dafny_run_mult_deepseek.txt

/home/namin/mambaforge/envs/trl/lib/python3.10/site-packages/trl/trainer/ppo_config.py:141: UserWarning: The `optimize_cuda_cache` arguement will be deprecated soon, please use `optimize_device_cache` instead.
  warnings.warn(
/home/namin/mambaforge/envs/trl/lib/python3.10/site-packages/transformers/models/auto/auto_factory.py:472: FutureWarning: The `use_auth_token` argument is deprecated and will be removed in v5 of Transformers. Please use `token` instead.
  warnings.warn(

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TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)

SCORE
1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0

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1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b

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1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{

  |
4 | {
  |  ^

ex.dfy(4,1): Error: rbrace expected
1 parse errors detected in ex.dfy

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None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO

  |
5 |   // TODO
  |          ^

ex.dfy(5,9): Error: rbrace expected
1 parse errors detected in ex.dfy

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None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}

SCORE
1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```
SCORE
1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

SCORE
1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```


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1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```
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1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny

SCORE
1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny
function MultiplyCancellation(a: nat, b: nat, n: nat)

  |
9 | function MultiplyCancellation(a: nat, b: nat, n: nat)
  |                                                      ^

ex.dfy(9,53): Error: colon expected
1 parse errors detected in ex.dfy

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None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny
function MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0

   |
10 |   requires n != 0
   |   ^

ex.dfy(10,2): Error: colon expected
1 parse errors detected in ex.dfy

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None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny
function MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b

   |
10 |   requires n != 0
   |   ^

ex.dfy(10,2): Error: colon expected
1 parse errors detected in ex.dfy

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None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny
function MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{

   |
10 |   requires n != 0
   |   ^

ex.dfy(10,2): Error: colon expected
   |
12 | {
   |  ^

ex.dfy(12,1): Error: invalid UnaryExpression
2 parse errors detected in ex.dfy

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-1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)

  |
9 | lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  |       ^^^^^^^^^^^^^^^^^^^^

ex.dfy(9,6): Error: Duplicate member name: MultiplyCancellation
1 resolution/type errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0

  |
9 | lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  |       ^^^^^^^^^^^^^^^^^^^^

ex.dfy(9,6): Error: Duplicate member name: MultiplyCancellation
1 resolution/type errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b

  |
9 | lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  |       ^^^^^^^^^^^^^^^^^^^^

ex.dfy(9,6): Error: Duplicate member name: MultiplyCancellation
1 resolution/type errors detected in ex.dfy

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-1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny

SCORE
1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)

  |
9 | lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  |       ^^^^^^^^^^^^^^^^^^^^

ex.dfy(9,6): Error: Duplicate member name: MultiplyCancellation
1 resolution/type errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0

  |
9 | lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  |       ^^^^^^^^^^^^^^^^^^^^

ex.dfy(9,6): Error: Duplicate member name: MultiplyCancellation
1 resolution/type errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b

  |
9 | lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  |       ^^^^^^^^^^^^^^^^^^^^

ex.dfy(9,6): Error: Duplicate member name: MultiplyCancellation
1 resolution/type errors detected in ex.dfy

SCORE
-1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)

  |
9 | lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  |       ^^^^^^^^^^^^^^^^^^^^

ex.dfy(9,6): Error: Duplicate member name: MultiplyCancellation
1 resolution/type errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0

  |
9 | lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  |       ^^^^^^^^^^^^^^^^^^^^

ex.dfy(9,6): Error: Duplicate member name: MultiplyCancellation
1 resolution/type errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b

  |
9 | lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  |       ^^^^^^^^^^^^^^^^^^^^

ex.dfy(9,6): Error: Duplicate member name: MultiplyCancellation
1 resolution/type errors detected in ex.dfy

SCORE
-1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

### Solution

SCORE
1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

### Solution
```
SCORE
1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

### Solution
```dafny

SCORE
1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

### Solution
```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)

  |
9 | lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  |       ^^^^^^^^^^^^^^^^^^^^

ex.dfy(9,6): Error: Duplicate member name: MultiplyCancellation
1 resolution/type errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

### Solution
```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0

  |
9 | lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  |       ^^^^^^^^^^^^^^^^^^^^

ex.dfy(9,6): Error: Duplicate member name: MultiplyCancellation
1 resolution/type errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

### Solution
```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b

  |
9 | lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  |       ^^^^^^^^^^^^^^^^^^^^

ex.dfy(9,6): Error: Duplicate member name: MultiplyCancellation
1 resolution/type errors detected in ex.dfy

SCORE
-1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it immediately.

```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  requires n != 0
  ensures a == b <==> n*a == n*b
{
  // TODO
}
```

### Solution
```dafny
lemma MultiplyCancellation(a: nat, b: nat, n: nat)

  |
9 | lemma MultiplyCancellation(a: nat, b: nat, n: nat)
  |       ^^^^^^^^^^^^^^^^^^^^

ex.dfy(9,6): Error: Duplicate member name: MultiplyCancellation
1 resolution/type errors detected in ex.dfy

SCORE
None
TEXT
/home/namin/mambaforge/envs/trl/lib/python3.10/site-packages/trl/trainer/ppo_config.py:141: UserWarning: The `optimize_cuda_cache` arguement will be deprecated soon, please use `optimize_device_cache` instead.
  warnings.warn(
/home/namin/mambaforge/envs/trl/lib/python3.10/site-packages/transformers/models/auto/auto_factory.py:472: FutureWarning: The `use_auth_token` argument is deprecated and will be removed in v5 of Transformers. Please use `token` instead.
  warnings.warn(

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/home/namin/mambaforge/envs/trl/lib/python3.10/site-packages/transformers/utils/hub.py:374: FutureWarning: The `use_auth_token` argument is deprecated and will be removed in v5 of Transformers. Please use `token` instead.
  warnings.warn(
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it right there, don't leave a TODO.

```dafny
lemma MultDistinct(a: nat, b: nat, n: nat)

SCORE
1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it right there, don't leave a TODO.

```dafny
lemma MultDistinct(a: nat, b: nat, n: nat)
    requires n != 0

SCORE
1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it right there, don't leave a TODO.

```dafny
lemma MultDistinct(a: nat, b: nat, n: nat)
    requires n != 0
    ensures a == b <==> n*a == n*b

SCORE
1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it right there, don't leave a TODO.

```dafny
lemma MultDistinct(a: nat, b: nat, n: nat)
    requires n != 0
    ensures a == b <==> n*a == n*b
{

  |
4 | {
  |  ^

ex.dfy(4,1): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it right there, don't leave a TODO.

```dafny
lemma MultDistinct(a: nat, b: nat, n: nat)
    requires n != 0
    ensures a == b <==> n*a == n*b
{
    if a == b {

  |
5 |     if a == b {
  |                ^

ex.dfy(5,15): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it right there, don't leave a TODO.

```dafny
lemma MultDistinct(a: nat, b: nat, n: nat)
    requires n != 0
    ensures a == b <==> n*a == n*b
{
    if a == b {
        assert n*a == n*b;

  |
6 |         assert n*a == n*b;
  |                           ^

ex.dfy(6,26): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it right there, don't leave a TODO.

```dafny
lemma MultDistinct(a: nat, b: nat, n: nat)
    requires n != 0
    ensures a == b <==> n*a == n*b
{
    if a == b {
        assert n*a == n*b;
    } else {

  |
7 |     } else {
  |             ^

ex.dfy(7,12): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it right there, don't leave a TODO.

```dafny
lemma MultDistinct(a: nat, b: nat, n: nat)
    requires n != 0
    ensures a == b <==> n*a == n*b
{
    if a == b {
        assert n*a == n*b;
    } else {
        assert n*a != n*b;

  |
8 |         assert n*a != n*b;
  |                           ^

ex.dfy(8,26): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it right there, don't leave a TODO.

```dafny
lemma MultDistinct(a: nat, b: nat, n: nat)
    requires n != 0
    ensures a == b <==> n*a == n*b
{
    if a == b {
        assert n*a == n*b;
    } else {
        assert n*a != n*b;
    }

  |
9 |     }
  |      ^

ex.dfy(9,5): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it right there, don't leave a TODO.

```dafny
lemma MultDistinct(a: nat, b: nat, n: nat)
    requires n != 0
    ensures a == b <==> n*a == n*b
{
    if a == b {
        assert n*a == n*b;
    } else {
        assert n*a != n*b;
    }
}

SCORE
1.0
TEXT
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it right there, don't leave a TODO.

```dafny
lemma MultDistinct(a: nat, b: nat, n: nat)
    requires n != 0
    ensures a == b <==> n*a == n*b
{
    if a == b {
        assert n*a == n*b;
    } else {
        assert n*a != n*b;
    }
}
```
SCORE
1.0
CHOSEN SOLUTION
### Spec: In Dafny, prove that for all natural numbers `a`, `b`, `n`, if `n` is not zero, then `n*a` equals `n*b` implies `a` equals `b`.
### Hint: Define the lemma only once and prove it right there, don't leave a TODO.

```dafny
lemma MultDistinct(a: nat, b: nat, n: nat)
    requires n != 0
    ensures a == b <==> n*a == n*b
{
    if a == b {
        assert n*a == n*b;
    } else {
        assert n*a != n*b;
    }
}
```