dafny_fact_run_diversity_deepseek.txt

[2023-12-06 23:21:31,113] [INFO] [real_accelerator.py:158:get_accelerator] Setting ds_accelerator to cuda (auto detect)
/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(

Loading checkpoint shards:   0%|          | 0/7 [00:00<?, ?it/s]
Loading checkpoint shards:  14%|█▍        | 1/7 [00:06<00:40,  6.69s/it]
Loading checkpoint shards:  29%|██▊       | 2/7 [00:13<00:34,  6.80s/it]
Loading checkpoint shards:  43%|████▎     | 3/7 [00:20<00:27,  6.83s/it]
Loading checkpoint shards:  57%|█████▋    | 4/7 [00:28<00:21,  7.31s/it]
Loading checkpoint shards:  71%|███████▏  | 5/7 [00:35<00:14,  7.18s/it]
Loading checkpoint shards:  86%|████████▌ | 6/7 [00:42<00:07,  7.03s/it]
Loading checkpoint shards: 100%|██████████| 7/7 [00:47<00:00,  6.46s/it]
Loading checkpoint shards: 100%|██████████| 7/7 [00:47<00:00,  6.78s/it]
/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(
['function Factorial(n: nat): nat\n', 'function factorial(n: nat): nat\n', 'function factorial(n: nat): nat\n', 'function factorial(n: nat): nat\n', 'function factorial(n: nat): nat\n', 'function factorial(n: nat): nat\n', 'function Factorial(n: nat): nat\n', 'function factorial(n: nat): nat\n', 'function Factorial(n: nat): nat\n', 'function factorial(n: nat): nat\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat

SCORE
1.0
['{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '  decreases n\n', '{\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n

SCORE
1.0
['{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{

  |
3 | {
  |  ^

ex.dfy(3,1): Error: invalid UnaryExpression
1 parse errors detected in ex.dfy

SCORE
None
['  if n == 0 then 1 else n * Factorial(n-1)\n', '  if n == 0 then 1 else n * Factorial(n-1)\n', '  if n == 0 then 1 else n * Factorial(n-1)\n', '  if n == 0 then 1 else n * Factorial(n-1)\n', '  if n == 0 then 1 else n * Factorial(n - 1)\n', '  if n == 0 then 1 else n * Factorial(n - 1)\n', '  if n == 0 then 1 else n * Factorial(n - 1)\n', '  if n == 0 then 1 else n * Factorial(n-1)\n', '  if n == 0 then 1 else n * Factorial(n-1)\n', '  if n == 0 then 1 else n * Factorial(n - 1)\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)

  |
4 |   if n == 0 then 1 else n * Factorial(n - 1)
  |                                             ^

ex.dfy(4,44): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

SCORE
1.0
['\n', '\n', '\n', '\n', '\n', '\n', '\n', '\n', '\n', '\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}


SCORE
1.0
['lemma AlwaysPositive()\n', 'lemma FactorialIsAlwaysPositive(n: nat)\n', 'lemma FactorialAlwaysPositive(n: nat)\n', 'lemma AlwaysStrictlyPositive()\n', 'lemma PositiveFactorial{n: nat}\n', 'method Main() {\n', 'lemma FactorialIsStrictlyPositive(n: nat)\n', 'lemma FactorialIsAlwaysStrictlyPositive(n: nat)\n', 'lemma FactorialIsAlwaysStrictlyPositive(n: nat)\n', 'lemma FactorialAlwaysPositive(n: nat)\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma PositiveFactorial{n: nat}

  |
7 | lemma PositiveFactorial{n: nat}
  |                        ^

ex.dfy(7,23): Error: invalid MethodDecl
  |
7 | lemma PositiveFactorial{n: nat}
  |                         ^

ex.dfy(7,24): Error: invalid statement beginning here (is a 'label' keyword missing? or a 'const' or 'var' keyword?)
  |
7 | lemma PositiveFactorial{n: nat}
  |                            ^

ex.dfy(7,27): Error: rbrace expected
3 parse errors detected in ex.dfy

SCORE
None
['  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', 'ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma PositiveFactorial{n: nat}
ensures Factorial(n) > 0

  |
7 | lemma PositiveFactorial{n: nat}
  |                        ^

ex.dfy(7,23): Error: invalid MethodDecl
  |
7 | lemma PositiveFactorial{n: nat}
  |                         ^

ex.dfy(7,24): Error: invalid statement beginning here (is a 'label' keyword missing? or a 'const' or 'var' keyword?)
  |
7 | lemma PositiveFactorial{n: nat}
  |                            ^

ex.dfy(7,27): Error: rbrace expected
3 parse errors detected in ex.dfy

SCORE
None
['{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma PositiveFactorial{n: nat}
ensures Factorial(n) > 0
{

  |
7 | lemma PositiveFactorial{n: nat}
  |                        ^

ex.dfy(7,23): Error: invalid MethodDecl
  |
7 | lemma PositiveFactorial{n: nat}
  |                         ^

ex.dfy(7,24): Error: invalid statement beginning here (is a 'label' keyword missing? or a 'const' or 'var' keyword?)
  |
7 | lemma PositiveFactorial{n: nat}
  |                            ^

ex.dfy(7,27): Error: rbrace expected
3 parse errors detected in ex.dfy

SCORE
-1.0
['lemma FactorialIsAlwaysPositive(n: nat)\n', 'lemma FactorialIsPositive(n: nat)\n', 'lemma FactorialIsAlwaysPositive(n: nat)\n', 'lemma FactorialAlwaysPositive(n: nat)\n', 'method Main()\n', 'method Main()\n', 'lemma FactorialIsPositive(n: nat)\n', 'lemma FactorialAlwaysStrictlyPositive(n: nat)\n', 'lemma PositiveFactorial(n: nat)\n', 'lemma FactorialAlwaysPositive(n: nat)\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

method Main()

SCORE
1.0
['{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

method Main()
{

  |
8 | {
  |  ^

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

SCORE
None
['  assert forall n :: 0 <= n < Factorial(n);\n', '  var n: nat := 5;\n', '  var n: nat;\n', '  assert Factorial(5) > 0;\n', '  assert Factorial(0) > 0;\n', '  var n: nat := 5;\n', '  var i: nat := 0;\n', '  var i: nat := 0;\n', '  assert Factorial(10) > 0;\n', '  var n: nat := 0;\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

method Main()
{
  var n: nat;

  |
9 |   var n: nat;
  |              ^

ex.dfy(9,13): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['  while n < 10 {\n', '  assert Factorial(n) > 0;\n', '  assert Factorial(n) > 0;\n', '  assert Factorial(n) >= 0;\n', '  assert Factorial(n) > 0;\n', '  assert Factorial(n) > 0;\n', '  assert Factorial(n) >= 1;\n', '  assert Factorial(n) > 0;\n', '  assert Factorial(n) > 0;\n', '  assert n >= 0; // Error: This assertion may not hold.\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

method Main()
{
  var n: nat;
  assert n >= 0; // Error: This assertion may not hold.

   |
10 |   assert n >= 0; // Error: This assertion may not hold.
   |                                                        ^

ex.dfy(10,55): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '  assert Factorial(n) >= 0;\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

method Main()
{
  var n: nat;
  assert n >= 0; // Error: This assertion may not hold.
  assert Factorial(n) >= 0;

   |
11 |   assert Factorial(n) >= 0;
   |                            ^

ex.dfy(11,27): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

method Main()
{
  var n: nat;
  assert n >= 0; // Error: This assertion may not hold.
  assert Factorial(n) >= 0;
}

ex.dfy(10,9): Error: variable 'n', which is subject to definite-assignment rules, might be uninitialized here
   |
10 |   assert n >= 0; // Error: This assertion may not hold.
   |          ^


Dafny program verifier finished with 1 verified, 1 error

SCORE
-1.0
['{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

method Main()
{

  |
8 | {
  |  ^

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

SCORE
None
['  for i in 0..100\n', '  var i: nat := 0;\n', '  var i: nat := 0;\n', '  var n: nat := 5;\n', '  assert Factorial(0) > 0;\n', '  var i: nat := 0;\n', '  var i: nat :| 0 <= i < 10;\n', '  var n: nat := 0;\n', '  var i: nat := 0;\n', '  assert Factorial(0) > 0;\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

method Main()
{
  for i in 0..100

  |
9 |   for i in 0..100
  |         ^

ex.dfy(9,8): Error: gets expected
  |
9 |   for i in 0..100
  |             ^

ex.dfy(9,12): Error: ident expected
  |
9 |   for i in 0..100
  |            ^

ex.dfy(9,11): Error: Expected 'to' or 'downto'
3 parse errors detected in ex.dfy

SCORE
None
['  {\n', '  {\n', '  {\n', '  {\n', '  {\n', '  {\n', '  {\n', '  {\n', '  {\n', '  {\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

method Main()
{
  for i in 0..100
  {

  |
9 |   for i in 0..100
  |         ^

ex.dfy(9,8): Error: gets expected
  |
9 |   for i in 0..100
  |             ^

ex.dfy(9,12): Error: ident expected
  |
9 |   for i in 0..100
  |            ^

ex.dfy(9,11): Error: Expected 'to' or 'downto'
3 parse errors detected in ex.dfy

SCORE
None
['    assert Factorial(i) > 0;\n', '    assert Factorial(i) > 0;\n', '    var fact := Factorial(i);\n', '    assert Factorial(i) > 0;\n', '    assert Factorial(i) > 0;\n', '    assert Factorial(i) > 0;\n', '    assert Factorial(i) > 0;\n', '    assert Factorial(i) > 0;\n', '    assert Factorial(i) > 0;\n', '    assert Factorial(i) > 0;\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

method Main()
{
  for i in 0..100
  {
    var fact := Factorial(i);

  |
9 |   for i in 0..100
  |         ^

ex.dfy(9,8): Error: gets expected
  |
9 |   for i in 0..100
  |             ^

ex.dfy(9,12): Error: ident expected
  |
9 |   for i in 0..100
  |            ^

ex.dfy(9,11): Error: Expected 'to' or 'downto'
3 parse errors detected in ex.dfy

SCORE
-1.0
['\n', '\n', '```', '\n', '\n', '\n', '\n', '\n', '\n', '\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}


SCORE
1.0
['lemma FactorialIsAlwaysStrictlyPositive(n: nat)\n', 'lemma AlwaysStrictlyPositive{n: nat}(factorial: Factorial(n)): 1 <= factorial\n', 'lemma FactorialIsAlwaysPositive(n: nat)\n', 'lemma FactorialIsAlwaysStrictlyPositive(n: nat)\n', 'lemma FactorialIsStrictlyPositive(n: nat)\n', 'lemma FactorialIsPositive(n: nat)\n', 'lemma FactorialAlwaysPositive()\n', 'method FactorialPositive(n: nat) returns (b: bool)\n', 'lemma FactorialIsPositive(n: nat)\n', 'lemma FactorialAlwaysStrictlyPositive()\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()

SCORE
1.0
['  ensures Factorial(n) > 0\n', '{\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(0) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures forall n :: 0 <= n < Factorial(n)\n', '  ensures Factorial(0) > 0\n', '  ensures Factorial(0) > 0\n', '{\n', '  ensures forall n :: 0 <= n < Factorial(n)\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)

SCORE
1.0
['{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{

  |
9 | {
  |  ^

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

SCORE
None
['  var n: nat :| 0 <= n;\n', '  var n := 0;\n', '  var n :| 0 <= n;\n', '  var n: nat := 0;\n', '  var n :| 0 <= n;\n', '}\n', '  var n :| true;\n', '}\n', '  var i: nat := 0;\n', '}\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  var n :| 0 <= n;

   |
10 |   var n :| 0 <= n;
   |                   ^

ex.dfy(10,18): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['  if n == 0 then {\n', '  if n == 0 {\n', '  var m :| m == Factorial(n);\n', '  {\n', '  {\n', '  {\n', '  if n == 0 then {\n', '  {\n', '  if n == 0 then {\n', '  {\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  var n :| 0 <= n;
  {

   |
11 |   {
   |    ^

ex.dfy(11,3): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    if n == 0 then\n', '    // Dafny requires a base case here.\n', '    if n == 0 {\n', '    // base case\n', '    var fact := Factorial(n);\n', '    if n == 0 then Factorial(n) == 1\n', '    if n == 0 then Factorial(n) == 1\n', '    if n == 0 then {\n', '    if n == 0 {\n', '    if n == 0 then 1 else n * Factorial(n - 1)\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  var n :| 0 <= n;
  {
    if n == 0 then

   |
12 |     if n == 0 then
   |               ^

ex.dfy(12,14): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['      assert 0 < Factorial(0);\n', '      assert 1 > 0;\n', '      assert Factorial(n) == 1;\n', '      assert Factorial(n) == 1;\n', '      assert 0 < Factorial(0);\n', '      return;\n', '      Factorial(n) == 1\n', '      // Base case: 0 is less than 1, so the property holds.\n', '      return;\n', '      assert Factorial(n) == 1 > 0;\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  var n :| 0 <= n;
  {
    if n == 0 then
      Factorial(n) == 1

   |
12 |     if n == 0 then
   |               ^

ex.dfy(12,14): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    else\n', '    else\n', '    else\n', '    else\n', '      decreases 0;\n', '      decreases 1\n', '      decreases 1 - n;\n', '    else\n', '      ensures Factorial(n) == 1\n', '      decreases 1\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  var n :| 0 <= n;
  {
    if n == 0 then
      Factorial(n) == 1
      decreases 1

   |
12 |     if n == 0 then
   |               ^

ex.dfy(12,14): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
-1.0
['{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{

  |
9 | {
  |  ^

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

SCORE
None
['  if n == 0 then\n', '  var i :| 0 <= i;\n', '  var n: nat :| 0 <= n;\n', '  var m: nat := n;\n', '  var n: nat :| 0 <= n;\n', '  var n :| 0 <= n;\n', '  var n :| 0 <= n;\n', '}\n', '  if n == 0 {\n', '  if n == 0 then 1 else n * Factorial(n - 1)\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  if n == 0 {

   |
10 |   if n == 0 {
   |              ^

ex.dfy(10,13): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    // base case\n', "    // factorial of 0 is 1, so it's positive.\n", '    assert 1 >= 0;\n', '    // base case\n', '    assert Factorial(0) > 0;\n', '    assert 0 < Factorial(0);\n', '    assert Factorial(0) > 0;\n', '    assert 1 > 0;\n', '    assert 0 < Factorial(0);\n', '    // base case\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  if n == 0 {
    assert 0 < Factorial(0);

   |
11 |     assert 0 < Factorial(0);
   |                             ^

ex.dfy(11,28): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    assert 0 < Factorial(1);\n', '  } else {\n', '    assert 0 < 1;\n', '    assert 0 < 1;\n', '  } else {\n', '  } else {\n', '  } else {\n', '  } else {\n', '  } else {\n', '    assert 0 < Factorial(0);\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  if n == 0 {
    assert 0 < Factorial(0);
  } else {

   |
12 |   } else {
   |           ^

ex.dfy(12,10): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    calc {\n', '    assert 0 < Factorial(n - 1);\n', '    assert Factorial(n) > 0;\n', '    assert 0 < Factorial(n - 1);\n', '    assert 0 < Factorial(n);\n', '    assert 0 < Factorial(n - 1);\n', '    assert Factorial(n - 1) < Factorial(n);\n', '    assert n > 0 ==> n * Factorial(n - 1) > Factorial(n - 1);\n', '    assert 0 < n;\n', '    assert 0 < Factorial(n - 1);\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  if n == 0 {
    assert 0 < Factorial(0);
  } else {
    calc {

   |
13 |     calc {
   |           ^

ex.dfy(13,10): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['      assert 0 < Factorial(n - 1);\n', '      assert n > 0;\n', '      assert n < Factorial(n);\n', '      assert 0 < n;\n', '      assert 0 < Factorial(n - 1);\n', '      assert 0 < Factorial(n);\n', '      assert 0 < 1;\n', '      assert n * Factorial(n - 1) > 0;\n', '      assert 0 < n;\n', '      assert 0 < n;\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  if n == 0 {
    assert 0 < Factorial(0);
  } else {
    calc {
      assert 0 < n;

   |
14 |       assert 0 < n;
   |                    ^

ex.dfy(14,19): Error: invalid UnaryExpression
1 parse errors detected in ex.dfy

SCORE
None
['      assert 0 < Factorial(n - 1);\n', '      assert n * Factorial(n - 1) == Factorial(n);\n', '      assert n * Factorial(n - 1) == Factorial(n);\n', '      assert Factorial(n) == n * Factorial(n - 1);\n', '      assert 0 < Factorial(n - 1);\n', '      assert n * Factorial(n - 1) < Factorial(n);\n', '      assert Factorial(n - 1) < Factorial(n);\n', '      assert 0 < Factorial(n - 1);\n', '      assert Factorial(n - 1) <= Factorial(n - 1);\n', '      assert n * Factorial(n - 1) == Factorial(n);\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  if n == 0 {
    assert 0 < Factorial(0);
  } else {
    calc {
      assert 0 < n;
      assert n * Factorial(n - 1) < Factorial(n);

   |
15 |       assert n * Factorial(n - 1) < Factorial(n);
   |                                                  ^

ex.dfy(15,49): Error: invalid UnaryExpression
1 parse errors detected in ex.dfy

SCORE
None
['      assert Factorial(n - 1) <= Factorial(n);\n', '      assert 0 < Factorial(n);\n', '    }\n', '      assert 0 < Factorial(n);\n', '    }\n', '      assert 0 < Factorial(n);\n', '    }\n', '    }\n', '      assert Factorial(n) == Factorial(n - 1) * n;\n', '      assert Factorial(n) == n * Factorial(n - 1);\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  if n == 0 {
    assert 0 < Factorial(0);
  } else {
    calc {
      assert 0 < n;
      assert n * Factorial(n - 1) < Factorial(n);
    }

   |
16 |     }
   |     ^

ex.dfy(16,4): Error: invalid UnaryExpression
1 parse errors detected in ex.dfy

SCORE
None
['    FactorialAlwaysPositive(n - 1);\n', '    apply ICP;\n', '    FactorialAlwaysPositive(n - 1);\n', '    FactorialAlwaysPositive(n - 1);\n', '    FactorialAlwaysPositive(n - 1);\n', '    FactorialAlwaysPositive(n - 1);\n', '    FactorialAlwaysPositive(n - 1);\n', '    apply FactorialAlwaysPositive;\n', '    apply FactorialAlwaysPositive;\n', '    apply IH;\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  if n == 0 {
    assert 0 < Factorial(0);
  } else {
    calc {
      assert 0 < n;
      assert n * Factorial(n - 1) < Factorial(n);
    }
    apply IH;

   |
16 |     }
   |     ^

ex.dfy(16,4): Error: invalid UnaryExpression
   |
17 |     apply IH;
   |     ^^^^^

ex.dfy(17,4): Error: missing semicolon at end of statement
   |
17 |     apply IH;
   |              ^

ex.dfy(17,13): Error: rbrace expected
3 parse errors detected in ex.dfy

SCORE
None
['  }\n', '  }\n', '  }\n', '  }\n', '  }\n', '  }\n', '  }\n', '  }\n', '  }\n', '  }\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
  ensures forall n :: 0 <= n < Factorial(n)
{
  if n == 0 {
    assert 0 < Factorial(0);
  } else {
    calc {
      assert 0 < n;
      assert n * Factorial(n - 1) < Factorial(n);
    }
    apply IH;
  }

   |
16 |     }
   |     ^

ex.dfy(16,4): Error: invalid UnaryExpression
   |
17 |     apply IH;
   |     ^^^^^

ex.dfy(17,4): Error: missing semicolon at end of statement
   |
18 |   }
   |    ^

ex.dfy(18,3): Error: rbrace expected
3 parse errors detected in ex.dfy

SCORE
-1.0
['{\n', '{\n', '  ensures forall n :: 0 <= n implies Factorial(n) > 0\n', '{\n', '  ensures Factorial(n) > 0\n', '{\n', '{\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
{

  |
8 | {
  |  ^

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

SCORE
None
['  var i: nat := 0;\n', '  assert forall n :: Factorial(n) > 0;\n', '  var n : nat :| 0 <= n;\n', '  var n : nat :| true;\n', '  assert forall n :: 0 <= n < Factorial(n) == 1;\n', '  var n: nat :| 0 <= n < 1000;\n', '  var n := 0;\n', '  var n: nat :| true;\n', '  assert forall n :: Factorial(n) > 0;\n', '  var n := 0;\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
{
  assert forall n :: 0 <= n < Factorial(n) == 1;

  |
9 |   assert forall n :: 0 <= n < Factorial(n) == 1;
  |                                                 ^

ex.dfy(9,48): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
{
  assert forall n :: 0 <= n < Factorial(n) == 1;
}

ex.dfy(9,9): Error: assertion might not hold
  |
9 |   assert forall n :: 0 <= n < Factorial(n) == 1;
  |          ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^


Dafny program verifier finished with 1 verified, 1 error

SCORE
None
['```', '```', '\n', '\n', '```', '```', '\n', '```', '```', '```']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialAlwaysPositive()
{
  assert forall n :: 0 <= n < Factorial(n) == 1;
}
```
ex.dfy(9,9): Error: assertion might not hold
  |
9 |   assert forall n :: 0 <= n < Factorial(n) == 1;
  |          ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^


Dafny program verifier finished with 1 verified, 1 error

SCORE
-1.0
['lemma FactorialIsPositive(n: nat)\n', 'lemma FactorialIsAlwaysStrictlyPositive(n: nat)\n', 'lemma FactorialAlwaysPositive(n: nat)\n', 'lemma Positivity(n: nat)\n', 'lemma FactorialIsStrictlyPositive(n: nat)\n', 'lemma FactorialIsStrictlyPositive(n: nat)\n', 'lemma FactorialIsAlwaysPositive(n: nat)\n', 'lemma AlwaysStrictlyPositive(n: nat)\n', 'method Main() {\n', 'lemma FactorialIsStrictlyPositive(n: nat)\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)

SCORE
1.0
['  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0

SCORE
1.0
['{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{

  |
9 | {
  |  ^

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

SCORE
None
['  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{
  if n == 0 then {

   |
10 |   if n == 0 then {
   |             ^

ex.dfy(10,12): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    // Base case\n', '    // base case: factorial of 0 is 1, which is positive\n', '    // Factorial(0) = 1 > 0\n', '    Factorial(0);\n', '    Factorial(0);\n', '    Factorial(n) == 1;\n', '    assert 1 > 0;\n', '    assert Factorial(0) > 0;\n', '    // Base case: Factorial(0) = 1 > 0\n', '    assert Factorial(n) == 1 > 0;\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{
  if n == 0 then {
    // Base case

   |
10 |   if n == 0 then {
   |             ^

ex.dfy(10,12): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    assert Factorial(0) == 1 > 0;\n', '    assert Factorial(0) == 1;\n', '    assert Factorial(0) > 0;\n', '    assert Factorial(0) == 1;\n', '    assert 1 > 0;\n', '    assert Factorial(0) > 0;\n', '    assert Factorial(0) > 0;\n', '    assert Factorial(0) == 1;\n', '    assert Factorial(0) > 0;\n', '    assert Factorial(0) == 1;\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{
  if n == 0 then {
    // Base case
    assert Factorial(0) == 1;

   |
10 |   if n == 0 then {
   |             ^

ex.dfy(10,12): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
-1.0
['{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{

  |
9 | {
  |  ^

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

SCORE
None
['  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n > 0 then \n', '  if n == 0 then {\n', '  if n == 0 then {\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 then 

   |
10 |   if n > 0 then
   |            ^

ex.dfy(10,11): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    calc {\n', '    {\n', '    calc {\n', '    Factorial(n) == n * Factorial(n - 1);\n', '    calc {\n', '    calc {\n', '    assert Factorial(n - 1) > 0; \n', '    calc {\n', '    calc {\n', '    Factorial(n) > 0\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 then 
    assert Factorial(n - 1) > 0; 

   |
10 |   if n > 0 then 
   |            ^

ex.dfy(10,11): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    assert Factorial(n) == n * Factorial(n - 1);\n', '    assert Factorial(n) > Factorial(n - 1);\n', '    assert Factorial(n - 1) * n > 0; \n', '    assert n > 0; \n', '    assert Factorial(n) == n * Factorial(n - 1); \n', '    calc {\n', '    // induction hypothesis\n', '    // Inductive hypothesis\n', '    assert n > 0 ==> n * Factorial(n - 1) > 0; \n', '    assert n > 0; \n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 then 
    assert Factorial(n - 1) > 0; 
    assert Factorial(n) == n * Factorial(n - 1);

   |
10 |   if n > 0 then 
   |            ^

ex.dfy(10,11): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
-1.0
['  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0

SCORE
1.0
['{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{

  |
9 | {
  |  ^

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

SCORE
None
['  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then\n', '  if n > 0 then\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n > 0 then\n', '  if n == 0 then {\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 then

   |
10 |   if n > 0 then
   |            ^

ex.dfy(10,11): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    calc {\n', '    Factorial(n) == 1 * Factorial(n - 1)\n', '    calc {\n', '    Factorial(n) == n * Factorial(n - 1)\n', '    calc {\n', '    Factorial(n) = n * Factorial(n - 1);\n', '    calc {\n', '    calc {\n', '    calc {\n', '    calc {\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 then
    Factorial(n) == 1 * Factorial(n - 1)

   |
10 |   if n > 0 then
   |            ^

ex.dfy(10,11): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['  else\n', '    Factorial(n) > 0\n', '    Factorial(n) > 0\n', '  else\n', '    // Write here the proof\n', '  else\n', '  else\n', '    Factorial(n) == 1 + (n - 1) + Factorial(n - 1)\n', '  else\n', '    calc {\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 then
    Factorial(n) == 1 * Factorial(n - 1)
    // Write here the proof

   |
10 |   if n > 0 then
   |            ^

ex.dfy(10,11): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
-1.0
['{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{

  |
9 | {
  |  ^

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

SCORE
None
['  if n == 0 then {\n', '  if n == 0 then\n', '  if n == 0 then {\n', '  if n == 0 then\n', '  if n == 0 then\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{
  if n == 0 then

   |
10 |   if n == 0 then
   |             ^

ex.dfy(10,12): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['  {\n', '    { assert Factorial(0) > 0; }\n', '  {\n', '  {\n', '    Factorial(0) is 1\n', '  {\n', '  {\n', '  {\n', '    Factorial(n) > 0\n', '  {\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{
  if n == 0 then
    Factorial(n) > 0

   |
10 |   if n == 0 then
   |             ^

ex.dfy(10,12): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['  else\n', '  else\n', '  else\n', '  else\n', '  else\n', '  else\n', '  else\n', '  else\n', '  else\n', '  else\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n - 1)
}

lemma FactorialIsAlwaysStrictlyPositive(n: nat)
  ensures Factorial(n) > 0
{
  if n == 0 then
    Factorial(n) > 0
  else

   |
10 |   if n == 0 then
   |             ^

ex.dfy(10,12): Error: lbrace expected
1 parse errors detected in ex.dfy

SCORE
-1.0
['{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{

  |
3 | {
  |  ^

ex.dfy(3,1): Error: invalid UnaryExpression
1 parse errors detected in ex.dfy

SCORE
None
['  if n == 0 then 1 else n * Factorial(n-1)\n', '  if n == 0 then 1 else n * Factorial(n-1)\n', '  if n == 0 then 1 else n * Factorial(n-1)\n', '  if n == 0 then 1 else n * Factorial(n-1)\n', '  if n == 0 then 1 else n * Factorial(n - 1)\n', '  if n == 0 then 1 else n * Factorial(n-1)\n', '  if n == 0 then 1 else n * Factorial(n - 1)\n', '  if n == 0 then 1 else n * Factorial(n - 1)\n', '  if n == 0 then 1 else n * Factorial(n-1)\n', '  if n == 0 then 1 else n * Factorial(n-1)\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)

  |
4 |   if n == 0 then 1 else n * Factorial(n-1)
  |                                           ^

ex.dfy(4,42): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

SCORE
1.0
['\n', '\n', '```', '\n', '\n', '\n', '\n', '\n', '\n', '\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}


SCORE
1.0
['lemma FactorialIsAlwaysPositive(n: nat)\n', 'lemma FactorialIsStrictlyPositive(n: nat)\n', 'lemma FactorialIsPositive(n: nat)\n', 'lemma PositiveFactorial(n: nat)\n', 'lemma FactorialIsAlwaysPositive(n: nat)\n', 'lemma FactorialIsStrictlyPositive(n: nat)\n', 'lemma FactorialIsPositive(n: nat)\n', 'lemma FactorialIsAlwaysPositive(n: nat)\n', 'lemma AlwaysPositiveFactorial(n: nat)\n', 'lemma FactorialIsStrictlyPositive(n: nat)\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)

SCORE
1.0
['  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n', '  ensures Factorial(n) > 0\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)
  ensures Factorial(n) > 0

SCORE
1.0
['{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n', '{\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)
  ensures Factorial(n) > 0
{

  |
9 | {
  |  ^

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

SCORE
None
['  if n == 0 then {\n', '  if n > 0 {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n', '  if n == 0 then {\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 {

   |
10 |   if n > 0 {
   |             ^

ex.dfy(10,12): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    assert Factorial(n-1) > 0;\n', '    assert Factorial(n) == n * Factorial(n-1);\n', '    assert Factorial(n) == n * Factorial(n-1);\n', '    assert Factorial(n) == n * Factorial(n-1);\n', '    calc {\n', '    assert Factorial(n) > 0;\n', '    assert Factorial(n) == n * Factorial(n-1);\n', '    assert Factorial(n) > 0;\n', '    assert Factorial(n) > 0;\n', '    assert Factorial(n) > 0;\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 {
    assert Factorial(n-1) > 0;

   |
11 |     assert Factorial(n-1) > 0;
   |                               ^

ex.dfy(11,30): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    calc {\n', '    calc {\n', '    assert Factorial(n) == n * Factorial(n-1);\n', '    assert Factorial(n) == n * Factorial(n-1);\n', '    calc {\n', '    assert Factorial(n) == n * Factorial(n-1);\n', '    assert Factorial(n) == n * Factorial(n-1);\n', '    assert Factorial(n) == n * Factorial(n-1);\n', '    assert Factorial(n) == n * Factorial(n-1);\n', '    assert Factorial(n) == n * Factorial(n-1);\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 {
    assert Factorial(n-1) > 0;
    assert Factorial(n) == n * Factorial(n-1);

   |
12 |     assert Factorial(n) == n * Factorial(n-1);
   |                                               ^

ex.dfy(12,46): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['  }\n', '    assert Factorial(n) > 0;\n', '    assert n * Factorial(n-1) > 0;\n', '    assert Factorial(n) > 0;\n', '    assert Factorial(n) > n;\n', '  } else {\n', '    assert Factorial(n) > 0;\n', '  }\n', '  }\n', '    calc {\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 {
    assert Factorial(n-1) > 0;
    assert Factorial(n) == n * Factorial(n-1);
    assert Factorial(n) > 0;

   |
13 |     assert Factorial(n) > 0;
   |                             ^

ex.dfy(13,28): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['  } else {\n', '  }\n', '    // prove the rest\n', '  } else {\n', '  } else {\n', '  } else {\n', '    // Now combine the above facts to prove that Factorial(n) > 0\n', '    calc {\n', '    // case n=0 is true, no need to add the case\n', '  } else {\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 {
    assert Factorial(n-1) > 0;
    assert Factorial(n) == n * Factorial(n-1);
    assert Factorial(n) > 0;
  }

   |
14 |   }
   |    ^

ex.dfy(14,3): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['}\n', '  if n == 0 {\n', '}\n', '}\n', '  else {\n', '  if n == 0 {\n', '  else {\n', '}\n', '  else {\n', '}\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 {
    assert Factorial(n-1) > 0;
    assert Factorial(n) == n * Factorial(n-1);
    assert Factorial(n) > 0;
  }
  else {

   |
15 |   else {
   |         ^

ex.dfy(15,8): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    assert Factorial(n) > 0;\n', '    assert Factorial(n) == 1 > 0;\n', '    assert Factorial(n) == 1;\n', '    assert Factorial(n) == 1 > 0;\n', '    assert Factorial(0) == 1;\n', '    assert Factorial(n) == 1;\n', '    assert Factorial(n) == 1;\n', '    assert Factorial(0) > 0;\n', '    assert Factorial(n) == 1 > 0;\n', '    assert Factorial(n) == 1;\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 {
    assert Factorial(n-1) > 0;
    assert Factorial(n) == n * Factorial(n-1);
    assert Factorial(n) > 0;
  }
  else {
    assert Factorial(n) == 1;

   |
16 |     assert Factorial(n) == 1;
   |                              ^

ex.dfy(16,29): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    assert Factorial(n) > 0;\n', '  }\n', '  }\n', '    assert Factorial(n) > 0;\n', '    assert Factorial(n) > 0;\n', '    assert Factorial(n) > 0;\n', '    assert Factorial(n) > 0;\n', '  }\n', '  }\n', '    assert Factorial(n) > 0;\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 {
    assert Factorial(n-1) > 0;
    assert Factorial(n) == n * Factorial(n-1);
    assert Factorial(n) > 0;
  }
  else {
    assert Factorial(n) == 1;
  }

   |
17 |   }
   |    ^

ex.dfy(17,3): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n', '}\n']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 {
    assert Factorial(n-1) > 0;
    assert Factorial(n) == n * Factorial(n-1);
    assert Factorial(n) > 0;
  }
  else {
    assert Factorial(n) == 1;
  }
}

SCORE
1.0
['```', '```', '```', '\n', '```', '```', '```', '```', '```', '```']
TEXT
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 {
    assert Factorial(n-1) > 0;
    assert Factorial(n) == n * Factorial(n-1);
    assert Factorial(n) > 0;
  }
  else {
    assert Factorial(n) == 1;
  }
}
```
SCORE
1.0
CHOSEN SOLUTION
### Spec: In Dafny, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Use a plain function, NOT a function method.
### Hint: Use a nat, NOT an int.

```dafny
function Factorial(n: nat): nat
  decreases n
{
  if n == 0 then 1 else n * Factorial(n-1)
}

lemma PositiveFactorial(n: nat)
  ensures Factorial(n) > 0
{
  if n > 0 {
    assert Factorial(n-1) > 0;
    assert Factorial(n) == n * Factorial(n-1);
    assert Factorial(n) > 0;
  }
  else {
    assert Factorial(n) == 1;
  }
}
```

STATS

number of nodes: 35
number of gen nodes: 18 (including leaves: 5)
number of back nodes: 17 (including leaves: 13)

expansion count: 27 (including failed: 10)