bench_coq_factorial_run_hints.txt

========
Experiment configuration: python experiments.py --experiment_name run.py --n_trials 10 --mins_timeout 10 --language Coq --problem_name problem_fact --seed 42 


To complete all experiments, 0 minutes and 0 seconds elapsed.




========
Experiment configuration: python experiments.py --experiment_name run.py --n_trials 10 --mins_timeout 10 --language Coq --problem_name problem_fact --seed 42 


[2024-01-13 02:43:49,658] [INFO] [real_accelerator.py:158:get_accelerator] Setting ds_accelerator to cuda (auto detect)
Trial  1
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.
SCORE
1.0
['From Coq Require Import Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\n\nFixpoint factorial (n:nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n\nTheorem factorial_pos : forall n, 0 < factorial n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\nProof.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - apply N0_lt_1.
File "./ex.v", line 14, characters 10-17:
Error: The variable N0_lt_1 was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - apply N0_lt_1."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - apply N0_lt_1.
  - apply Nat.
File "./ex.v", line 14, characters 10-17:
Error: The variable N0_lt_1 was not found in the current environment.


SCORE
-1.0
['\n  - apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - apply Nat.
File "./ex.v", line 14, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - apply Nat.lt_0_1.
  - apply Nat.
File "./ex.v", line 15, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n  - apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - apply Nat.lt_0_1.
  - apply Nat.lt_lt_add_r with (1 * factorial n') as (1 + n' * factorial n').
    + apply Nat.
File "./ex.v", line 15, characters 50-52:
Error:
Syntax error: [ltac_use_default] expected after [tactic] (in [tactic_command]).


SCORE
-1.0
["lt_lt_add_r with (1 * factorial n') as (1 + n' * factorial n').\n    + apply Nat."]
['\n  - apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - apply Nat.lt_0_1.
  - apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - apply Nat.lt_0_1.
  - apply Nat.lt_lt_add_r.
    apply IHn'.
    apply Nat.
File "./ex.v", line 17, characters 4-14:
Error: No such goal.


SCORE
-1.0
["\n    apply IHn'.\n    apply Nat."]
["\n    apply IHn'.\n    apply Nat."]
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    now apply Nat.
File "./ex.v", line 15, characters 4-17:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - (* base case *)\n    now apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    now apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    now apply Nat.lt_0_1.
  - (* inductive case *)
    now apply Nat.
File "./ex.v", line 17, characters 4-17:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n  - (* inductive case *)\n    now apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    now apply Nat.lt_0_1.
  - (* inductive case *)
    now apply Nat.lt_lt_add_r; apply IHn'; ring.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
["lt_lt_add_r; apply IHn'; ring."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    now apply Nat.lt_0_1.
  - (* inductive case *)
    now apply Nat.lt_lt_add_r; apply IHn'; ring.
Qed.
SCORE
1.0
['\nQed.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    now apply Nat.lt_0_1.
  - (* inductive case *)
    now apply Nat.lt_lt_add_r; apply IHn'; ring.
Qed.
```

This Coq script defines a factorial function on natural numbers and proves that the factorial is always strictly positive.
SCORE
1.0
['\n```\n\nThis Coq script defines a factorial function on natural numbers and proves that the factorial is always strictly positive.']
CHOSEN SOLUTION
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    now apply Nat.lt_0_1.
  - (* inductive case *)
    now apply Nat.lt_lt_add_r; apply IHn'; ring.
Qed.
```

This Coq script defines a factorial function on natural numbers and proves that the factorial is always strictly positive.

STATS

number of nodes: 25
number of gen nodes: 13 (including leaves: 2)
number of back nodes: 12 (including leaves: 11)

expansion count: 16 (including failed: 4)

cache stats {'hit': 3, 'miss': 19}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run.py for 10 trials

Trial times: [38]

Trial calls: [22]


Time Statistics:

  Min: 38.00 sec, Max: 38.00 sec, Average: 38.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 22 calls, Max: 22 calls, Average: 22.00 calls, Standard Deviation: 0.00 calls

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

Experiment: run.py for 10 trials

Trial times: [38]

Trial calls: [22]


Time Statistics:

  Min: 38.00 sec, Max: 38.00 sec, Average: 38.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 22 calls, Max: 22 calls, Average: 22.00 calls, Standard Deviation: 0.00 calls

Trial  2
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.
File "./ex.v", line 2, characters 0-19:
Error: Cannot find a physical path bound to logical path Coq.


SCORE
None
['Require Import Coq.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.
File "./ex.v", line 2, characters 0-25:
Error: Cannot find a physical path bound to logical path Coq.Arith.


SCORE
None
['Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
SCORE
1.0
['Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.
File "./ex.v", line 3, characters 0-19:
Error: Cannot find a physical path bound to logical path Coq.


SCORE
None
['\nRequire Import Coq.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.
File "./ex.v", line 3, characters 0-25:
Error: Cannot find a physical path bound to logical path Coq.Lists.


SCORE
None
['Lists.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
SCORE
1.0
['List.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.
SCORE
1.0
['\nImport ListNotations.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\n\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n\nTheorem factorial_pos : forall n, 0 < factorial n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\nProof.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now constructor.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
["\n  induction n as [|n' IHn'].\n  - now constructor."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now constructor.
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  - simpl.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now constructor.
  - simpl.
    rewrite IHn'.
    apply Nat.
File "./ex.v", line 19, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n    rewrite IHn'.\n    apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now constructor.
  - simpl.
    rewrite IHn'.
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now constructor.
  - simpl.
    rewrite IHn'.
    apply Nat.lt_lt_add_r.
Qed.
File "./ex.v", line 20, characters 0-4:
Error:  (in proof factorial_pos): Attempt to save an incomplete proof


SCORE
-1.0
['\nQed.']
['\nQed.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now constructor.
  - simpl.
    rewrite IHn'.
    now apply Nat.
File "./ex.v", line 19, characters 4-17:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n    rewrite IHn'.\n    now apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now constructor.
  - simpl.
    rewrite IHn'.
    now apply Nat.lt_lt_add_r.
File "./ex.v", line 19, characters 4-29:
Error: Tactic failure: Cannot solve this goal.


SCORE
-1.0
['lt_lt_add_r.']
['\n  - simpl.']
["\n    rewrite IHn'.\n    apply Nat."]
['lt_lt_add_r.']
['\nQed.']
['\nQed.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now constructor.
  - simpl.
    rewrite IHn'.
    constructor.
File "./ex.v", line 19, characters 4-15:
Error: In environment
n' : nat
IHn' : 0 < factorial n'
Unable to unify "?M1098 <= S ?M1099" with
 "S (factorial n') <= factorial n' + n' * factorial n'".


SCORE
-1.0
["\n    rewrite IHn'.\n    constructor."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now apply Nat.
File "./ex.v", line 16, characters 4-17:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - now apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now apply Nat.lt_0_1.
  - now apply Nat.
File "./ex.v", line 17, characters 4-17:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n  - now apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now apply Nat.lt_0_1.
  - now apply Nat.lt_lt_add_r with (n * factorial n') as Hn.
File "./ex.v", line 17, characters 54-56:
Error:
Syntax error: [ltac_use_default] expected after [tactic] (in [tactic_command]).


SCORE
-1.0
["lt_lt_add_r with (n * factorial n') as Hn."]
['\nQed.']
['\n  intros n.']
["\n  induction n as [|n' IHn'].\n  - now apply Nat."]
['lt_0_1.']
['\n  - now apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now apply Nat.lt_0_1.
  - now apply Nat.lt_lt_add_r with (n * factorial n') and 1.
File "./ex.v", line 17, characters 4-59:
Error: The variable n was not found in the current environment.


SCORE
None
["lt_lt_add_r with (n * factorial n') and 1."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now apply Nat.lt_0_1.
  - now apply Nat.lt_lt_add_r with (n * factorial n') and 1.
    apply IHn'.
Qed.
File "./ex.v", line 17, characters 4-59:
Error: The variable n was not found in the current environment.


SCORE
-1.0
["\n    apply IHn'.\nQed."]
['\n  - now apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now apply Nat.lt_0_1.
  - now apply Nat.lt_lt_add_r with (n' * factorial n') as H and 1 as a.
File "./ex.v", line 17, characters 55-57:
Error:
Syntax error: [ltac_use_default] expected after [tactic] (in [tactic_command]).


SCORE
-1.0
["lt_lt_add_r with (n' * factorial n') as H and 1 as a."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now apply Nat.lt_0_1.
  - apply Nat.
File "./ex.v", line 17, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n  - apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now apply Nat.lt_0_1.
  - apply Nat.lt_lt_add_r with (factorial n') as H.
File "./ex.v", line 17, characters 46-48:
Error:
Syntax error: [ltac_use_default] expected after [tactic] (in [tactic_command]).


SCORE
-1.0
["lt_lt_add_r with (factorial n') as H."]
['\nProof.']
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now rewrite Nat.
File "./ex.v", line 16, characters 4-19:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - now rewrite Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now rewrite Nat.lt_0_1.
File "./ex.v", line 16, characters 4-26:
Error:
setoid rewrite failed: Unable to satisfy the following constraints:
UNDEFINED EVARS:
 ?X13==[ |- Relation_Definitions.relation nat] (internal placeholder) {?r}
 ?X14==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper (Morphisms.respectful lt ?r) factorial]
         (internal placeholder) {?p}
 ?X15==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper
           (Morphisms.respectful lt
              (Morphisms.respectful ?r (Basics.flip Basics.impl))) lt]
         (internal placeholder) {?p0}
TYPECLASSES:?X13 ?X14 ?X15
SHELF:||
FUTURE GOALS STACK:?X15 ?X14 ?X13||



SCORE
-1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - now rewrite <- Nat.
File "./ex.v", line 16, characters 4-22:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [| n' IHn'].\n  - now rewrite <- Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - now rewrite <- Nat.lt_0_1.
File "./ex.v", line 16, characters 4-29:
Error: Tactic failure: Cannot solve this goal.


SCORE
-1.0
['lt_0_1.']
['\n\nTheorem factorial_pos : forall n, 0 < factorial n.']
['\nProof.']
['\n  intros n.']
["\n  induction n as [|n' IHn'].\n  - now rewrite Nat."]
['lt_0_1.']
["\n  induction n as [|n' IHn'].\n  - now rewrite Nat."]
['lt_0_1.']
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    apply lt_0_1.
File "./ex.v", line 17, characters 10-16:
Error: The variable lt_0_1 was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - (* base case *)\n    apply lt_0_1."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    apply lt_0_1.
  - (* inductive case *)
    apply lt_lt_add_r, IHn', mult_pos_pos.
File "./ex.v", line 17, characters 10-16:
Error: The variable lt_0_1 was not found in the current environment.


SCORE
-1.0
["\n  - (* inductive case *)\n    apply lt_lt_add_r, IHn', mult_pos_pos."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - now rewrite Nat.
File "./ex.v", line 16, characters 4-19:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [| n' IHn'].\n  - now rewrite Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - now rewrite Nat.lt_0_1.
File "./ex.v", line 16, characters 4-26:
Error:
setoid rewrite failed: Unable to satisfy the following constraints:
UNDEFINED EVARS:
 ?X13==[ |- Relation_Definitions.relation nat] (internal placeholder) {?r}
 ?X14==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper (Morphisms.respectful lt ?r) factorial]
         (internal placeholder) {?p}
 ?X15==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper
           (Morphisms.respectful lt
              (Morphisms.respectful ?r (Basics.flip Basics.impl))) lt]
         (internal placeholder) {?p0}
TYPECLASSES:?X13 ?X14 ?X15
SHELF:||
FUTURE GOALS STACK:?X15 ?X14 ?X13||



SCORE
-1.0
['lt_0_1.']
["\n  induction n as [|n' IHn'].\n  - now apply Nat."]
['lt_0_1.']
['\n  - apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now apply Nat.lt_0_1.
  - apply Nat.lt_lt_add_r with (n * factorial n') as n' and 1 as m.
File "./ex.v", line 17, characters 50-52:
Error:
Syntax error: [ltac_use_default] expected after [tactic] (in [tactic_command]).


SCORE
-1.0
["lt_lt_add_r with (n * factorial n') as n' and 1 as m."]
['\n  - apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
Import ListNotations.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now apply Nat.lt_0_1.
  - apply Nat.lt_lt_add_r with (n * factorial n').
    + apply IHn'.
    + apply Nat.
File "./ex.v", line 17, characters 32-33:
Error: The variable n was not found in the current environment.


SCORE
-1.0
["lt_lt_add_r with (n * factorial n').\n    + apply IHn'.\n    + apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Definition factorial (n : nat) :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
File "./ex.v", line 8, characters 16-25:
Error: The reference factorial was not found in the current environment.


SCORE
-1.0
["\n\nDefinition factorial (n : nat) :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
SCORE
1.0
['\nImport Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | O => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\n\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | O => 1\n  | S n' => n * factorial n'\n  end."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | O => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n\nTheorem factorial_positive : forall n, 0 < factorial n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | O => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\nProof.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | O => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | O => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - now rewrite Nat.
File "./ex.v", line 15, characters 4-19:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [| n' IHn'].\n  - now rewrite Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | O => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - now rewrite Nat.lt_0_1.
File "./ex.v", line 15, characters 4-26:
Error:
setoid rewrite failed: Unable to satisfy the following constraints:
UNDEFINED EVARS:
 ?X13==[ |- Relation_Definitions.relation nat] (internal placeholder) {?r}
 ?X14==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper (Morphisms.respectful Peano.lt ?r) factorial]
         (internal placeholder) {?p}
 ?X15==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper
           (Morphisms.respectful Peano.lt
              (Morphisms.respectful ?r (Basics.flip Basics.impl))) Peano.lt]
         (internal placeholder) {?p0}
TYPECLASSES:?X13 ?X14 ?X15
SHELF:||
FUTURE GOALS STACK:?X15 ?X14 ?X13||



SCORE
-1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | O => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* n = 0 *)
    apply Nat.
File "./ex.v", line 16, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - (* n = 0 *)\n    apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | O => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* n = 0 *)
    apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | O => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* n = 0 *)
    apply Nat.lt_0_1.
  - (* n = S n' *)
    apply Nat.
File "./ex.v", line 18, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  - (* n = S n' *)\n    apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | O => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* n = 0 *)
    apply Nat.lt_0_1.
  - (* n = S n' *)
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | O => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* n = 0 *)
    apply Nat.lt_0_1.
  - (* n = S n' *)
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
SCORE
1.0
["\n    apply IHn'.\nQed."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | O => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* n = 0 *)
    apply Nat.lt_0_1.
  - (* n = S n' *)
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

In this Coq script, we first define the factorial function using pattern matching on the input `n`. If `n` is 0, we return 1.
SCORE
1.0
['\n```\n\nIn this Coq script, we first define the factorial function using pattern matching on the input `n`. If `n` is 0, we return 1.']
CHOSEN SOLUTION
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | O => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* n = 0 *)
    apply Nat.lt_0_1.
  - (* n = S n' *)
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

In this Coq script, we first define the factorial function using pattern matching on the input `n`. If `n` is 0, we return 1.

STATS

number of nodes: 63
number of gen nodes: 32 (including leaves: 9)
number of back nodes: 31 (including leaves: 23)

expansion count: 52 (including failed: 21)

cache stats {'hit': 26, 'miss': 52}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run.py for 10 trials

Trial times: [38, 118]

Trial calls: [22, 78]


Time Statistics:

  Min: 38.00 sec, Max: 118.00 sec, Average: 78.00 sec, Standard Deviation: 40.00 sec

Call Statistics:

  Min: 22 calls, Max: 78 calls, Average: 50.00 calls, Standard Deviation: 28.00 calls

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

Experiment: run.py for 10 trials

Trial times: [38, 118]

Trial calls: [22, 78]


Time Statistics:

  Min: 38.00 sec, Max: 118.00 sec, Average: 78.00 sec, Standard Deviation: 40.00 sec

Call Statistics:

  Min: 22 calls, Max: 78 calls, Average: 50.00 calls, Standard Deviation: 28.00 calls

Trial  3
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.
File "./ex.v", line 2, characters 0-19:
Error: Cannot find a physical path bound to logical path Coq.


SCORE
None
['Require Import Coq.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.
File "./ex.v", line 2, characters 0-25:
Error: Cannot find a physical path bound to logical path Coq.Arith.


SCORE
None
['Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
SCORE
1.0
['Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
SCORE
1.0
['\nImport Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.
SCORE
1.0
['\nImport PeanoNat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\n\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n\nTheorem factorial_positive : forall n, 0 < factorial n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\nProof.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    now rewrite Nat.
File "./ex.v", line 17, characters 4-19:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - (* base case: n = 0 *)\n    now rewrite Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    now rewrite Nat.lt_0_1.
File "./ex.v", line 17, characters 4-26:
Error:
setoid rewrite failed: Unable to satisfy the following constraints:
UNDEFINED EVARS:
 ?X13==[ |- Relation_Definitions.relation nat] (internal placeholder) {?r}
 ?X14==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper (Morphisms.respectful Peano.lt ?r) factorial]
         (internal placeholder) {?p}
 ?X15==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper
           (Morphisms.respectful Peano.lt
              (Morphisms.respectful ?r (Basics.flip Basics.impl))) Peano.lt]
         (internal placeholder) {?p0}
TYPECLASSES:?X13 ?X14 ?X15
SHELF:||
FUTURE GOALS STACK:?X15 ?X14 ?X13||



SCORE
-1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now constructor.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
["\n  induction n as [|n' IHn'].\n  - now constructor."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now constructor.
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n  - simpl.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now constructor.
  - simpl.
    rewrite IHn'.
    now constructor.
File "./ex.v", line 19, characters 4-19:
Error: In environment
n' : nat
IHn' : 0 < factorial n'
Unable to unify "?M1098 <= S ?M1099" with
 "S (factorial n') <= factorial n' + n' * factorial n'".


SCORE
-1.0
["\n    rewrite IHn'.\n    now constructor."]
["\n    rewrite IHn'.\n    now constructor."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now constructor.
  - now constructor.
File "./ex.v", line 17, characters 4-19:
Error: In environment
n' : nat
IHn' : 0 < factorial n'
Unable to unify "?M1096 <= S ?M1097" with "1 <= factorial (S n')".


SCORE
-1.0
['\n  - now constructor.']
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
["\n  induction n as [| n' IHn'].\n  - simpl."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.
File "./ex.v", line 17, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n  - simpl.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.
File "./ex.v", line 19, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
SCORE
1.0
["\n    apply IHn'.\nQed."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

In this Coq script, we first define a factorial function using a fixpoint.
SCORE
1.0
['\n```\n\nIn this Coq script, we first define a factorial function using a fixpoint.']
CHOSEN SOLUTION
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
Import PeanoNat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

In this Coq script, we first define a factorial function using a fixpoint.

STATS

number of nodes: 33
number of gen nodes: 17 (including leaves: 2)
number of back nodes: 16 (including leaves: 15)

expansion count: 20 (including failed: 4)

cache stats {'hit': 2, 'miss': 23}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run.py for 10 trials

Trial times: [38, 118, 37]

Trial calls: [22, 78, 25]


Time Statistics:

  Min: 37.00 sec, Max: 118.00 sec, Average: 64.33 sec, Standard Deviation: 37.95 sec

Call Statistics:

  Min: 22 calls, Max: 78 calls, Average: 41.67 calls, Standard Deviation: 25.72 calls

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

Experiment: run.py for 10 trials

Trial times: [38, 118, 37]

Trial calls: [22, 78, 25]


Time Statistics:

  Min: 37.00 sec, Max: 118.00 sec, Average: 64.33 sec, Standard Deviation: 37.95 sec

Call Statistics:

  Min: 22 calls, Max: 78 calls, Average: 41.67 calls, Standard Deviation: 25.72 calls

Trial  4
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.
SCORE
1.0
['Require Import Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\n\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n\nTheorem factorial_pos : forall n, 0 < factorial n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\nProof.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now rewrite Nat.
File "./ex.v", line 14, characters 4-19:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - now rewrite Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now rewrite Nat.lt_0_1.
File "./ex.v", line 14, characters 4-26:
Error:
setoid rewrite failed: Unable to satisfy the following constraints:
UNDEFINED EVARS:
 ?X13==[ |- Relation_Definitions.relation nat] (internal placeholder) {?r}
 ?X14==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper (Morphisms.respectful lt ?r) factorial]
         (internal placeholder) {?p}
 ?X15==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper
           (Morphisms.respectful lt
              (Morphisms.respectful ?r (Basics.flip Basics.impl))) lt]
         (internal placeholder) {?p0}
TYPECLASSES:?X13 ?X14 ?X15
SHELF:||
FUTURE GOALS STACK:?X15 ?X14 ?X13||



SCORE
-1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
["\n  induction n as [|n' IHn'].\n  - simpl."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.
File "./ex.v", line 15, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  - simpl.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.
File "./ex.v", line 17, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
SCORE
1.0
["\n    apply IHn'.\nQed."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

In this Coq script, we first define a `factorial` function that computes the factorial of a natural number `n`. This function is defined by recursion on `n`: the base case is `0 => 1`, and the inductive case is `S n' => n * factorial n'`.

Then we prove `factorial_pos`, which states that the factorial of any natural number `n` is strictly positive.
SCORE
1.0
["\n```\n\nIn this Coq script, we first define a `factorial` function that computes the factorial of a natural number `n`. This function is defined by recursion on `n`: the base case is `0 => 1`, and the inductive case is `S n' => n * factorial n'`.\n\nThen we prove `factorial_pos`, which states that the factorial of any natural number `n` is strictly positive."]
CHOSEN SOLUTION
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

In this Coq script, we first define a `factorial` function that computes the factorial of a natural number `n`. This function is defined by recursion on `n`: the base case is `0 => 1`, and the inductive case is `S n' => n * factorial n'`.

Then we prove `factorial_pos`, which states that the factorial of any natural number `n` is strictly positive.

STATS

number of nodes: 23
number of gen nodes: 12 (including leaves: 1)
number of back nodes: 11 (including leaves: 11)

expansion count: 12 (including failed: 1)

cache stats {'hit': 0, 'miss': 15}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31]

Trial calls: [22, 78, 25, 15]


Time Statistics:

  Min: 31.00 sec, Max: 118.00 sec, Average: 56.00 sec, Standard Deviation: 35.90 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 35.00 calls, Standard Deviation: 25.09 calls

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

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31]

Trial calls: [22, 78, 25, 15]


Time Statistics:

  Min: 31.00 sec, Max: 118.00 sec, Average: 56.00 sec, Standard Deviation: 35.90 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 35.00 calls, Standard Deviation: 25.09 calls

Trial  5
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.
SCORE
1.0
['From Coq Require Import Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Definition factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
File "./ex.v", line 7, characters 16-25:
Error: The reference factorial was not found in the current environment.


SCORE
-1.0
["\n\nDefinition factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\n\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n\nTheorem factorial_pos : forall n, 0 < factorial n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\nProof.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
["\n  induction n as [| n' IHn'].\n  - simpl."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.
File "./ex.v", line 15, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  - simpl.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.
File "./ex.v", line 17, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
    apply Nat.
File "./ex.v", line 19, characters 4-14:
Error: No such goal.


SCORE
-1.0
["\n    apply IHn'.\n    apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
SCORE
1.0
["\n    apply IHn'.\nQed."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

This script defines a factorial function in Coq, then proves that the factorial of any natural number is strictly positive.
SCORE
1.0
['\n```\n\nThis script defines a factorial function in Coq, then proves that the factorial of any natural number is strictly positive.']
CHOSEN SOLUTION
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
From Coq Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

This script defines a factorial function in Coq, then proves that the factorial of any natural number is strictly positive.

STATS

number of nodes: 23
number of gen nodes: 12 (including leaves: 1)
number of back nodes: 11 (including leaves: 11)

expansion count: 13 (including failed: 2)

cache stats {'hit': 0, 'miss': 15}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31, 27]

Trial calls: [22, 78, 25, 15, 15]


Time Statistics:

  Min: 27.00 sec, Max: 118.00 sec, Average: 50.20 sec, Standard Deviation: 34.14 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 31.00 calls, Standard Deviation: 23.82 calls

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

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31, 27]

Trial calls: [22, 78, 25, 15, 15]


Time Statistics:

  Min: 27.00 sec, Max: 118.00 sec, Average: 50.20 sec, Standard Deviation: 34.14 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 31.00 calls, Standard Deviation: 23.82 calls

Trial  6
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.
File "./ex.v", line 2, characters 0-19:
Error: Cannot find a physical path bound to logical path Coq.


SCORE
None
['Require Import Coq.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.
File "./ex.v", line 2, characters 0-25:
Error: Cannot find a physical path bound to logical path Coq.Arith.


SCORE
None
['Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
SCORE
1.0
['Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Coq.
File "./ex.v", line 3, characters 7-10:
Error: Cannot find module Coq


SCORE
-1.0
['\nImport Coq.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.
File "./ex.v", line 3, characters 0-19:
Error: Cannot find a physical path bound to logical path Coq.


SCORE
None
['\nRequire Import Coq.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.
File "./ex.v", line 3, characters 0-25:
Error: Cannot find a physical path bound to logical path Coq.Lists.


SCORE
None
['Lists.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.
SCORE
1.0
['List.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\n\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n\nTheorem factorial_pos : forall n, 0 < factorial n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\nProof.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    now apply Nat.
File "./ex.v", line 16, characters 4-17:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - (* base case: n = 0 *)\n    now apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    now apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    now apply Nat.lt_0_1.
  - (* inductive case: n = S n' *)
    apply Nat.
File "./ex.v", line 18, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  - (* inductive case: n = S n' *)\n    apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    now apply Nat.lt_0_1.
  - (* inductive case: n = S n' *)
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    now apply Nat.lt_0_1.
  - (* inductive case: n = S n' *)
    apply Nat.lt_lt_add_r.
    - (* n' < factorial n' *)
      apply IHn'.
    - (* S n' <= S n' *)
      now apply Nat.
File "./ex.v", line 19, characters 4-5:
Error: [Focus] Wrong bullet -: Current bullet - is not finished.


SCORE
-1.0
["\n    - (* n' < factorial n' *)\n      apply IHn'.\n    - (* S n' <= S n' *)\n      now apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    now apply Nat.lt_0_1.
  - (* inductive case: n = S n' *)
    apply Nat.lt_lt_add_r.
    - (* n' < factorial n' *)
      apply IHn'.
    - (* n < S n' *)
      apply Nat.
File "./ex.v", line 19, characters 4-5:
Error: [Focus] Wrong bullet -: Current bullet - is not finished.


SCORE
-1.0
["\n    - (* n' < factorial n' *)\n      apply IHn'.\n    - (* n < S n' *)\n      apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    now apply Nat.lt_0_1.
  - (* inductive case: n = S n' *)
    now apply Nat.
File "./ex.v", line 18, characters 4-17:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  - (* inductive case: n = S n' *)\n    now apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    now apply Nat.lt_0_1.
  - (* inductive case: n = S n' *)
    now apply Nat.lt_lt_add_r with (factorial n') as [|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
File "./ex.v", line 18, characters 50-52:
Error:
Syntax error: [ltac_use_default] expected after [tactic] (in [tactic_command]).


SCORE
-1.0
["lt_lt_add_r with (factorial n') as [|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _"]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now rewrite Nat.
File "./ex.v", line 15, characters 4-19:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - now rewrite Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now rewrite Nat.lt_0_1.
File "./ex.v", line 15, characters 4-26:
Error:
setoid rewrite failed: Unable to satisfy the following constraints:
UNDEFINED EVARS:
 ?X13==[ |- Relation_Definitions.relation nat] (internal placeholder) {?r}
 ?X14==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper (Morphisms.respectful lt ?r) factorial]
         (internal placeholder) {?p}
 ?X15==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper
           (Morphisms.respectful lt
              (Morphisms.respectful ?r (Basics.flip Basics.impl))) lt]
         (internal placeholder) {?p0}
TYPECLASSES:?X13 ?X14 ?X15
SHELF:||
FUTURE GOALS STACK:?X15 ?X14 ?X13||



SCORE
-1.0
['lt_0_1.']
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - apply lt_0_1.
File "./ex.v", line 15, characters 10-16:
Error: The variable lt_0_1 was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - apply lt_0_1."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - apply lt_0_1.
  - simpl.
File "./ex.v", line 15, characters 10-16:
Error: The variable lt_0_1 was not found in the current environment.


SCORE
-1.0
['\n  - simpl.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* n = 0 *)
    now rewrite Nat.
File "./ex.v", line 16, characters 4-19:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - (* n = 0 *)\n    now rewrite Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* n = 0 *)
    now rewrite Nat.lt_0_1.
File "./ex.v", line 16, characters 4-26:
Error:
setoid rewrite failed: Unable to satisfy the following constraints:
UNDEFINED EVARS:
 ?X13==[ |- Relation_Definitions.relation nat] (internal placeholder) {?r}
 ?X14==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper (Morphisms.respectful lt ?r) factorial]
         (internal placeholder) {?p}
 ?X15==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper
           (Morphisms.respectful lt
              (Morphisms.respectful ?r (Basics.flip Basics.impl))) lt]
         (internal placeholder) {?p0}
TYPECLASSES:?X13 ?X14 ?X15
SHELF:||
FUTURE GOALS STACK:?X15 ?X14 ?X13||



SCORE
-1.0
['lt_0_1.']
['\nProof.']
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
["\n  induction n as [|n' IHn'].\n  - simpl."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.
File "./ex.v", line 16, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  - simpl.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    rewrite IHn'.
    apply Nat.
File "./ex.v", line 19, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n    rewrite IHn'.\n    apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    rewrite IHn'.
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    rewrite IHn'.
    apply Nat.lt_lt_add_r.
Qed.
File "./ex.v", line 20, characters 0-4:
Error:  (in proof factorial_pos): Attempt to save an incomplete proof


SCORE
-1.0
['\nQed.']
['\nQed.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.
File "./ex.v", line 18, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
SCORE
1.0
["\n    apply IHn'.\nQed."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

The `factorial` function is defined as a recursive function on `nat`. The base case is when `n` is `0`, it returns `1`. In the inductive case, `n` is `S n'`, so it returns `n * factorial n'`.

The `factorial_pos` theorem states that for all `nat` values `n`, `0` is less than `factorial n`. The proof is by induction
SCORE
1.0
["\n```\n\nThe `factorial` function is defined as a recursive function on `nat`. The base case is when `n` is `0`, it returns `1`. In the inductive case, `n` is `S n'`, so it returns `n * factorial n'`.\n\nThe `factorial_pos` theorem states that for all `nat` values `n`, `0` is less than `factorial n`. The proof is by induction"]
CHOSEN SOLUTION
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Require Import Coq.Lists.List.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

The `factorial` function is defined as a recursive function on `nat`. The base case is when `n` is `0`, it returns `1`. In the inductive case, `n` is `S n'`, so it returns `n * factorial n'`.

The `factorial_pos` theorem states that for all `nat` values `n`, `0` is less than `factorial n`. The proof is by induction

STATS

number of nodes: 37
number of gen nodes: 19 (including leaves: 4)
number of back nodes: 18 (including leaves: 15)

expansion count: 27 (including failed: 9)

cache stats {'hit': 4, 'miss': 36}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31, 27, 78]

Trial calls: [22, 78, 25, 15, 15, 40]


Time Statistics:

  Min: 27.00 sec, Max: 118.00 sec, Average: 54.83 sec, Standard Deviation: 32.84 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 32.50 calls, Standard Deviation: 22.01 calls

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

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31, 27, 78]

Trial calls: [22, 78, 25, 15, 15, 40]


Time Statistics:

  Min: 27.00 sec, Max: 118.00 sec, Average: 54.83 sec, Standard Deviation: 32.84 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 32.50 calls, Standard Deviation: 22.01 calls

Trial  7
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.
File "./ex.v", line 2, characters 0-19:
Error: Cannot find a physical path bound to logical path Coq.


SCORE
None
['Require Import Coq.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.
File "./ex.v", line 2, characters 0-25:
Error: Cannot find a physical path bound to logical path Coq.Arith.


SCORE
None
['Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
SCORE
1.0
['Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Coq.
File "./ex.v", line 3, characters 7-10:
Error: Cannot find module Coq


SCORE
-1.0
['\nImport Coq.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.
SCORE
1.0
['\nImport Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\n\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n\nTheorem factorial_pos : forall n, 0 < factorial n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\nProof.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now rewrite Nat.
File "./ex.v", line 15, characters 4-19:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - now rewrite Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - now rewrite Nat.lt_0_1.
File "./ex.v", line 15, characters 4-26:
Error:
setoid rewrite failed: Unable to satisfy the following constraints:
UNDEFINED EVARS:
 ?X13==[ |- Relation_Definitions.relation nat] (internal placeholder) {?r}
 ?X14==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper (Morphisms.respectful Peano.lt ?r) factorial]
         (internal placeholder) {?p}
 ?X15==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper
           (Morphisms.respectful Peano.lt
              (Morphisms.respectful ?r (Basics.flip Basics.impl))) Peano.lt]
         (internal placeholder) {?p0}
TYPECLASSES:?X13 ?X14 ?X15
SHELF:||
FUTURE GOALS STACK:?X15 ?X14 ?X13||



SCORE
-1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
["\n  induction n as [|n' IHn'].\n  - simpl."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.
File "./ex.v", line 16, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  - simpl.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.
File "./ex.v", line 18, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r in IHn'.
    apply IHn'.
Qed.
File "./ex.v", line 18, characters 4-33:
Error: Unable to find an instance for the variable p.


SCORE
-1.0
["lt_lt_add_r in IHn'.\n    apply IHn'.\nQed."]
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
    apply Nat.
File "./ex.v", line 20, characters 4-14:
Error: No such goal.


SCORE
-1.0
["\n    apply IHn'.\n    apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
SCORE
1.0
["\n    apply IHn'.\nQed."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

This Coq script defines a factorial function on natural numbers and proves that the factorial is always strictly positive.
SCORE
1.0
['\n```\n\nThis Coq script defines a factorial function on natural numbers and proves that the factorial is always strictly positive.']
CHOSEN SOLUTION
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

This Coq script defines a factorial function on natural numbers and proves that the factorial is always strictly positive.

STATS

number of nodes: 25
number of gen nodes: 13 (including leaves: 1)
number of back nodes: 12 (including leaves: 12)

expansion count: 16 (including failed: 4)

cache stats {'hit': 1, 'miss': 21}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31, 27, 78, 33]

Trial calls: [22, 78, 25, 15, 15, 40, 22]


Time Statistics:

  Min: 27.00 sec, Max: 118.00 sec, Average: 51.71 sec, Standard Deviation: 31.35 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 31.00 calls, Standard Deviation: 20.70 calls

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

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31, 27, 78, 33]

Trial calls: [22, 78, 25, 15, 15, 40, 22]


Time Statistics:

  Min: 27.00 sec, Max: 118.00 sec, Average: 51.71 sec, Standard Deviation: 31.35 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 31.00 calls, Standard Deviation: 20.70 calls

Trial  8
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.
File "./ex.v", line 2, characters 0-19:
Error: Cannot find a physical path bound to logical path Coq.


SCORE
None
['Require Import Coq.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.
File "./ex.v", line 2, characters 0-25:
Error: Cannot find a physical path bound to logical path Coq.Arith.


SCORE
None
['Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
SCORE
1.0
['Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\n\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n\nTheorem factorial_pos : forall n, 0 < factorial n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\nProof.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
["\n  induction n as [|n' IHn'].\n  - simpl."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.
File "./ex.v", line 15, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  - simpl.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.
File "./ex.v", line 17, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
SCORE
1.0
["\n    apply IHn'.\nQed."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

In this Coq script, we first define the factorial function using a fixpoint.
SCORE
1.0
['\n```\n\nIn this Coq script, we first define the factorial function using a fixpoint.']
CHOSEN SOLUTION
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

In this Coq script, we first define the factorial function using a fixpoint.

STATS

number of nodes: 23
number of gen nodes: 12 (including leaves: 1)
number of back nodes: 11 (including leaves: 11)

expansion count: 11 (including failed: 0)

cache stats {'hit': 0, 'miss': 15}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31, 27, 78, 33, 22]

Trial calls: [22, 78, 25, 15, 15, 40, 22, 15]


Time Statistics:

  Min: 22.00 sec, Max: 118.00 sec, Average: 48.00 sec, Standard Deviation: 30.93 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 29.00 calls, Standard Deviation: 20.07 calls

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

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31, 27, 78, 33, 22]

Trial calls: [22, 78, 25, 15, 15, 40, 22, 15]


Time Statistics:

  Min: 22.00 sec, Max: 118.00 sec, Average: 48.00 sec, Standard Deviation: 30.93 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 29.00 calls, Standard Deviation: 20.07 calls

Trial  9
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.
File "./ex.v", line 2, characters 0-19:
Error: Cannot find a physical path bound to logical path Coq.


SCORE
None
['Require Import Coq.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.
File "./ex.v", line 2, characters 0-25:
Error: Cannot find a physical path bound to logical path Coq.Arith.


SCORE
None
['Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
SCORE
1.0
['Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import Coq.
File "./ex.v", line 3, characters 7-10:
Error: Cannot find module Coq


SCORE
-1.0
['\nImport Coq.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Coq.Arith.Arith.
Import ListNotations.
File "./ex.v", line 3, characters 7-20:
Error: Cannot find module ListNotations


SCORE
-1.0
['\nImport ListNotations.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.
SCORE
1.0
['Require Import Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\n\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n\nLemma factorial_positive : forall n, 0 < factorial n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\nProof.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    now rewrite Nat.
File "./ex.v", line 15, characters 4-19:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - (* base case: n = 0 *)\n    now rewrite Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    now rewrite Nat.lt_0_1.
File "./ex.v", line 15, characters 4-26:
Error:
setoid rewrite failed: Unable to satisfy the following constraints:
UNDEFINED EVARS:
 ?X13==[ |- Relation_Definitions.relation nat] (internal placeholder) {?r}
 ?X14==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper (Morphisms.respectful lt ?r) factorial]
         (internal placeholder) {?p}
 ?X15==[(do_subrelation:=Morphisms.do_subrelation) |-
         Morphisms.Proper
           (Morphisms.respectful lt
              (Morphisms.respectful ?r (Basics.flip Basics.impl))) lt]
         (internal placeholder) {?p0}
TYPECLASSES:?X13 ?X14 ?X15
SHELF:||
FUTURE GOALS STACK:?X15 ?X14 ?X13||



SCORE
-1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
["\n  induction n as [|n' IHn'].\n  - simpl."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.
File "./ex.v", line 15, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n  - simpl.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.
File "./ex.v", line 17, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
SCORE
1.0
["\n    apply IHn'.\nQed."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

In this script, we first define the factorial function using the `Fixpoint` keyword.
SCORE
1.0
['\n```\n\nIn this script, we first define the factorial function using the `Fixpoint` keyword.']
CHOSEN SOLUTION
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

In this script, we first define the factorial function using the `Fixpoint` keyword.

STATS

number of nodes: 25
number of gen nodes: 13 (including leaves: 2)
number of back nodes: 12 (including leaves: 11)

expansion count: 15 (including failed: 3)

cache stats {'hit': 0, 'miss': 20}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31, 27, 78, 33, 22, 30]

Trial calls: [22, 78, 25, 15, 15, 40, 22, 15, 20]


Time Statistics:

  Min: 22.00 sec, Max: 118.00 sec, Average: 46.00 sec, Standard Deviation: 29.70 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 28.00 calls, Standard Deviation: 19.14 calls

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

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31, 27, 78, 33, 22, 30]

Trial calls: [22, 78, 25, 15, 15, 40, 22, 15, 20]


Time Statistics:

  Min: 22.00 sec, Max: 118.00 sec, Average: 46.00 sec, Standard Deviation: 29.70 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 28.00 calls, Standard Deviation: 19.14 calls

Trial  10
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.
SCORE
1.0
['Require Import Arith.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\n\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n\nTheorem factorial_pos : forall n, 0 < factorial n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\nProof.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - now simpl.
File "./ex.v", line 14, characters 4-13:
Error: Tactic failure: Cannot solve this goal.


SCORE
-1.0
["\n  induction n as [| n' IHn'].\n  - now simpl."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - now apply Nat.
File "./ex.v", line 14, characters 4-17:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [| n' IHn'].\n  - now apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - now apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - now apply Nat.lt_0_1.
  - now apply Nat.
File "./ex.v", line 15, characters 4-17:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n  - now apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - now apply Nat.lt_0_1.
  - now apply Nat.lt_lt_add_r with (n * factorial n') and 1.
File "./ex.v", line 15, characters 4-59:
Error: The variable n was not found in the current environment.


SCORE
None
["lt_lt_add_r with (n * factorial n') and 1."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - now apply Nat.lt_0_1.
  - now apply Nat.lt_lt_add_r with (n * factorial n') and 1.
    apply IHn'.
Qed.
File "./ex.v", line 15, characters 4-59:
Error: The variable n was not found in the current environment.


SCORE
-1.0
["\n    apply IHn'.\nQed."]
['\n  - now apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - now apply Nat.lt_0_1.
  - now apply Nat.lt_lt_add_r with (1 + factorial n') and n'.
    + exact IHn'.
    + reflexivity.
File "./ex.v", line 15, characters 4-60:
Error: Not the right number of missing arguments (expected 0).


SCORE
-1.0
["lt_lt_add_r with (1 + factorial n') and n'.\n    + exact IHn'.\n    + reflexivity."]
['\n  intros n.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    apply Nat.
File "./ex.v", line 15, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n  induction n as [|n' IHn'].\n  - (* base case *)\n    apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    apply Nat.lt_0_1.
  - (* inductive case *)
    apply Nat.
File "./ex.v", line 17, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n  - (* inductive case *)\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    apply Nat.lt_0_1.
  - (* inductive case *)
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    apply Nat.lt_0_1.
  - (* inductive case *)
    apply Nat.lt_lt_add_r.
    rewrite IHn'.
    apply Nat.
File "./ex.v", line 19, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n    rewrite IHn'.\n    apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    apply Nat.lt_0_1.
  - (* inductive case *)
    apply Nat.lt_lt_add_r.
    rewrite IHn'.
    apply Nat.lt_0_1.
File "./ex.v", line 19, characters 10-20:
Error: In environment
n' : nat
IHn' : 0 < factorial n'
Unable to unify "0 < 1" with
 "factorial n' <
  (fix factorial (n : nat) : nat :=
     match n with
     | 0 => 1
     | S n' => n * factorial n'
     end) n'".


SCORE
-1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    apply Nat.lt_0_1.
  - (* inductive case *)
    apply Nat.lt_lt_add_r.
    + (* left sub-goal *)
      exact IHn'.
    + (* right sub-goal *)
      exact Nat.
File "./ex.v", line 20, characters 4-5:
Error: [Focus] Wrong bullet +: No more goals.


SCORE
-1.0
["\n    + (* left sub-goal *)\n      exact IHn'.\n    + (* right sub-goal *)\n      exact Nat."]
['\n  - (* inductive case *)\n    apply Nat.']
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    apply Nat.lt_0_1.
  - (* inductive case *)
    apply Nat.lt_lt_add_r.
    + (* n' >= 1 *)
      apply IHn'.
    + (* n' * factorial n' >= 1 *)
      apply Nat.
File "./ex.v", line 20, characters 4-5:
Error: [Focus] Wrong bullet +: No more goals.


SCORE
-1.0
["\n    + (* n' >= 1 *)\n      apply IHn'.\n    + (* n' * factorial n' >= 1 *)\n      apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case *)
    apply Nat.lt_0_1.
  - (* inductive case *)
    apply Nat.lt_lt_add_r.
    apply IHn'.
    apply Nat.
File "./ex.v", line 19, characters 4-14:
Error: No such goal.


SCORE
-1.0
["\n    apply IHn'.\n    apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
["\n  induction n as [|n' IHn'].\n  - simpl."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.
File "./ex.v", line 15, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_0_1.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['\n  - simpl.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    rewrite IHn'.
    apply Nat.
File "./ex.v", line 18, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n    rewrite IHn'.\n    apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    rewrite IHn'.
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    rewrite IHn'.
    apply Nat.lt_lt_add_r.
Qed.
File "./ex.v", line 19, characters 0-4:
Error:  (in proof factorial_pos): Attempt to save an incomplete proof


SCORE
-1.0
['\nQed.']
['\nQed.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.
File "./ex.v", line 17, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
['\n    apply Nat.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['lt_lt_add_r.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    rewrite IHn'.
    apply Nat.
File "./ex.v", line 19, characters 10-13:
Error: The variable Nat was not found in the current environment.


SCORE
None
["\n    rewrite IHn'.\n    apply Nat."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    rewrite IHn'.
    apply Nat.lt_trans.
File "./ex.v", line 19, characters 10-22:
Error: Unable to find an instance for the variable m.


SCORE
-1.0
['lt_trans.']
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
SCORE
1.0
["\n    apply IHn'.\nQed."]
TEXT
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

This Coq script defines a factorial function on natural numbers, and proves that the factorial is always strictly positive.
SCORE
1.0
['\n```\n\nThis Coq script defines a factorial function on natural numbers, and proves that the factorial is always strictly positive.']
CHOSEN SOLUTION
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.
### Hint: Don't forget to import the Arith module.
### Hint: use `Nat.lt_0_1` in the base case of the proof.
### Hint: use `Nat.lt_lt_add_r` in the inductive case of the proof.

```coq
Require Import Arith.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - simpl.
    apply Nat.lt_0_1.
  - simpl.
    apply Nat.lt_lt_add_r.
    apply IHn'.
Qed.
```

This Coq script defines a factorial function on natural numbers, and proves that the factorial is always strictly positive.

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)

cache stats {'hit': 5, 'miss': 34}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31, 27, 78, 33, 22, 30, 65]

Trial calls: [22, 78, 25, 15, 15, 40, 22, 15, 20, 39]


Time Statistics:

  Min: 22.00 sec, Max: 118.00 sec, Average: 47.90 sec, Standard Deviation: 28.75 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 29.10 calls, Standard Deviation: 18.45 calls

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

Experiment: run.py for 10 trials

Trial times: [38, 118, 37, 31, 27, 78, 33, 22, 30, 65]

Trial calls: [22, 78, 25, 15, 15, 40, 22, 15, 20, 39]


Time Statistics:

  Min: 22.00 sec, Max: 118.00 sec, Average: 47.90 sec, Standard Deviation: 28.75 sec

Call Statistics:

  Min: 15 calls, Max: 78 calls, Average: 29.10 calls, Standard Deviation: 18.45 calls

To complete all experiments, 8 minutes and 42 seconds elapsed.