unzip are the same as the length of the original list *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l : list (nat * nat),
  let (la, lb) := unzip l in
  length la = length l /\ length lb = length l.
Proof.
  intros l.
  induction l as [| hd tl l].
  - simpl. apply (@helper0 ).
  - simpl.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['  - simpl.']
CHOSEN SOLUTION

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

0 = 0 /\ 0 = 0

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

From Coq Require Import Lists.List.

Import ListNotations.
Import Nat.
Import List.

(* Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (la, lb) := unzip tl in
    (a :: la, b :: lb)
  end.

(* Part 2: Prove that the lengths of the lists in the pair returned by unzip are the same as the length of the original list *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l : list (nat * nat),
  let (la, lb) := unzip l in
  length la = length l /\ length lb = length l.
Proof.
  intros l.
  induction l as [| hd tl l].
  - simpl. apply (@helper0 ).
  - simpl.```

STATS

number of nodes: 35
number of gen nodes: 18 (including leaves: 3)
number of back nodes: 17 (including leaves: 15)

expansion count: 23 (including failed: 6)

cache stats {'hit': 0, 'miss': 23}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 602, 57, 60]

Trial calls: [8, 43, 94, 21, 23]


Time Statistics:

  Min: 20.00 sec, Max: 602.00 sec, Average: 171.00 sec, Standard Deviation: 217.67 sec

Call Statistics:

  Min: 8 calls, Max: 94 calls, Average: 37.80 calls, Standard Deviation: 30.25 calls

=====SUMMARY STATISTICS FOR TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [602]

Trial calls: [94]


Time Statistics:

  Min: 602.00 sec, Max: 602.00 sec, Average: 602.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 94 calls, Max: 94 calls, Average: 94.00 calls, Standard Deviation: 0.00 calls

=====SUMMARY STATISTICS FOR NON_TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 57, 60]

Trial calls: [8, 43, 21, 23]


Time Statistics:

  Min: 20.00 sec, Max: 116.00 sec, Average: 63.25 sec, Standard Deviation: 34.29 sec

Call Statistics:

  Min: 8 calls, Max: 43 calls, Average: 23.75 calls, Standard Deviation: 12.52 calls

Trial  6
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
SCORE
1.0
['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.
SCORE
1.0
["\n\n(* (1) Define the function unzip *)\nFixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n  | nil => (nil, nil)\n  | (x, y) :: l' =>\n    let (xs, ys) := unzip l' in\n    (x :: xs, y :: ys)\n  end."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n\n(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)\nLemma unzip_length : forall l, length (fst (unzip l)) = length l /\\ length (snd (unzip l)) = length l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\nProof.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  intros l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| x y l' IH].
  - simpl.
File "./ex.v", line 23, characters 2-31:
Warning: Unused introduction pattern: IH
[unused-intro-pattern,tactics,default]
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
["\n  induction l as [| x y l' IH].\n  - simpl."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

0 = 0 /\ 0 = 0

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| x y l' IH].
  - simpl. reflexivity.
File "./ex.v", line 23, characters 2-31:
Warning: Unused introduction pattern: IH
[unused-intro-pattern,tactics,default]
File "./ex.v", line 24, characters 11-22:
Error:
 The relation and is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
[' reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall l : list (nat * nat), Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
(* do not use induction, try using hammer *)
destruct l; hammer.

SCORE
-1.0
['destruct l; hammer.']
[' reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall l : list (nat * nat), Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
(* do not use induction, try using hammer *)
hammer.

SCORE
-1.0
['hammer.']
['hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
["\n  induction l as [| p l' IH]; simpl; auto."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length
  (fst
     (let (x, y) := p in
      let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (x, y) := p in
      let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

p, l', IH as follows.

p : (nat * nat)%type
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  - destruct p as [x y]; simpl; split.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.
### Need to show

Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    + apply Nat.
File "./ex.v", line 25, characters 12-15:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    + apply Nat.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    + apply Nat.eq_refl.
File "./ex.v", line 25, characters 12-23:
Error:
In environment
x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) = Datatypes.length l' /\
     Datatypes.length (snd (unzip l')) = Datatypes.length l'
Unable to unify "RelationClasses.Reflexive eq" with
 "Datatypes.length (fst (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
  S (Datatypes.length l')".


SCORE
-1.0
['eq_refl.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    + apply Nat.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > The variable Nat was not found in the current environment.
   The offending chunk is delimited by >>>…<<< below:
     >   induction l as [| p l' IH]; simpl; auto.
     >   - destruct p as [x y]; simpl; split.
     >     + apply >>>Nat<<<. simpl.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.
### Need to show

Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    + reflexivity.
File "./ex.v", line 25, characters 6-17:
Error:
In environment
x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) = Datatypes.length l' /\
     Datatypes.length (snd (unzip l')) = Datatypes.length l'
Unable to unify "S (Datatypes.length l')" with
 "Datatypes.length (fst (let (xs, ys) := unzip l' in (x :: xs, y :: ys)))".


SCORE
-1.0
['\n    + reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    +
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall (x y : nat) (l' : list (nat * nat)),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: hauto 


## Context
### Need to show

forall (x y : nat) (l' : list (nat * nat)),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.
SCORE
1.0
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: hauto 


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    + apply (@helper0 x y l' IH).
  - split.
File "./ex.v", line 39, characters 2-3:
Error: [Focus] Wrong bullet -: Expecting +.


SCORE
-1.0
['  - split.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    +
</CODE>
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: hauto 


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    + apply (@helper0 x y l' IH).
  - destruct p as [x y]; simpl; split.
File "./ex.v", line 39, characters 2-3:
Error: [Focus] Wrong bullet -: Expecting +.


SCORE
-1.0
['  - destruct p as [x y]; simpl; split.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    +
</CODE>
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: hauto 


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    + apply (@helper0 x y l' IH).
    + apply (@helper0 x y l' IH).
  - subst; split.
File "./ex.v", line 39, characters 13-31:
Error:
In environment
x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) = Datatypes.length l' /\
     Datatypes.length (snd (unzip l')) = Datatypes.length l'
Unable to unify
 "Datatypes.length (fst (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
  S (Datatypes.length l')" with
 "Datatypes.length (snd (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
  S (Datatypes.length l')".


SCORE
-1.0
["    + apply (@helper0 x y l' IH).\n  - subst; split."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    +
</CODE>
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: hauto 


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    + apply (@helper0 x y l' IH).
    + apply (@helper0 x y l' IH).
  - apply IH.
File "./ex.v", line 39, characters 13-31:
Error:
In environment
x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) = Datatypes.length l' /\
     Datatypes.length (snd (unzip l')) = Datatypes.length l'
Unable to unify
 "Datatypes.length (fst (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
  S (Datatypes.length l')" with
 "Datatypes.length (snd (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
  S (Datatypes.length l')".


SCORE
-1.0
["    + apply (@helper0 x y l' IH).\n  - apply IH."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    +
</CODE>
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.
### Need to show

Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    + rewrite IH.
File "./ex.v", line 25, characters 6-16:
Error:
Found no subterm matching "Datatypes.length (fst (unzip l')) =
                           Datatypes.length l'" in the current goal.


SCORE
-1.0
['\n    + rewrite IH.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    +
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall (x y : nat) (l' : list (nat * nat)),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
["destruct l'; hammer."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: hauto 


## Context
### Need to show

forall (x y : nat) (l' : list (nat * nat)),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.
SCORE
1.0
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: hauto 


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    + apply (@helper0 x y l' IH).
    + apply (@helper0 x y l' IH).
  - destruct p as [x y]; simpl; split.
File "./ex.v", line 39, characters 13-31:
Error:
In environment
x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) = Datatypes.length l' /\
     Datatypes.length (snd (unzip l')) = Datatypes.length l'
Unable to unify
 "Datatypes.length (fst (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
  S (Datatypes.length l')" with
 "Datatypes.length (snd (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
  S (Datatypes.length l')".


SCORE
-1.0
["    + apply (@helper0 x y l' IH).\n  - destruct p as [x y]; simpl; split."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    +
</CODE>
["    + apply (@helper0 x y l' IH).\n  - destruct p as [x y]; simpl; split."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    +
</CODE>
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: hauto 


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    + apply (@helper0 x y l' IH).
    + apply (@helper0 x y l' IH).
  - simpl; split.
File "./ex.v", line 39, characters 13-31:
Error:
In environment
x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) = Datatypes.length l' /\
     Datatypes.length (snd (unzip l')) = Datatypes.length l'
Unable to unify
 "Datatypes.length (fst (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
  S (Datatypes.length l')" with
 "Datatypes.length (snd (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
  S (Datatypes.length l')".


SCORE
-1.0
["    + apply (@helper0 x y l' IH).\n  - simpl; split."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    +
</CODE>
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: hauto 


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

x, y, l', IH as follows.

x, y : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    + apply (@helper0 x y l' IH).
  - split; apply IH.
File "./ex.v", line 39, characters 2-3:
Error: [Focus] Wrong bullet -: Expecting +.


SCORE
-1.0
['  - split; apply IH.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p as [x y]; simpl; split.
    +
</CODE>
[' reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
</CODE>
Created Lemma helper0.
['hammer.']
['\n  intros l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| x y l' IH].
  - (* nil case *)
    reflexivity.
File "./ex.v", line 23, characters 2-31:
Warning: Unused introduction pattern: IH
[unused-intro-pattern,tactics,default]
File "./ex.v", line 25, characters 4-15:
Error:
 The relation and is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
["\n  induction l as [| x y l' IH].\n  - (* nil case *)\n    reflexivity."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
</CODE>
Created Lemma helper0.
['hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [ | x y l' IH].
  - simpl.
File "./ex.v", line 23, characters 2-32:
Warning: Unused introduction pattern: IH
[unused-intro-pattern,tactics,default]
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
["\n  induction l as [ | x y l' IH].\n  - simpl."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

0 = 0 /\ 0 = 0

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [ | x y l' IH].
  - simpl. reflexivity.
File "./ex.v", line 23, characters 2-32:
Warning: Unused introduction pattern: IH
[unused-intro-pattern,tactics,default]
File "./ex.v", line 24, characters 11-22:
Error:
 The relation and is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
[' reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall l : list (nat * nat), Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l.
Proof.
(* do not use induction, try using hammer *)
destruct l as [| p l']; rewrite IHl'; hammer.
File "./ex.v", line 26, characters 32-36:
Error: The variable IHl' was not found in the current environment.


SCORE
-1.0
["destruct l as [| p l']; rewrite IHl'; hammer."]
[' reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
</CODE>
Created Lemma helper0.
['hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length
  (fst
     (let (x, y) := p in
      let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (x, y) := p in
      let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

p, l', IH as follows.

p : (nat * nat)%type
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p; simpl.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  - destruct p; simpl.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l')

### given assumptions

n, n0, l', IH as follows.

n, n0 : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p; simpl.
    rewrite IH.
File "./ex.v", line 25, characters 4-14:
Error:
Found no subterm matching "Datatypes.length (fst (unzip l')) =
                           Datatypes.length l'" in the current goal.


SCORE
-1.0
['\n    rewrite IH.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p; simpl.
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall (n n0 : nat) (l' : list (nat * nat)),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall n n0 : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
["destruct l'; hammer."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: hauto 


## Context
### Need to show

forall (n n0 : nat) (l' : list (nat * nat)),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall n n0 : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.
SCORE
1.0
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: hauto 


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l')

### given assumptions

n, n0, l', IH as follows.

n, n0 : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall n n0 : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p; simpl. apply (@helper0 n n0 l' IH).
  - destruct p; simpl.
File "./ex.v", line 42, characters 2-3:
Error: [Focus] Wrong bullet -: No more goals.


SCORE
-1.0
['  - destruct p; simpl.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall n n0 : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p; simpl.
</CODE>
['  - destruct p; simpl.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall n n0 : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p; simpl.
</CODE>
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: hauto 


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l')

### given assumptions

n, n0, l', IH as follows.

n, n0 : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall n n0 : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p; simpl. apply (@helper0 n n0 l' IH).
Qed.
SCORE
1.0
['Qed.']
CHOSEN SOLUTION

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: hauto 


## Context
### Need to show

Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l')

### given assumptions

n, n0, l', IH as follows.

n, n0 : nat
l' : list (nat * nat)
IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma about the length of the lists in the returned pair of unzip *)

Lemma helper0: forall n n0 : nat, forall l' : list (nat * nat), forall IH : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (n :: xs, n0 :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
destruct l'; hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IH]; simpl; auto.
  - destruct p; simpl. apply (@helper0 n n0 l' IH).
Qed.```

STATS

number of nodes: 63
number of gen nodes: 32 (including leaves: 13)
number of back nodes: 31 (including leaves: 19)

expansion count: 49 (including failed: 18)

cache stats {'hit': 13, 'miss': 36}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 602, 57, 60, 124]

Trial calls: [8, 43, 94, 21, 23, 49]


Time Statistics:

  Min: 20.00 sec, Max: 602.00 sec, Average: 163.17 sec, Standard Deviation: 199.48 sec

Call Statistics:

  Min: 8 calls, Max: 94 calls, Average: 39.67 calls, Standard Deviation: 27.93 calls

=====SUMMARY STATISTICS FOR TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [602]

Trial calls: [94]


Time Statistics:

  Min: 602.00 sec, Max: 602.00 sec, Average: 602.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 94 calls, Max: 94 calls, Average: 94.00 calls, Standard Deviation: 0.00 calls

=====SUMMARY STATISTICS FOR NON_TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 57, 60, 124]

Trial calls: [8, 43, 21, 23, 49]


Time Statistics:

  Min: 20.00 sec, Max: 124.00 sec, Average: 75.40 sec, Standard Deviation: 39.13 sec

Call Statistics:

  Min: 8 calls, Max: 49 calls, Average: 28.80 calls, Standard Deviation: 15.08 calls

Trial  7
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
SCORE
1.0
['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: t => let (l1, l2) := unzip t in (a :: l1, b :: l2)
  end.
File "./ex.v", line 14, characters 35-40:
Error: The reference unzip was not found in the current environment.


SCORE
-1.0
['\n\n(* (1) Define unzip function *)\nDefinition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=\n  match l with\n  | nil => (nil, nil)\n  | (a, b) :: t => let (l1, l2) := unzip t in (a :: l1, b :: l2)\n  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let rec aux acc1 acc2 = function
    | [] => (acc1, acc2)
    | (a, b) :: t => aux (a :: acc1) (b :: acc2) t
  in
    aux [] [] l.
File "./ex.v", line 11, characters 24-25:
Error:
Syntax error: [let_type_cstr] expected after [binders] (in [binder_constr]).


SCORE
-1.0
['\n\nDefinition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=\n  let rec aux acc1 acc2 = function\n    | [] => (acc1, acc2)\n    | (a, b) :: t => aux (a :: acc1) (b :: acc2) t\n  in\n    aux [] [] l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: l' =>
    let (la, lb) := unzip l' in
    (a :: la, b :: lb)
  end.
File "./ex.v", line 14, characters 20-25:
Error: The reference unzip was not found in the current environment.


SCORE
-1.0
["\n\nDefinition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=\n  match l with\n  | nil => (nil, nil)\n  | (a, b) :: l' =>\n    let (la, lb) := unzip l' in\n    (a :: la, b :: lb)\n  end."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define the unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: tl =>
    let (al, bl) := unzip tl in
    (a :: al, b :: bl)
  end.
File "./ex.v", line 13, characters 4-5:
Error: Syntax error: 'end' expected after [branches] (in [term_match]).


SCORE
-1.0
['\n\n(* (1) Define the unzip function *)\nFixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n  | [] => ([], [])\n  | (a, b) :: tl =>\n    let (al, bl) := unzip tl in\n    (a :: al, b :: bl)\n  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {X Y : Type} (l : list (X * Y)) : (list X) * (list Y) :=
  match l with
  | [] => ([], [])
  | (x, y) :: tl =>
    let (xs, ys) := unzip tl in
    (x :: xs, y :: ys)
  end.
File "./ex.v", line 12, characters 4-5:
Error: Syntax error: 'end' expected after [branches] (in [term_match]).


SCORE
-1.0
['\n\nDefinition unzip {X Y : Type} (l : list (X * Y)) : (list X) * (list Y) :=\n  match l with\n  | [] => ([], [])\n  | (x, y) :: tl =>\n    let (xs, ys) := unzip tl in\n    (x :: xs, y :: ys)\n  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let rec aux l A_list B_list :=
    match l with
    | nil => (A_list, B_list)
    | (a, b) :: l' => aux l' (a :: A_list) (b :: B_list)
    end
  in
  aux l
File "./ex.v", line 17, characters 7-8:
Error: Syntax error: '.' expected after [gallina] (in [vernac_aux]).


SCORE
None
["\n\nDefinition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=\n  let rec aux l A_list B_list :=\n    match l with\n    | nil => (A_list, B_list)\n    | (a, b) :: l' => aux l' (a :: A_list) (b :: B_list)\n    end\n  in\n  aux l"]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  let rec aux l A_list B_list :=
    match l with
    | nil => (A_list, B_list)
    | (a, b) :: l' => aux l' (a :: A_list) (b :: B_list)
    end
  in
  aux l nil nil.
File "./ex.v", line 17, characters 2-5:
Error: The reference aux was not found in the current environment.


SCORE
-1.0
[' nil nil.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: t => (a :: (fst (unzip t)), b :: (snd (unzip t)))
  end.
File "./ex.v", line 12, characters 4-5:
Error: Syntax error: 'end' expected after [branches] (in [term_match]).


SCORE
-1.0
['\n\nDefinition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n  | [] => ([], [])\n  | (a, b) :: t => (a :: (fst (unzip t)), b :: (snd (unzip t)))\n  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import ListNotations.
File "./ex.v", line 8, characters 7-20:
Error: Cannot find module ListNotations


SCORE
-1.0
['Import ListNotations.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

[' nil nil.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

[' nil nil.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
SCORE
1.0
['\nImport ListNotations.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.
SCORE
1.0
["\n\nFixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n  | [] => ([], [])\n  | (x, y) :: l' =>\n    let (xs, ys) := unzip l' in\n    (x :: xs, y :: ys)\n  end."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

Lemma unzip_length : forall (l : list (nat * nat)),
  let (xs, ys) := unzip l in
  length xs = length ys = length l.
File "./ex.v", line 21, characters 2-34:
Error: Unknown interpretation for notation "_ = _ = _".


SCORE
-1.0
['\n\nLemma unzip_length : forall (l : list (nat * nat)),\n  let (xs, ys) := unzip l in\n  length xs = length ys = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >     let (xs, ys) := unzip l' in
     >     (x :: xs, y :: ys)
     >   end. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

Lemma unzip_length l :
  let (xs, ys) := unzip l in
  length xs = length ys.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n\nLemma unzip_length l :\n  let (xs, ys) := unzip l in\n  length xs = length ys.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

let (xs, ys) := unzip l in
Datatypes.length xs = Datatypes.length ys

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

Lemma unzip_length l :
  let (xs, ys) := unzip l in
  length xs = length ys.
Proof.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\nProof.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

let (xs, ys) := unzip l in
Datatypes.length xs = Datatypes.length ys

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

Lemma unzip_length l :
  let (xs, ys) := unzip l in
  length xs = length ys.
Proof.
  induction l as [ | p l' IHl'].
  - reflexivity.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
["\n  induction l as [ | p l' IHl'].\n  - reflexivity."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

let (xs, ys) := unzip [] in
Datatypes.length xs = Datatypes.length ys

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

Lemma unzip_length l :
  let (xs, ys) := unzip l in
  length xs = length ys.
Proof.
  induction l as [ | p l' IHl'].
  - reflexivity.
  - simpl.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  - simpl.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

let (xs, ys) :=
  let (x, y) := p in
  let (xs, ys) := unzip l' in (x :: xs, y :: ys) in
Datatypes.length xs = Datatypes.length ys

### given assumptions

p, l', IHl' as follows.

p : (nat * nat)%type
l' : list (nat * nat)
IHl' : let (xs, ys) := unzip l' in
Datatypes.length xs = Datatypes.length ys.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

Lemma unzip_length l :
  let (xs, ys) := unzip l in
  length xs = length ys.
Proof.
  induction l as [ | p l' IHl'].
  - reflexivity.
  - simpl.
    destruct p as [x y].
    destruct (unzip l') as [xs' ys'].
    rewrite IHl'.
    reflexivity.
File "./ex.v", line 28, characters 4-16:
Error: Found no subterm matching "Datatypes.length xs'" in the current goal.


SCORE
-1.0
["\n    destruct p as [x y].\n    destruct (unzip l') as [xs' ys'].\n    rewrite IHl'.\n    reflexivity."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

Lemma unzip_length l :
  let (xs, ys) := unzip l in
  length xs = length ys.
Proof.
  induction l as [ | p l' IHl'].
  - reflexivity.
  - simpl.
    destruct p as [x y].
    destruct (unzip l') as [xs' ys'].
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

nat ->
nat ->
list (nat * nat) ->
forall xs' ys' : list nat,
Datatypes.length xs' = Datatypes.length ys' ->
S (Datatypes.length xs') = S (Datatypes.length ys')

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.


Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall xs' ys' : list nat, forall IHl' : Datatypes.length xs' = Datatypes.length ys', S (Datatypes.length xs') = S (Datatypes.length ys').
Proof.
(* do not use induction, try using hammer *)
intros.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['intros.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

S (Datatypes.length xs') = S (Datatypes.length ys')

### given assumptions

x, y, l', xs', ys', IHl' as follows.

x, y : nat
l' : list (nat * nat)
xs', ys' : list nat
IHl' : Datatypes.length xs' = Datatypes.length ys'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.


Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall xs' ys' : list nat, forall IHl' : Datatypes.length xs' = Datatypes.length ys', S (Datatypes.length xs') = S (Datatypes.length ys').
Proof.
(* do not use induction, try using hammer *)
intros.
hammer.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['\nhammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

S (Datatypes.length xs') = S (Datatypes.length ys')

### given assumptions

x, y, l', xs', ys', IHl' as follows.

x, y : nat
l' : list (nat * nat)
xs', ys' : list nat
IHl' : Datatypes.length xs' = Datatypes.length ys'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.


Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall xs' ys' : list nat, forall IHl' : Datatypes.length xs' = Datatypes.length ys', S (Datatypes.length xs') = S (Datatypes.length ys').
Proof.
(* do not use induction, try using hammer *)
intros.
hammer.
Qed.
SCORE
1.0
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (x :: xs') =
Datatypes.length (y :: ys')

### given assumptions

x, y, l', xs', ys', IHl' as follows.

x, y : nat
l' : list (nat * nat)
xs', ys' : list nat
IHl' : Datatypes.length xs' = Datatypes.length ys'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.


Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall xs' ys' : list nat, forall IHl' : Datatypes.length xs' = Datatypes.length ys', S (Datatypes.length xs') = S (Datatypes.length ys').
Proof.
(* do not use induction, try using hammer *)
intros.
hammer.
Qed.

Lemma unzip_length l :
  let (xs, ys) := unzip l in
  length xs = length ys.
Proof.
  induction l as [ | p l' IHl'].
  - reflexivity.
  - simpl.
    destruct p as [x y].
    destruct (unzip l') as [xs' ys']. apply (@helper0 x y l' xs' ys' IHl').
    reflexivity.
File "./ex.v", line 36, characters 4-16:
Error: No such goal.


SCORE
-1.0
['    reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.


Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall xs' ys' : list nat, forall IHl' : Datatypes.length xs' = Datatypes.length ys', S (Datatypes.length xs') = S (Datatypes.length ys').
Proof.
(* do not use induction, try using hammer *)
intros.
hammer.
Qed.

Lemma unzip_length l :
  let (xs, ys) := unzip l in
  length xs = length ys.
Proof.
  induction l as [ | p l' IHl'].
  - reflexivity.
  - simpl.
    destruct p as [x y].
    destruct (unzip l') as [xs' ys']. apply (@helper0 x y l' xs' ys' IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >   - simpl.
     >     destruct p as [x y].
     >     destruct (unzip l') as [xs' ys']. apply (@helper0 x y l' xs' ys' IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

['    reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.


Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall xs' ys' : list nat, forall IHl' : Datatypes.length xs' = Datatypes.length ys', S (Datatypes.length xs') = S (Datatypes.length ys').
Proof.
(* do not use induction, try using hammer *)
intros.
hammer.
Qed.

Lemma unzip_length l :
  let (xs, ys) := unzip l in
  length xs = length ys.
Proof.
  induction l as [ | p l' IHl'].
  - reflexivity.
  - simpl.
    destruct p as [x y].
    destruct (unzip l') as [xs' ys']. apply (@helper0 x y l' xs' ys' IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >   - simpl.
     >     destruct p as [x y].
     >     destruct (unzip l') as [xs' ys']. apply (@helper0 x y l' xs' ys' IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

['\nQed.']
['    reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.


Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall xs' ys' : list nat, forall IHl' : Datatypes.length xs' = Datatypes.length ys', S (Datatypes.length xs') = S (Datatypes.length ys').
Proof.
(* do not use induction, try using hammer *)
intros.
hammer.
Qed.

Lemma unzip_length l :
  let (xs, ys) := unzip l in
  length xs = length ys.
Proof.
  induction l as [ | p l' IHl'].
  - reflexivity.
  - simpl.
    destruct p as [x y].
    destruct (unzip l') as [xs' ys']. apply (@helper0 x y l' xs' ys' IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >   - simpl.
     >     destruct p as [x y].
     >     destruct (unzip l') as [xs' ys']. apply (@helper0 x y l' xs' ys' IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (x :: xs') =
Datatypes.length (y :: ys')

### given assumptions

x, y, l', xs', ys', IHl' as follows.

x, y : nat
l' : list (nat * nat)
xs', ys' : list nat
IHl' : Datatypes.length xs' = Datatypes.length ys'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.


Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall xs' ys' : list nat, forall IHl' : Datatypes.length xs' = Datatypes.length ys', S (Datatypes.length xs') = S (Datatypes.length ys').
Proof.
(* do not use induction, try using hammer *)
intros.
hammer.
Qed.

Lemma unzip_length l :
  let (xs, ys) := unzip l in
  length xs = length ys.
Proof.
  induction l as [ | p l' IHl'].
  - reflexivity.
  - simpl.
    destruct p as [x y].
    destruct (unzip l') as [xs' ys']. apply (@helper0 x y l' xs' ys' IHl').
Qed.
SCORE
1.0
['Qed.']
CHOSEN SOLUTION

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (x :: xs') =
Datatypes.length (y :: ys')

### given assumptions

x, y, l', xs', ys', IHl' as follows.

x, y : nat
l' : list (nat * nat)
xs', ys' : list nat
IHl' : Datatypes.length xs' = Datatypes.length ys'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.


Lemma helper0: forall x y : nat, forall l' : list (nat * nat), forall xs' ys' : list nat, forall IHl' : Datatypes.length xs' = Datatypes.length ys', S (Datatypes.length xs') = S (Datatypes.length ys').
Proof.
(* do not use induction, try using hammer *)
intros.
hammer.
Qed.

Lemma unzip_length l :
  let (xs, ys) := unzip l in
  length xs = length ys.
Proof.
  induction l as [ | p l' IHl'].
  - reflexivity.
  - simpl.
    destruct p as [x y].
    destruct (unzip l') as [xs' ys']. apply (@helper0 x y l' xs' ys' IHl').
Qed.```

STATS

number of nodes: 31
number of gen nodes: 16 (including leaves: 3)
number of back nodes: 15 (including leaves: 13)

expansion count: 29 (including failed: 14)

cache stats {'hit': 6, 'miss': 23}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 602, 57, 60, 124, 90]

Trial calls: [8, 43, 94, 21, 23, 49, 29]


Time Statistics:

  Min: 20.00 sec, Max: 602.00 sec, Average: 152.71 sec, Standard Deviation: 186.45 sec

Call Statistics:

  Min: 8 calls, Max: 94 calls, Average: 38.14 calls, Standard Deviation: 26.12 calls

=====SUMMARY STATISTICS FOR TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [602]

Trial calls: [94]


Time Statistics:

  Min: 602.00 sec, Max: 602.00 sec, Average: 602.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 94 calls, Max: 94 calls, Average: 94.00 calls, Standard Deviation: 0.00 calls

=====SUMMARY STATISTICS FOR NON_TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 57, 60, 124, 90]

Trial calls: [8, 43, 21, 23, 49, 29]


Time Statistics:

  Min: 20.00 sec, Max: 124.00 sec, Average: 77.83 sec, Standard Deviation: 36.13 sec

Call Statistics:

  Min: 8 calls, Max: 49 calls, Average: 28.83 calls, Standard Deviation: 13.76 calls

Trial  8
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
SCORE
1.0
['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  match l with
    | nil => (nil, nil)
    | (a, b) :: l' =>
      let (la, lb) := unzip l' in
      (a :: la, b :: lb)
  end.
File "./ex.v", line 14, characters 22-27:
Error: The reference unzip was not found in the current environment.


SCORE
-1.0
["\n\nDefinition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=\n  match l with\n    | nil => (nil, nil)\n    | (a, b) :: l' =>\n      let (la, lb) := unzip l' in\n      (a :: la, b :: lb)\n  end."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: l' =>
    let (a_list, b_list) := unzip l' in
    (a :: a_list, b :: b_list)
  end.
File "./ex.v", line 14, characters 28-33:
Error: The reference unzip was not found in the current environment.


SCORE
-1.0
["\n\nDefinition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=\n  match l with\n  | nil => (nil, nil)\n  | (a, b) :: l' =>\n    let (a_list, b_list) := unzip l' in\n    (a :: a_list, b :: b_list)\n  end."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).

Lemma unzip_length {A B} (l : list (A * B)) :
  (length (map fst l)) = (length (map snd l)) = (length l).
Proof.
File "./ex.v", line 14, characters 2-58:
Error: Unknown interpretation for notation "_ = _ = _".


SCORE
-1.0
['\n\nDefinition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=\n  (map fst l, map snd l).\n\nLemma unzip_length {A B} (l : list (A * B)) :\n  (length (map fst l)) = (length (map snd l)) = (length l).\nProof.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
  (map fst l, map snd l).
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > Definition unzip {A B} (l : list (A * B)) : (list A) * (list B) :=
     >   (map fst l, map snd l). >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists *)
Definition unzip {A B} (l : (A * B) list) : (A list * B list) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: l' =>
    let (as, bs) := unzip l' in
    (a :: as
File "./ex.v", line 15, characters 9-11:
Error: Syntax error: [name] expected after '(' (in [binder_constr]).


SCORE
None
["\n\n(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists *)\nDefinition unzip {A B} (l : (A * B) list) : (A list * B list) :=\n  match l with\n  | nil => (nil, nil)\n  | (a, b) :: l' =>\n    let (as, bs) := unzip l' in\n    (a :: as"]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Write a function `unzip` that takes a list of pairs and returns a pair of lists *)
Definition unzip {A B} (l : (A * B) list) : (A list * B list) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: l' =>
    let (as, bs) := unzip l' in
    (a :: as, b :: bs)
  end.
File "./ex.v", line 15, characters 9-11:
Error: Syntax error: [name] expected after '(' (in [binder_constr]).


SCORE
-1.0
[', b :: bs)\n  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(** ******************************************* *)
(** unzip function                             *)
(** ******************************************* *)

Fixpoint unzip {A B:Type} (l: list (A * B)) : (list A) * (list B) :=
  match l with
  | [] => ([], [])
  | (a, b) :: l' =>
    let (la, lb) := unzip l'
File "./ex.v", line 16, characters 4-5:
Error: Syntax error: 'end' expected after [branches] (in [term_match]).


SCORE
None
["\n\n(** ******************************************* *)\n(** unzip function                             *)\n(** ******************************************* *)\n\nFixpoint unzip {A B:Type} (l: list (A * B)) : (list A) * (list B) :=\n  match l with\n  | [] => ([], [])\n  | (a, b) :: l' =>\n    let (la, lb) := unzip l'"]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(** ******************************************* *)
(** unzip function                             *)
(** ******************************************* *)

Fixpoint unzip {A B:Type} (l: list (A * B)) : (list A) * (list B) :=
  match l with
  | [] => ([], [])
  | (a, b) :: l' =>
    let (la, lb) := unzip l' in
    (a :: la, b :: lb)
  end.
File "./ex.v", line 16, characters 4-5:
Error: Syntax error: 'end' expected after [branches] (in [term_match]).


SCORE
-1.0
[' in\n    (a :: la, b :: lb)\n  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
SCORE
1.0
['\nImport ListNotations.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (a, b) :: l' =>
    let (l1, l2) := unzip l' in
    (a :: l1, b :: l2)
  end.
File "./ex.v", line 15, characters 20-25:
Error: The reference unzip was not found in the current environment.


SCORE
-1.0
["\n\nDefinition unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n  | [] => ([], [])\n  | (a, b) :: l' =>\n    let (l1, l2) := unzip l' in\n    (a :: l1, b :: l2)\n  end."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Fixpoint unzip (l : list (A * B)) : (list A) * (list B) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: t =>
    let (l1, l2) := unzip t in
    (a :: l1, b :: l2)
  end.
File "./ex.v", line 11, characters 26-27:
Error: The reference A was not found in the current environment.


SCORE
-1.0
['\n\nFixpoint unzip (l : list (A * B)) : (list A) * (list B) :=\n  match l with\n  | nil => (nil, nil)\n  | (a, b) :: t =>\n    let (l1, l2) := unzip t in\n    (a :: l1, b :: l2)\n  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import ListNotations.
File "./ex.v", line 8, characters 7-20:
Error: Cannot find module ListNotations


SCORE
-1.0
['Import ListNotations.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

[' in\n    (a :: la, b :: lb)\n  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.
SCORE
1.0
['\n\n(* (1) Define unzip function *)\nFixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n  | nil => (nil, nil)\n  | (a, b) :: tl =>\n    let (l1, l2) := unzip tl in\n    (a :: l1, b :: l2)\n  end.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list.
File "./ex.v", line 19, characters 0-141:
Error: Syntax Error: Lexer: Unterminated comment


SCORE
None
['\n\n(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
[' *)\nLemma unzip_length : forall l, length (fst (unzip l)) = length l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\nProof.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  intros.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
[" induction l as [| p l' IHl'].\n  - (* case l = nil *)\n    reflexivity."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip nil)) =
Datatypes.length nil

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
["\n  - (* case l = (p :: l') *)\n    simpl."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l')

### given assumptions

p, l', IHl' as follows.

p : (nat * nat)%type
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl. rewrite IHl'. reflexivity.
File "./ex.v", line 26, characters 11-23:
Error:
Found no subterm matching "Datatypes.length (fst (unzip l'))" in the current goal.


SCORE
-1.0
[" rewrite IHl'. reflexivity."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall (p : nat * nat) (l' : list (nat * nat)),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' ->
Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: hauto lq: on 


## Context
### Need to show

forall (p : nat * nat) (l' : list (nat * nat)),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' ->
Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.
SCORE
1.0
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: hauto lq: on 


## Context
### Need to show

Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l')

### given assumptions

p, l', IHl' as follows.

p : (nat * nat)%type
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl. apply (@helper0 p l' IHl').
    reflexivity.
File "./ex.v", line 39, characters 4-16:
Error: No such goal.


SCORE
-1.0
['    reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl. apply (@helper0 p l' IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >     reflexivity.
     >   - (* case l = (p :: l') *)
     >     simpl. apply (@helper0 p l' IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: hauto lq: on 


## Context
### Need to show

Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l')

### given assumptions

p, l', IHl' as follows.

p : (nat * nat)%type
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl. apply (@helper0 p l' IHl').
    (* now we have to prove that fst (unzip l') = fst (unzip l) *)
    reflexivity.
File "./ex.v", line 40, characters 4-16:
Error: No such goal.


SCORE
-1.0
["    (* now we have to prove that fst (unzip l') = fst (unzip l) *)\n    reflexivity."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl. apply (@helper0 p l' IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >     reflexivity.
     >   - (* case l = (p :: l') *)
     >     simpl. apply (@helper0 p l' IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

['\nQed.']
['    reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl. apply (@helper0 p l' IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >     reflexivity.
     >   - (* case l = (p :: l') *)
     >     simpl. apply (@helper0 p l' IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

['    reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl. apply (@helper0 p l' IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >     reflexivity.
     >   - (* case l = (p :: l') *)
     >     simpl. apply (@helper0 p l' IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l')

### given assumptions

p, l', IHl' as follows.

p : (nat * nat)%type
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
["\n    destruct p as [a b].\n    destruct (unzip l') as [l1 l2].\n    simpl."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

S (Datatypes.length l1) = S (Datatypes.length l')

### given assumptions

a, b, l', l1, l2, IHl' as follows.

a, b : nat
l' : list (nat * nat)
l1, l2 : list nat
IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl.
    rewrite IHl'.
    reflexivity.
File "./ex.v", line 30, characters 4-16:
Error:
Found no subterm matching "Datatypes.length (fst (l1, l2))" in the current goal.


SCORE
-1.0
["\n    rewrite IHl'.\n    reflexivity."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl.
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

nat ->
nat ->
forall (l' : list (nat * nat)) (l1 l2 : list nat),
Datatypes.length (fst (l1, l2)) = Datatypes.length l' ->
S (Datatypes.length l1) = S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall a b : nat, forall l' : list (nat * nat), forall l1 l2 : list nat, forall IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l', S (Datatypes.length l1) = S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

nat ->
nat ->
forall (l' : list (nat * nat)) (l1 l2 : list nat),
Datatypes.length (fst (l1, l2)) = Datatypes.length l' ->
S (Datatypes.length l1) = S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall a b : nat, forall l' : list (nat * nat), forall l1 l2 : list nat, forall IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l', S (Datatypes.length l1) = S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.
SCORE
1.0
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

S (Datatypes.length l1) = S (Datatypes.length l')

### given assumptions

a, b, l', l1, l2, IHl' as follows.

a, b : nat
l' : list (nat * nat)
l1, l2 : list nat
IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall a b : nat, forall l' : list (nat * nat), forall l1 l2 : list nat, forall IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l', S (Datatypes.length l1) = S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl. apply (@helper0 a b l' l1 l2 IHl').
    reflexivity.
File "./ex.v", line 38, characters 4-16:
Error: No such goal.


SCORE
-1.0
['    reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall a b : nat, forall l' : list (nat * nat), forall l1 l2 : list nat, forall IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l', S (Datatypes.length l1) = S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl. apply (@helper0 a b l' l1 l2 IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >     destruct p as [a b].
     >     destruct (unzip l') as [l1 l2].
     >     simpl. apply (@helper0 a b l' l1 l2 IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

['    reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall a b : nat, forall l' : list (nat * nat), forall l1 l2 : list nat, forall IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l', S (Datatypes.length l1) = S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl. apply (@helper0 a b l' l1 l2 IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >     destruct p as [a b].
     >     destruct (unzip l') as [l1 l2].
     >     simpl. apply (@helper0 a b l' l1 l2 IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

['\nQed.']
['    reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall a b : nat, forall l' : list (nat * nat), forall l1 l2 : list nat, forall IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l', S (Datatypes.length l1) = S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl. apply (@helper0 a b l' l1 l2 IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >     destruct p as [a b].
     >     destruct (unzip l') as [l1 l2].
     >     simpl. apply (@helper0 a b l' l1 l2 IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

['    reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall a b : nat, forall l' : list (nat * nat), forall l1 l2 : list nat, forall IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l', S (Datatypes.length l1) = S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl. apply (@helper0 a b l' l1 l2 IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >     destruct p as [a b].
     >     destruct (unzip l') as [l1 l2].
     >     simpl. apply (@helper0 a b l' l1 l2 IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

["\n    rewrite IHl'.\n    reflexivity."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl.
</CODE>
Created Lemma helper0.
['hammer.']
['\nQed.']
['    reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall a b : nat, forall l' : list (nat * nat), forall l1 l2 : list nat, forall IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l', S (Datatypes.length l1) = S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl. apply (@helper0 a b l' l1 l2 IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >     destruct p as [a b].
     >     destruct (unzip l') as [l1 l2].
     >     simpl. apply (@helper0 a b l' l1 l2 IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

S (Datatypes.length l1) = S (Datatypes.length l')

### given assumptions

a, b, l', l1, l2, IHl' as follows.

a, b : nat
l' : list (nat * nat)
l1, l2 : list nat
IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall a b : nat, forall l' : list (nat * nat), forall l1 l2 : list nat, forall IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l', S (Datatypes.length l1) = S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl. apply (@helper0 a b l' l1 l2 IHl').
    (* the hammer tactic is used to discharge the subgoal *)
    hammer.
File "./ex.v", line 39, characters 4-11:
Error: No such goal.


SCORE
-1.0
['    (* the hammer tactic is used to discharge the subgoal *)\n    hammer.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall a b : nat, forall l' : list (nat * nat), forall l1 l2 : list nat, forall IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l', S (Datatypes.length l1) = S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl. apply (@helper0 a b l' l1 l2 IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >     destruct p as [a b].
     >     destruct (unzip l') as [l1 l2].
     >     simpl. apply (@helper0 a b l' l1 l2 IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

['\nQed.']
['    reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall a b : nat, forall l' : list (nat * nat), forall l1 l2 : list nat, forall IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l', S (Datatypes.length l1) = S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl. apply (@helper0 a b l' l1 l2 IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >     destruct p as [a b].
     >     destruct (unzip l') as [l1 l2].
     >     simpl. apply (@helper0 a b l' l1 l2 IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

['    reflexivity.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall a b : nat, forall l' : list (nat * nat), forall l1 l2 : list nat, forall IHl' : Datatypes.length (fst (l1, l2)) =
Datatypes.length l', S (Datatypes.length l1) = S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl.
    destruct p as [a b].
    destruct (unzip l') as [l1 l2].
    simpl. apply (@helper0 a b l' l1 l2 IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >     destruct p as [a b].
     >     destruct (unzip l') as [l1 l2].
     >     simpl. apply (@helper0 a b l' l1 l2 IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall (p : nat * nat) (l' : list (nat * nat)),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' ->
Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
  intros p l' IHl'.
  hammer.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
["  intros p l' IHl'.\n  hammer."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: hauto lq: on 


## Context
### Need to show

Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l')

### given assumptions

p, l', IHl' as follows.

p : (nat * nat)%type
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
  intros p l' IHl'.
  hammer.
Qed.
SCORE
1.0
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: hauto lq: on 


## Context
### Need to show

Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l')

### given assumptions

p, l', IHl' as follows.

p : (nat * nat)%type
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
  intros p l' IHl'.
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl. apply (@helper0 p l' IHl').
    apply IHl'.
Qed.
File "./ex.v", line 40, characters 4-15:
Error: No such goal.


SCORE
-1.0
["    apply IHl'.\nQed."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
  intros p l' IHl'.
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl. apply (@helper0 p l' IHl').
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > No such goal.
   The offending chunk is delimited by >>>…<<< below:
     >     reflexivity.
     >   - (* case l = (p :: l') *)
     >     simpl. apply (@helper0 p l' IHl'). >>>simpl.<<<
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: hauto lq: on 


## Context
### Need to show

Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l')

### given assumptions

p, l', IHl' as follows.

p : (nat * nat)%type
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
  intros p l' IHl'.
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl. apply (@helper0 p l' IHl').
Qed.
SCORE
1.0
['Qed.']
CHOSEN SOLUTION

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: hauto lq: on 


## Context
### Need to show

Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l')

### given assumptions

p, l', IHl' as follows.

p : (nat * nat)%type
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.

(* (1) Define unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (a, b) :: tl =>
    let (l1, l2) := unzip tl in
    (a :: l1, b :: l2)
  end.

(* (2) Define the lemma that proves the lengths of the lists in the returned pair of `unzip` are the same as the length of the original list. *)

Lemma helper0: forall p : (nat * nat)%type, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (a, b) := p in
      let (l1, l2) := unzip l' in (a :: l1, b :: l2))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
  intros p l' IHl'.
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l.
Proof.
  intros. induction l as [| p l' IHl'].
  - (* case l = nil *)
    reflexivity.
  - (* case l = (p :: l') *)
    simpl. apply (@helper0 p l' IHl').
Qed.```

STATS

number of nodes: 57
number of gen nodes: 29 (including leaves: 10)
number of back nodes: 28 (including leaves: 19)

expansion count: 50 (including failed: 22)

cache stats {'hit': 16, 'miss': 34}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 602, 57, 60, 124, 90, 134]

Trial calls: [8, 43, 94, 21, 23, 49, 29, 50]


Time Statistics:

  Min: 20.00 sec, Max: 602.00 sec, Average: 150.38 sec, Standard Deviation: 174.51 sec

Call Statistics:

  Min: 8 calls, Max: 94 calls, Average: 39.62 calls, Standard Deviation: 24.75 calls

=====SUMMARY STATISTICS FOR TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [602]

Trial calls: [94]


Time Statistics:

  Min: 602.00 sec, Max: 602.00 sec, Average: 602.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 94 calls, Max: 94 calls, Average: 94.00 calls, Standard Deviation: 0.00 calls

=====SUMMARY STATISTICS FOR NON_TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 57, 60, 124, 90, 134]

Trial calls: [8, 43, 21, 23, 49, 29, 50]


Time Statistics:

  Min: 20.00 sec, Max: 134.00 sec, Average: 85.86 sec, Standard Deviation: 38.80 sec

Call Statistics:

  Min: 8 calls, Max: 50 calls, Average: 31.86 calls, Standard Deviation: 14.74 calls

Trial  9
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
SCORE
1.0
['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import Nat.
SCORE
1.0
['\nImport Nat.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import Nat.

Lemma unzip_length : forall (l : list (nat * nat))
, (fun_map fst l) ++ (fun_map snd l) = l.
File "./ex.v", line 12, characters 3-10:
Error: The reference fun_map was not found in the current environment.


SCORE
-1.0
['\n\nLemma unzip_length : forall (l : list (nat * nat))\n, (fun_map fst l) ++ (fun_map snd l) = l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import Nat.

(* Define the unzip function *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: t =>
    let (xs, ys) := unzip t in
    (x :: xs, y :: ys)
  end.
File "./ex.v", line 14, characters 4-5:
Error: Syntax error: 'end' expected after [branches] (in [term_match]).


SCORE
-1.0
['\n\n(* Define the unzip function *)\nFixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n  | [] => ([], [])\n  | (x, y) :: t =>\n    let (xs, ys) := unzip t in\n    (x :: xs, y :: ys)\n  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

['\nImport Nat.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import Nat.

(** Part 1: Define unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.
File "./ex.v", line 15, characters 4-5:
Error: Syntax error: 'end' expected after [branches] (in [term_match]).


SCORE
-1.0
["\n\n(** Part 1: Define unzip function *)\n\nFixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n  | [] => ([], [])\n  | (x, y) :: l' =>\n    let (xs, ys) := unzip l' in\n    (x :: xs, y :: ys)\n  end."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import Nat.

(** ******************************************* *)
(** ******************************************* *)

Definition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=
  match l with
  | [] => ([], [])
  | (a, b) :: t =>
    let (la, lb) := unzip t in
    (a :: la, b :: lb)

File "./ex.v", line 16, characters 4-5:
Error: Syntax error: 'end' expected after [branches] (in [term_match]).


SCORE
None
['\n\n(** ******************************************* *)\n(** ******************************************* *)\n\nDefinition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=\n  match l with\n  | [] => ([], [])\n  | (a, b) :: t =>\n    let (la, lb) := unzip t in\n    (a :: la, b :: lb)\n']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import Nat.

(** ******************************************* *)
(** ******************************************* *)

Definition unzip {A B : Type} (l : list (A * B)) : (list A) * (list B) :=
  match l with
  | [] => ([], [])
  | (a, b) :: t =>
    let (la, lb) := unzip t in
    (a :: la, b :: lb)
  end.
File "./ex.v", line 16, characters 4-5:
Error: Syntax error: 'end' expected after [branches] (in [term_match]).


SCORE
-1.0
['  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
SCORE
1.0
['\nImport ListNotations.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
    | nil => (nil, nil)
    | (a, b) :: tl =>
      let (l1, l2) := unzip tl in
      (a :: l1, b :: l2)
  end.
SCORE
1.0
['\n\n(* Function unzip *)\nFixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n    | nil => (nil, nil)\n    | (a, b) :: tl =>\n      let (l1, l2) := unzip tl in\n      (a :: l1, b :: l2)\n  end.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
    | nil => (nil, nil)
    | (a, b) :: tl =>
      let (l1, l2) := unzip tl in
      (a :: l1, b :: l2)
  end.

(* Lemma length_unzip *)
Lemma length_unzip : forall l, length (unzip l).1 = length l /\ length (unzip l).2 = length l.
File "./ex.v", line 21, characters 47-49:
Error: Syntax Error: Lexer: Undefined token


SCORE
-1.0
['\n\n(* Lemma length_unzip *)\nLemma length_unzip : forall l, length (unzip l).1 = length l /\\ length (unzip l).2 = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
    | nil => (nil, nil)
    | (a, b) :: tl =>
      let (l1, l2) := unzip tl in
      (a :: l1, b :: l2)
  end.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >       let (l1, l2) := unzip tl in
     >       (a :: l1, b :: l2)
     >   end. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
    | nil => (nil, nil)
    | (a, b) :: tl =>
      let (l1, l2) := unzip tl in
      (a :: l1, b :: l2)
  end.

(* Lemma: The lengths of the lists in the pair returned by unzip are the same as the length of the original list *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length (snd (unzip l)) = length l.
File "./ex.v", line 21, characters 31-89:
Error: Unknown interpretation for notation "_ = _ = _".


SCORE
-1.0
['\n\n(* Lemma: The lengths of the lists in the pair returned by unzip are the same as the length of the original list *)\nLemma unzip_length : forall l, length (fst (unzip l)) = length (snd (unzip l)) = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
    | nil => (nil, nil)
    | (a, b) :: tl =>
      let (l1, l2) := unzip tl in
      (a :: l1, b :: l2)
  end.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >       let (l1, l2) := unzip tl in
     >       (a :: l1, b :: l2)
     >   end. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.
SCORE
1.0
["\n\n(* (1) Define the unzip function *)\n\nFixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n  | nil => (nil, nil)\n  | (x, y) :: l' =>\n    let (xs, ys) := unzip l' in\n    (x :: xs, y :: ys)\n  end."]
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define a lemma to prove that the lengths of the lists in the returned pair are the same as the length of the original list *)

Lemma unzip_length : forall l : list (nat * nat),
  let (xs, ys) := unzip l in
  length xs = length ys = length l.
File "./ex.v", line 25, characters 2-34:
Error: Unknown interpretation for notation "_ = _ = _".


SCORE
-1.0
['\n\n(* (2) Define a lemma to prove that the lengths of the lists in the returned pair are the same as the length of the original list *)\n\nLemma unzip_length : forall l : list (nat * nat),\n  let (xs, ys) := unzip l in\n  length xs = length ys = length l.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >     let (xs, ys) := unzip l' in
     >     (x :: xs, y :: ys)
     >   end. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n\n(* (2) Define the lemma *)\n\nLemma unzip_length : forall l, length (fst (unzip l)) = length l /\\ length (snd (unzip l)) = length l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\nProof.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  intros l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l. induction l as [| (x, y) l' IHl'].
  - reflexivity.
File "./ex.v", line 26, characters 4-15:
Error:
 The relation and is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
[" induction l as [| (x, y) l' IHl'].\n  - reflexivity."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l. induction l as [| (x, y) l' IHl'].
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

0 = 0 /\ 0 = 0

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

0 = 0 /\ 0 = 0

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.
SCORE
1.0
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip ((x, y) :: l'))) =
Datatypes.length ((x, y) :: l') /\
Datatypes.length (snd (unzip ((x, y) :: l'))) =
Datatypes.length ((x, y) :: l')

### given assumptions

x, y, l', IHl' as follows.

x, y : nat
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l. induction l as [| (x, y) l' IHl']. apply (@helper0 ).
  (* use the induction hypothesis IHl' *)
  apply (IHl' ).
  (* use the constructor of the pair type *)
  split.
File "./ex.v", line 34, characters 9-13:
Error:
In environment
x, y : nat
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) = Datatypes.length l' /\
       Datatypes.length (snd (unzip l')) = Datatypes.length l'
Unable to unify
 "Datatypes.length (fst (unzip l')) = Datatypes.length l' /\
  Datatypes.length (snd (unzip l')) = Datatypes.length l'" with
 "Datatypes.length (fst (unzip ((x, y) :: l'))) =
  Datatypes.length ((x, y) :: l') /\
  Datatypes.length (snd (unzip ((x, y) :: l'))) =
  Datatypes.length ((x, y) :: l')".


SCORE
-1.0
["  (* use the induction hypothesis IHl' *)\n  apply (IHl' ).\n  (* use the constructor of the pair type *)\n  split."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l. induction l as [| (x, y) l' IHl'].
</CODE>
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip ((x, y) :: l'))) =
Datatypes.length ((x, y) :: l') /\
Datatypes.length (snd (unzip ((x, y) :: l'))) =
Datatypes.length ((x, y) :: l')

### given assumptions

x, y, l', IHl' as follows.

x, y : nat
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l. induction l as [| (x, y) l' IHl']. apply (@helper0 ).
  (* now we need to prove the two goals *)
  (* we use the induction hypothesis *)
  rewrite IHl'.
  (* we can use the rewrite tactic to simplify the goals *)
  rewrite <- Nat.
File "./ex.v", line 35, characters 2-14:
Error:
Found no subterm matching "Datatypes.length (fst (unzip l')) =
                           Datatypes.length l'" in the current goal.


SCORE
-1.0
["  (* now we need to prove the two goals *)\n  (* we use the induction hypothesis *)\n  rewrite IHl'.\n  (* we can use the rewrite tactic to simplify the goals *)\n  rewrite <- Nat."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l. induction l as [| (x, y) l' IHl']. apply (@helper0 ).
</CODE>
Created Lemma helper1.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

forall (x y : nat) (l' : list (nat * nat)),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.


Lemma helper1: forall x y : nat, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
  hammer.
Error: There are pending proofs in file ./ex.v: helper1.

SCORE
1.0
['  hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: hauto 


## Context
### Need to show

forall (x y : nat) (l' : list (nat * nat)),
Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l' ->
Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l')

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.


Lemma helper1: forall x y : nat, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.
SCORE
1.0
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: hauto 


## Context
### Need to show

Datatypes.length (fst (unzip ((x, y) :: l'))) =
Datatypes.length ((x, y) :: l') /\
Datatypes.length (snd (unzip ((x, y) :: l'))) =
Datatypes.length ((x, y) :: l')

### given assumptions

x, y, l', IHl' as follows.

x, y : nat
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.


Lemma helper1: forall x y : nat, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l. induction l as [| (x, y) l' IHl']. apply (@helper0 ). apply (@helper1 x y l' IHl').
Qed.
SCORE
1.0
['Qed.']
CHOSEN SOLUTION

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 
Replace the hammer tactic with: hauto 


## Context
### Need to show

Datatypes.length (fst (unzip ((x, y) :: l'))) =
Datatypes.length ((x, y) :: l') /\
Datatypes.length (snd (unzip ((x, y) :: l'))) =
Datatypes.length ((x, y) :: l')

### given assumptions

x, y, l', IHl' as follows.

x, y : nat
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

(* (1) Define the unzip function *)

Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: l' =>
    let (xs, ys) := unzip l' in
    (x :: xs, y :: ys)
  end.

(* (2) Define the lemma *)


Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.


Lemma helper1: forall x y : nat, forall l' : list (nat * nat), forall IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l', Datatypes.length
  (fst
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l') /\
Datatypes.length
  (snd
     (let (xs, ys) := unzip l' in (x :: xs, y :: ys))) =
S (Datatypes.length l').
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l. induction l as [| (x, y) l' IHl']. apply (@helper0 ). apply (@helper1 x y l' IHl').
Qed.```

STATS

number of nodes: 37
number of gen nodes: 19 (including leaves: 4)
number of back nodes: 18 (including leaves: 15)

expansion count: 26 (including failed: 8)

cache stats {'hit': 2, 'miss': 24}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 602, 57, 60, 124, 90, 134, 89]

Trial calls: [8, 43, 94, 21, 23, 49, 29, 50, 26]


Time Statistics:

  Min: 20.00 sec, Max: 602.00 sec, Average: 143.56 sec, Standard Deviation: 165.66 sec

Call Statistics:

  Min: 8 calls, Max: 94 calls, Average: 38.11 calls, Standard Deviation: 23.72 calls

=====SUMMARY STATISTICS FOR TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [602]

Trial calls: [94]


Time Statistics:

  Min: 602.00 sec, Max: 602.00 sec, Average: 602.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 94 calls, Max: 94 calls, Average: 94.00 calls, Standard Deviation: 0.00 calls

=====SUMMARY STATISTICS FOR NON_TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 57, 60, 124, 90, 134, 89]

Trial calls: [8, 43, 21, 23, 49, 29, 50, 26]


Time Statistics:

  Min: 20.00 sec, Max: 134.00 sec, Average: 86.25 sec, Standard Deviation: 36.31 sec

Call Statistics:

  Min: 8 calls, Max: 50 calls, Average: 31.12 calls, Standard Deviation: 13.92 calls

Trial  10
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Section Unzip.
Error: The section Unzip needs to be closed.


SCORE
-1.0
['Section Unzip.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import ListNotations.
File "./ex.v", line 8, characters 7-20:
Error: Cannot find module ListNotations


SCORE
-1.0
['Import ListNotations.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Coq.Strings.String. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
SCORE
1.0
['Import List.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
SCORE
1.0
['\nImport ListNotations.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.

Definition unzip {X Y} (l : list (X * Y)) : (list X) * (list Y) :=
  match l with
  | nil => (nil, nil)
  | (x, y) :: tl =>
    let (xs, ys) := unzip tl in
    (x :: xs, y :: ys)
  end.
File "./ex.v", line 15, characters 20-25:
Error: The reference unzip was not found in the current environment.


SCORE
-1.0
['\n\nDefinition unzip {X Y} (l : list (X * Y)) : (list X) * (list Y) :=\n  match l with\n  | nil => (nil, nil)\n  | (x, y) :: tl =>\n    let (xs, ys) := unzip tl in\n    (x :: xs, y :: ys)\n  end.']
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     > Require Import Coq.Strings.String.
     >
     > Import List. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.
SCORE
1.0
['\nImport Nat.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: tl => (x :: (fst (unzip tl)), y :: (snd (unzip tl)))
  end.
SCORE
1.0
['\n\n(* Function unzip *)\nFixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=\n  match l with\n  | [] => ([], [])\n  | (x, y) :: tl => (x :: (fst (unzip tl)), y :: (snd (unzip tl)))\n  end.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context


[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: tl => (x :: (fst (unzip tl)), y :: (snd (unzip tl)))
  end.

(* Lemma: length of the lists in the returned pair of unzip is the same as the length of the original list *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n\n(* Lemma: length of the lists in the returned pair of unzip is the same as the length of the original list *)\nLemma unzip_length : forall l, length (fst (unzip l)) = length l /\\ length (snd (unzip l)) = length l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: tl => (x :: (fst (unzip tl)), y :: (snd (unzip tl)))
  end.

(* Lemma: length of the lists in the returned pair of unzip is the same as the length of the original list *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\nProof.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

forall l : list (nat * nat),
Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: tl => (x :: (fst (unzip tl)), y :: (snd (unzip tl)))
  end.

(* Lemma: length of the lists in the returned pair of unzip is the same as the length of the original list *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['\n  intros l.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

Datatypes.length (fst (unzip l)) = Datatypes.length l /\
Datatypes.length (snd (unzip l)) = Datatypes.length l

### given assumptions

l as follows.

l : list (nat * nat).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: tl => (x :: (fst (unzip tl)), y :: (snd (unzip tl)))
  end.

(* Lemma: length of the lists in the returned pair of unzip is the same as the length of the original list *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IHl'].
  - reflexivity.
File "./ex.v", line 24, characters 4-15:
Error:
 The relation and is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
["\n  induction l as [| p l' IHl'].\n  - reflexivity."]
<CODE> 
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: tl => (x :: (fst (unzip tl)), y :: (snd (unzip tl)))
  end.

(* Lemma: length of the lists in the returned pair of unzip is the same as the length of the original list *)
Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IHl'].
</CODE>
Created Lemma helper0.
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out


## Context
### Need to show

0 = 0 /\ 0 = 0

### given assumptions

(none).

[/INST]
## Code

```Coq

From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: tl => (x :: (fst (unzip tl)), y :: (snd (unzip tl)))
  end.

(* Lemma: length of the lists in the returned pair of unzip is the same as the length of the original list *)

Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  hammer.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

0 = 0 /\ 0 = 0

### given assumptions

(none).

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: tl => (x :: (fst (unzip tl)), y :: (snd (unzip tl)))
  end.

(* Lemma: length of the lists in the returned pair of unzip is the same as the length of the original list *)

Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.
SCORE
1.0
['\nQed.']
TEXT

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip (p :: l'))) =
Datatypes.length (p :: l') /\
Datatypes.length (snd (unzip (p :: l'))) =
Datatypes.length (p :: l')

### given assumptions

p, l', IHl' as follows.

p : (nat * nat)%type
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: tl => (x :: (fst (unzip tl)), y :: (snd (unzip tl)))
  end.

(* Lemma: length of the lists in the returned pair of unzip is the same as the length of the original list *)

Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IHl']. apply (@helper0 ).
  (* do not use induction, try using hammer *)
  hammer.
Error: There are pending proofs in file ./ex.v: unzip_length.

SCORE
1.0
['  (* do not use induction, try using hammer *)\n  hammer.']
CHOSEN SOLUTION

 [INST] <<SYS>>
You are a Coq programmer that writes functional code and prove properties about it. When you are unsure of which lemmas to use, you use the `Search` function, for example `Search (0 < _).`. You can see the output of the Coq verifier in the Out section, and the context of the current proof, comprising the current goal and assumptions, in the Context section. The assumptions have names that you can use in your proofs.

You can use Coq Hammer, including the tactic `hammer` to attempt to discharge a goal automatically.
To use Coq Hammer effectively, combine it with destruct using `;`: `destruct e1; destruct e2; hammer`.

You take a single step and will be given feedback -- listen to the feedback in the instructions.
<</SYS>>

## Instructions
### Spec: In Coq:
(1) Write a function `unzip` that takes a list of pairs and returns a pair of lists, where the first element is a list of all the first elements of the original list, and the second elemens is a list of all the second elements of the original list.
(2) Then write a lemma that proves that the length of the lists in the returned pair of `unzip` are the same as the length of the original list.
### Hint: Import `Coq.Lists.List`.

## Out
Replace the hammer tactic with: sfirstorder 


## Context
### Need to show

Datatypes.length (fst (unzip (p :: l'))) =
Datatypes.length (p :: l') /\
Datatypes.length (snd (unzip (p :: l'))) =
Datatypes.length (p :: l')

### given assumptions

p, l', IHl' as follows.

p : (nat * nat)%type
l' : list (nat * nat)
IHl' : Datatypes.length (fst (unzip l')) =
Datatypes.length l' /\
Datatypes.length (snd (unzip l')) =
Datatypes.length l'.

[/INST]
## Code

```Coq
From Hammer Require Import Tactics.
From Hammer Require Import Hammer.

From Coq Require Import Arith.Arith.
Require Import Coq.Strings.String.

Import List.
Import ListNotations.
Import Nat.

(* Function unzip *)
Fixpoint unzip (l : list (nat * nat)) : (list nat) * (list nat) :=
  match l with
  | [] => ([], [])
  | (x, y) :: tl => (x :: (fst (unzip tl)), y :: (snd (unzip tl)))
  end.

(* Lemma: length of the lists in the returned pair of unzip is the same as the length of the original list *)

Lemma helper0: 0 = 0 /\ 0 = 0.
Proof.
(* do not use induction, try using hammer *)
  hammer.
Qed.

Lemma unzip_length : forall l, length (fst (unzip l)) = length l /\ length (snd (unzip l)) = length l.
Proof.
  intros l.
  induction l as [| p l' IHl']. apply (@helper0 ).
  (* do not use induction, try using hammer *)
  hammer.```

STATS

number of nodes: 23
number of gen nodes: 12 (including leaves: 1)
number of back nodes: 11 (including leaves: 11)

expansion count: 14 (including failed: 3)

cache stats {'hit': 0, 'miss': 14}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 602, 57, 60, 124, 90, 134, 89, 38]

Trial calls: [8, 43, 94, 21, 23, 49, 29, 50, 26, 14]


Time Statistics:

  Min: 20.00 sec, Max: 602.00 sec, Average: 133.00 sec, Standard Deviation: 160.32 sec

Call Statistics:

  Min: 8 calls, Max: 94 calls, Average: 35.70 calls, Standard Deviation: 23.64 calls

=====SUMMARY STATISTICS FOR TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [602]

Trial calls: [94]


Time Statistics:

  Min: 602.00 sec, Max: 602.00 sec, Average: 602.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 94 calls, Max: 94 calls, Average: 94.00 calls, Standard Deviation: 0.00 calls

=====SUMMARY STATISTICS FOR NON_TIMED OUT TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [20, 116, 57, 60, 124, 90, 134, 89, 38]

Trial calls: [8, 43, 21, 23, 49, 29, 50, 26, 14]


Time Statistics:

  Min: 20.00 sec, Max: 134.00 sec, Average: 80.89 sec, Standard Deviation: 37.44 sec

Call Statistics:

  Min: 8 calls, Max: 50 calls, Average: 29.22 calls, Standard Deviation: 14.19 calls

To complete all experiments, 22 minutes and 44 seconds elapsed.