dafny_fact_run_diversity_alt.txt

[2023-12-06 21:54:25,533] [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(

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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(
You are using the default legacy behaviour of the <class 'transformers.models.llama.tokenization_llama.LlamaTokenizer'>. This is expected, and simply means that the `legacy` (previous) behavior will be used so nothing changes for you. If you want to use the new behaviour, set `legacy=False`. This should only be set if you understand what it means, and thouroughly read the reason why this was added as explained in https://github.com/huggingface/transformers/pull/24565
['function method Factorial(n: int) returns (r: int)\n', 'function factorial(n: nat): nat\n', 'function method Factorial(n: nat): nat {\n', 'function factorial(n: int): int\n', 'function Factorial(n: nat): nat {\n', 'function factorial(n: int): int {\n', 'function factorial(n: int): int\n', 'function factorial(n: int): int \n', 'function factorial(n: nat): nat {\n', 'function method Factorial(n: int): int\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.

```dafny
function method Factorial(n: int) returns (r: int)

  |
1 | function method Factorial(n: int) returns (r: int)
  |                                   ^

ex.dfy(1,34): Error: colon expected
  |
1 | function method Factorial(n: int) returns (r: int)
  | ^^^^^^^^

ex.dfy(1,0): Error: the phrase 'function method' is not allowed when using --function-syntax:4; to declare a compiled function, use just 'function'
2 parse errors detected in ex.dfy

SCORE
None
['    requires n >= 0\n', '    requires n >= 0\n', '{\n', '{\n', '    ensures r >= 0\n', '{\n', '    requires n >= 0\n', '{\n', '{\n', '    requires 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.

```dafny
function method Factorial(n: int) returns (r: int)
    ensures r >= 0

  |
1 | function method Factorial(n: int) returns (r: int)
  |                                   ^

ex.dfy(1,34): Error: colon expected
  |
1 | function method Factorial(n: int) returns (r: int)
  | ^^^^^^^^

ex.dfy(1,0): Error: the phrase 'function method' is not allowed when using --function-syntax:4; to declare a compiled function, use just 'function'
2 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.

```dafny
function method Factorial(n: int) returns (r: int)
    ensures r >= 0
{

  |
1 | function method Factorial(n: int) returns (r: int)
  |                                   ^

ex.dfy(1,34): Error: colon expected
  |
3 | {
  |  ^

ex.dfy(3,1): Error: invalid UnaryExpression
  |
1 | function method Factorial(n: int) returns (r: int)
  | ^^^^^^^^

ex.dfy(1,0): Error: the phrase 'function method' is not allowed when using --function-syntax:4; to declare a compiled function, use just 'function'
3 parse errors detected in ex.dfy

SCORE
-1.0
['function Factorial(n: int): int\n', 'function factorial(n: int): int \n', 'function factorial(n: nat): nat {\n', 'function factorial(n: nat): nat\n', 'function factorial(n: int) returns (res: int)\n', 'function factorial(n: int): int\n', 'function factorial(n: nat): nat {\n', 'function factorial(n: int): int\n', 'function Factorial(n: nat): nat\n', 'function factorial(n: int) returns (res: int)\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.

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

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.

```dafny
function factorial(n: nat): nat
{

  |
2 | {
  |  ^

ex.dfy(2,1): 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.

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

  |
3 |   if n == 0 then 1 else n * factorial(n - 1)
  |                                             ^

ex.dfy(3,44): 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.

```dafny
function factorial(n: nat): nat
{
  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.

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


SCORE
1.0
['lemma FactorialAlwaysPositive(n: nat)\n', 'lemma FactorialIsPositive(n: nat)\n', 'lemma {:induction} FactorialPositive(n: nat) \n', 'lemma {:induction} "factorial(n) > 0"\n', 'lemma {:induction n} "FactorialPositive"(n: nat)\n', 'lemma {:induction n} FactorialPositive(n: nat) \n', 'lemma FactorialAlwaysPositive(n: nat)\n', 'lemma {:induction} "FactorialPositive(n: nat)"\n', 'lemma FactorialPositive(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.

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

lemma {:induction} FactorialPositive(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', '  requires 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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0

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) >= 1\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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures 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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{

  |
9 | {
  |  ^

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

SCORE
None
['  if n == 0 {\n', '  if n == 0 {\n', '  if n == 0 {\n', '  if n == 0 {\n', '  if n == 0 {\n', '  if n == 0 {\n', '  if n == 0 {\n', '  if n == 0 {\n', '  if n == 0 {\n', '  if 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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  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', '  } else {\n', '  } else {\n', '    // base case\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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  if n == 0 {
    // base case

   |
11 |     // base case
   |                 ^

ex.dfy(11,16): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['  } else {\n', '    assert factorial(0) == 1;\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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  if n == 0 {
    // base case
  } else {

   |
12 |   } else {
   |           ^

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

SCORE
None
['    // induction step\n', '    // inductive step\n', '    FactorialPositive(n - 1);\n', '    // inductive step\n', '    FactorialPositive(n - 1);\n', '    // induction step\n', '    FactorialPositive(n - 1);\n', '    FactorialPositive(n - 1);\n', '    FactorialPositive(n - 1);\n', '    FactorialPositive(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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  if n == 0 {
    // base case
  } else {
    // inductive step

   |
13 |     // inductive step
   |                      ^

ex.dfy(13,21): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    FactorialPositive(n - 1);\n', '    FactorialPositive(n - 1);\n', '    FactorialPositive(n - 1);\n', '    FactorialPositive(n - 1);\n', '    FactorialPositive(n - 1);\n', '    FactorialPositive(n - 1);\n', '    FactorialPositive(n - 1);\n', '    FactorialPositive(n - 1);\n', '    FactorialPositive(n - 1);\n', '    FactorialPositive(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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  if n == 0 {
    // base case
  } else {
    // inductive step
    FactorialPositive(n - 1);

   |
14 |     FactorialPositive(n - 1);
   |                              ^

ex.dfy(14,29): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    // assertion\n', '    assert factorial(n - 1) >= 1;\n', '    assert factorial(n - 1) >= 1;\n', '    // assertion that proves factorial(n) is positive\n', '  }\n', '    assert factorial(n - 1) >= 1;\n', '    assert factorial(n - 1) >= 1;\n', '    assert factorial(n - 1) >= 1;\n', '    assert factorial(n - 1) >= 1;\n', '    assert factorial(n - 1) >= 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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  if n == 0 {
    // base case
  } else {
    // inductive step
    FactorialPositive(n - 1);
    // assertion

   |
15 |     // assertion
   |                 ^

ex.dfy(15,16): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    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', '    assert factorial(n) == n * factorial(n - 1);\n', '    assert factorial(n - 1) >= 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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  if n == 0 {
    // base case
  } else {
    // inductive step
    FactorialPositive(n - 1);
    // assertion
    assert factorial(n) == n * factorial(n - 1);

   |
16 |     assert factorial(n) == n * factorial(n - 1);
   |                                                 ^

ex.dfy(16,48): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    assert factorial(n - 1) >= 1;\n', '    assert factorial(n - 1) >= 1; // from inductive step\n', '    assert factorial(n - 1) >= 1;\n', '  }\n', '  }\n', '    assert n * factorial(n - 1) > 0;\n', '  }\n', '    assert factorial(n - 1) >= 1; // inductive hypothesis\n', '    assert factorial(n - 1) >= 1;\n', '    assert factorial(n - 1) >= 1; // by induction hypothesis\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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  if n == 0 {
    // base case
  } else {
    // inductive step
    FactorialPositive(n - 1);
    // assertion
    assert factorial(n) == n * factorial(n - 1);
    assert factorial(n - 1) >= 1; // from inductive step

   |
17 |     assert factorial(n - 1) >= 1; // from inductive step
   |                                                         ^

ex.dfy(17,56): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
['    assert n * factorial(n - 1) > 0; // since both n and factorial(n - 1) >= 1\n', '    assert n * factorial(n - 1) >= 1; // since n > 0 and factorial(n - 1) >= 1\n', '    assert n * factorial(n - 1) >= 1 * factorial(n - 1);\n', '    assert n * factorial(n - 1) >= 1 * factorial(n - 1);\n', '    assert n * factorial(n - 1) > 0; // because n > 0 and factorial(n - 1) >= 1\n', '    assert n * factorial(n - 1) > 0; // because both n and factorial(n - 1) >= 1\n', '    assert n * factorial(n - 1) >= 1 * factorial(n - 1);\n', '    assert n * factorial(n - 1) >= 1 * factorial(n - 1);\n', '    assert n * factorial(n - 1) >= 1;\n', '    assert n * factorial(n - 1) >= 1 * factorial(n - 1); // from definition of multiplication\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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  if n == 0 {
    // base case
  } else {
    // inductive step
    FactorialPositive(n - 1);
    // assertion
    assert factorial(n) == n * factorial(n - 1);
    assert factorial(n - 1) >= 1; // from inductive step
    assert n * factorial(n - 1) >= 1 * factorial(n - 1); // from definition of multiplication

   |
18 |     assert n * factorial(n - 1) >= 1 * factorial(n - 1); // from definition of multiplication
   |                                                                                              ^

ex.dfy(18,93): Error: rbrace expected
1 parse errors detected in ex.dfy

SCORE
None
["    assert n * factorial(n - 1) >= 1; // from the lemma's inductive hypothesis\n", '  }\n', '    assert n * factorial(n - 1) >= factorial(n - 1); // cancel the 1\n', '    assert n * factorial(n - 1) >= 1; // from above line and the fact that multiplication is monotonic\n', '    assert n * factorial(n - 1) >= 1; // from our goal\n', '    assert n * factorial(n - 1) >= 1; // factorial(n - 1) >= 1\n', '    assert n * factorial(n - 1) >= factorial(n - 1); // from previous line\n', '    assert n * factorial(n - 1) >= 1;\n', '    assert n * factorial(n - 1) >= 1;\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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  if n == 0 {
    // base case
  } else {
    // inductive step
    FactorialPositive(n - 1);
    // assertion
    assert factorial(n) == n * factorial(n - 1);
    assert factorial(n - 1) >= 1; // from inductive step
    assert n * factorial(n - 1) >= 1 * factorial(n - 1); // from definition of multiplication
    assert n * factorial(n - 1) >= 1; // factorial(n - 1) >= 1

   |
19 |     assert n * factorial(n - 1) >= 1; // factorial(n - 1) >= 1
   |                                                               ^

ex.dfy(19,62): 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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  if n == 0 {
    // base case
  } else {
    // inductive step
    FactorialPositive(n - 1);
    // assertion
    assert factorial(n) == n * factorial(n - 1);
    assert factorial(n - 1) >= 1; // from inductive step
    assert n * factorial(n - 1) >= 1 * factorial(n - 1); // from definition of multiplication
    assert n * factorial(n - 1) >= 1; // factorial(n - 1) >= 1
  }

   |
20 |   }
   |    ^

ex.dfy(20,3): 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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  if n == 0 {
    // base case
  } else {
    // inductive step
    FactorialPositive(n - 1);
    // assertion
    assert factorial(n) == n * factorial(n - 1);
    assert factorial(n - 1) >= 1; // from inductive step
    assert n * factorial(n - 1) >= 1 * factorial(n - 1); // from definition of multiplication
    assert n * factorial(n - 1) >= 1; // factorial(n - 1) >= 1
  }
}

SCORE
1.0
['``` #pragma once\n', '``` <?php\n', '``` #include "../include/stdio.h"\n', '```\n', '```\n', '``` \n', '\n', '``` package nablarch.fw.web.handler.health;\n', '```\n', '``` //go:build go1.18\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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  if n == 0 {
    // base case
  } else {
    // inductive step
    FactorialPositive(n - 1);
    // assertion
    assert factorial(n) == n * factorial(n - 1);
    assert factorial(n - 1) >= 1; // from inductive step
    assert n * factorial(n - 1) >= 1 * factorial(n - 1); // from definition of multiplication
    assert n * factorial(n - 1) >= 1; // factorial(n - 1) >= 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.

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

lemma {:induction} FactorialPositive(n: nat) 
  requires n >= 0
  ensures factorial(n) >= 1
{
  if n == 0 {
    // base case
  } else {
    // inductive step
    FactorialPositive(n - 1);
    // assertion
    assert factorial(n) == n * factorial(n - 1);
    assert factorial(n - 1) >= 1; // from inductive step
    assert n * factorial(n - 1) >= 1 * factorial(n - 1); // from definition of multiplication
    assert n * factorial(n - 1) >= 1; // factorial(n - 1) >= 1
  }
}
```


STATS

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

expansion count: 9 (including failed: 1)