bench_coq_factorial_lemma_lia.txt

========
Experiment configuration: python experiments.py --experiment_name run_meta.py --n_trials 10 --mins_timeout 10 --problem_name problem_fact --language Coq --remove_hints True --use_hammer False --extract_lemma_depth 1 


[2024-01-12 14:28:43,919] [INFO] [real_accelerator.py:158:get_accelerator] Setting ds_accelerator to cuda (auto detect)
Trial  1
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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
[" induction n as [|n' IHn'].\n  - simpl."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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. reflexivity.
File "./ex.v", line 14, characters 11-22:
Error:
 The relation lt is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
[' reflexivity.']
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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.
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.
SCORE
1.0
['\nQed.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n. induction n as [|n' IHn'].
  - simpl. apply (@helper0 ).
  - simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['  - simpl.']
CHOSEN SOLUTION

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n. induction n as [|n' IHn'].
  - simpl. apply (@helper0 ).
  - simpl.```

STATS

number of nodes: 19
number of gen nodes: 10 (including leaves: 1)
number of back nodes: 9 (including leaves: 9)

expansion count: 9 (including failed: 0)

cache stats {'hit': 0, 'miss': 9}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [23]

Trial calls: [9]


Time Statistics:

  Min: 23.00 sec, Max: 23.00 sec, Average: 23.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 9 calls, Max: 9 calls, Average: 9.00 calls, Standard Deviation: 0.00 calls

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

Experiment: run_meta.py for 10 trials

Trial times: [23]

Trial calls: [9]


Time Statistics:

  Min: 23.00 sec, Max: 23.00 sec, Average: 23.00 sec, Standard Deviation: 0.00 sec

Call Statistics:

  Min: 9 calls, Max: 9 calls, Average: 9.00 calls, Standard Deviation: 0.00 calls

Trial  2
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import Lia.
SCORE
1.0
['Import Lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import Lia.

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'].
  - reflexivity.
File "./ex.v", line 16, characters 4-15:
Error:
 The relation lt is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
["\n  induction n as [|n' IHn'].\n  - reflexivity."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import Lia.

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'].
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.
Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.
SCORE
1.0
['\nQed.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn']. apply (@helper0 ).
  simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['  simpl.']
CHOSEN SOLUTION

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn']. apply (@helper0 ).
  simpl.```

STATS

number of nodes: 19
number of gen nodes: 10 (including leaves: 1)
number of back nodes: 9 (including leaves: 9)

expansion count: 9 (including failed: 0)

cache stats {'hit': 0, 'miss': 9}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [23, 24]

Trial calls: [9, 9]


Time Statistics:

  Min: 23.00 sec, Max: 24.00 sec, Average: 23.50 sec, Standard Deviation: 0.50 sec

Call Statistics:

  Min: 9 calls, Max: 9 calls, Average: 9.00 calls, Standard Deviation: 0.00 calls

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

Experiment: run_meta.py for 10 trials

Trial times: [23, 24]

Trial calls: [9, 9]


Time Statistics:

  Min: 23.00 sec, Max: 24.00 sec, Average: 23.50 sec, Standard Deviation: 0.50 sec

Call Statistics:

  Min: 9 calls, Max: 9 calls, Average: 9.00 calls, Standard Deviation: 0.00 calls

Trial  3
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

(* Define factorial *)
Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\n(* Define factorial *)\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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

SCORE
1.0
['\n\n(* Lemma: factorial is positive *)\nLemma factorial_positive : forall n, 0 < factorial n.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
["\n  induction n as [|n' IHn'].\n  - (* base case: n = 0 *)\n    simpl."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n    lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
["\n  - (* inductive case: n = S n' *)\n    simpl."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n' + n' * factorial n'

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
    rewrite IHn'.
    lia.
File "./ex.v", line 23, characters 4-7:
Error: Tactic failure:  Cannot find witness.


SCORE
-1.0
["\n    rewrite IHn'.\n    lia."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
    rewrite IHn'.
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n' : nat,
0 < factorial n' ->
factorial n' < factorial n' + n' * factorial n'

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)

Lemma helper0: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  induction n' as [|n' IHn'].
  - (* base case *)
    simpl.
File "./ex.v", line 18, characters 2-28:
Error: IHn' is already used.


SCORE
-1.0
["  intros n' IHn'.\n  induction n' as [|n' IHn'].\n  - (* base case *)\n    simpl."]
["\n    rewrite IHn'.\n    lia."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
    rewrite IHn'.
</CODE>
Created Lemma helper0.
["  intros n' IHn'.\n  induction n' as [|n' IHn'].\n  - (* base case *)\n    simpl."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n' : nat,
0 < factorial n' ->
factorial n' < factorial n' + n' * factorial n'

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)

Lemma helper0: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  induction n' as [|n' IHn'].
  - simpl; lia.
File "./ex.v", line 18, characters 2-28:
Error: IHn' is already used.


SCORE
-1.0
["  intros n' IHn'.\n  induction n' as [|n' IHn'].\n  - simpl; lia."]
["\n  - (* inductive case: n = S n' *)\n    simpl."]
["\n    rewrite IHn'.\n    lia."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
    rewrite IHn'.
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n' : nat,
0 < factorial n' ->
factorial n' < factorial n' + n' * factorial n'

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)

Lemma helper0: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  rewrite <- (Nat.
File "./ex.v", line 18, characters 17-18:
Error: Syntax error: ',' or ')' expected after [term level 200] (in [term]).


SCORE
None
["  intros n' IHn'.\n  rewrite <- (Nat."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

factorial n' < factorial n' + n' * factorial n'

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)

Lemma helper0: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  rewrite <- (Nat.mul_one_r n').
  (* Now we need to show that n' * factorial n' is positive *)
  assert (H1 : 0 < n' * factorial n') as H1.
File "./ex.v", line 17, characters 14-27:
Error: The reference Nat.mul_one_r was not found in the current environment.


SCORE
-1.0
["mul_one_r n').\n  (* Now we need to show that n' * factorial n' is positive *)\n  assert (H1 : 0 < n' * factorial n') as H1."]
["\n    rewrite IHn'.\n    lia."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
    rewrite IHn'.
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n' : nat,
0 < factorial n' ->
factorial n' < factorial n' + n' * factorial n'

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)

Lemma helper0: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  rewrite (IHn').
  lia.
File "./ex.v", line 18, characters 2-16:
Error: Found no subterm matching "0" in the current goal.


SCORE
-1.0
["  intros n' IHn'.\n  rewrite (IHn').\n  lia."]
["  intros n' IHn'.\n  induction n' as [|n' IHn'].\n  - (* base case *)\n    simpl."]
['\n    lia.']
["\n  - (* inductive case: n = S n' *)\n    simpl."]
["\n    rewrite IHn'.\n    lia."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
    rewrite IHn'.
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n' : nat,
0 < factorial n' ->
factorial n' < factorial n' + n' * factorial n'

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)

Lemma helper0: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  rewrite -> add_comm.
File "./ex.v", line 18, characters 13-21:
Error: The variable add_comm was not found in the current environment.


SCORE
None
["  intros n' IHn'.\n  rewrite -> add_comm."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

factorial n' < factorial n' + n' * factorial n'

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)

Lemma helper0: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  rewrite -> add_comm.
  apply (mul_lt_mono_l IHn' _ _).
  apply IHn'.
  apply lt_mul_pos_right.
File "./ex.v", line 17, characters 13-21:
Error: The variable add_comm was not found in the current environment.


SCORE
-1.0
["\n  apply (mul_lt_mono_l IHn' _ _).\n  apply IHn'.\n  apply lt_mul_pos_right."]
["\n    rewrite IHn'.\n    lia."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
    rewrite IHn'.
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n' : nat,
0 < factorial n' ->
factorial n' < factorial n' + n' * factorial n'

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)

Lemma helper0: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  induction n' as [|n' IHn'].
  - simpl.
File "./ex.v", line 18, characters 2-28:
Error: IHn' is already used.


SCORE
-1.0
["  intros n' IHn'.\n  induction n' as [|n' IHn'].\n  - simpl."]
["  intros n' IHn'.\n  induction n' as [|n' IHn'].\n  - (* base case *)\n    simpl."]
["\n    rewrite IHn'.\n    lia."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
    rewrite IHn'.
</CODE>
Created Lemma helper0.
["  intros n' IHn'.\n  induction n' as [|n' IHn'].\n  - simpl."]
["  intros n' IHn'.\n  rewrite <- (Nat."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

factorial n' < factorial n' + n' * factorial n'

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)

Lemma helper0: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  rewrite <- (Nat.mul_one_r (factorial n')).
  apply (lt_add_r IHn').
  apply (lt_mul_of_pos_right IHn').
Qed.
File "./ex.v", line 17, characters 14-27:
Error: The reference Nat.mul_one_r was not found in the current environment.


SCORE
-1.0
["mul_one_r (factorial n')).\n  apply (lt_add_r IHn').\n  apply (lt_mul_of_pos_right IHn').\nQed."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    reflexivity.
File "./ex.v", line 18, characters 4-15:
Error:
 The relation lt is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
["\n  induction n as [|n' IHn'].\n  - (* base case: n = 0 *)\n    reflexivity."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)

Lemma helper0: 0 < 1.
Proof.
  lia.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)

Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.
SCORE
1.0
['\nQed.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)

Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn']. apply (@helper0 ).
  simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['  simpl.']
CHOSEN SOLUTION

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

(* Lemma: factorial is positive *)

Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn']. apply (@helper0 ).
  simpl.```

STATS

number of nodes: 49
number of gen nodes: 25 (including leaves: 8)
number of back nodes: 24 (including leaves: 17)

expansion count: 35 (including failed: 11)

cache stats {'hit': 14, 'miss': 21}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100]

Trial calls: [9, 9, 35]


Time Statistics:

  Min: 23.00 sec, Max: 100.00 sec, Average: 49.00 sec, Standard Deviation: 36.06 sec

Call Statistics:

  Min: 9 calls, Max: 35 calls, Average: 17.67 calls, Standard Deviation: 12.26 calls

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

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100]

Trial calls: [9, 9, 35]


Time Statistics:

  Min: 23.00 sec, Max: 100.00 sec, Average: 49.00 sec, Standard Deviation: 36.06 sec

Call Statistics:

  Min: 9 calls, Max: 35 calls, Average: 17.67 calls, Standard Deviation: 12.26 calls

Trial  4
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

Fixpoint factorial (n: nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\nFixpoint factorial (n: nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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

SCORE
1.0
['\n\nLemma factorial_pos : forall n, 0 < factorial n.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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

SCORE
1.0
['\nProof.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

Lemma 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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

Lemma factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n. induction n as [| n' IHn'].
  - reflexivity.
File "./ex.v", line 14, characters 4-15:
Error:
 The relation lt is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
[" induction n as [| n' IHn'].\n  - reflexivity."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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

Lemma factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n. induction n as [| n' IHn'].
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.
SCORE
1.0
['\nQed.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n. induction n as [| n' IHn']. apply (@helper0 ).
  simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['  simpl.']
CHOSEN SOLUTION

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n. induction n as [| n' IHn']. apply (@helper0 ).
  simpl.```

STATS

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

expansion count: 8 (including failed: 0)

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

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22]

Trial calls: [9, 9, 35, 8]


Time Statistics:

  Min: 22.00 sec, Max: 100.00 sec, Average: 42.25 sec, Standard Deviation: 33.35 sec

Call Statistics:

  Min: 8 calls, Max: 35 calls, Average: 15.25 calls, Standard Deviation: 11.41 calls

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

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22]

Trial calls: [9, 9, 35, 8]


Time Statistics:

  Min: 22.00 sec, Max: 100.00 sec, Average: 42.25 sec, Standard Deviation: 33.35 sec

Call Statistics:

  Min: 8 calls, Max: 35 calls, Average: 15.25 calls, Standard Deviation: 11.41 calls

Trial  5
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => S n' * factorial n'
  end.
SCORE
1.0
["\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => S n' * factorial n'\n  end."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n. induction n as [|n' IHn'].
  - (* n = 0 *)
    reflexivity.
File "./ex.v", line 15, characters 4-15:
Error:
 The relation lt is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
[" induction n as [|n' IHn'].\n  - (* n = 0 *)\n    reflexivity."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n. induction n as [|n' IHn'].
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.
SCORE
1.0
['\nQed.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n. induction n as [|n' IHn']. apply (@helper0 ).
  simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['  simpl.']
CHOSEN SOLUTION

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n. induction n as [|n' IHn']. apply (@helper0 ).
  simpl.```

STATS

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

expansion count: 8 (including failed: 0)

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

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22, 22]

Trial calls: [9, 9, 35, 8, 8]


Time Statistics:

  Min: 22.00 sec, Max: 100.00 sec, Average: 38.20 sec, Standard Deviation: 30.91 sec

Call Statistics:

  Min: 8 calls, Max: 35 calls, Average: 13.80 calls, Standard Deviation: 10.61 calls

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

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22, 22]

Trial calls: [9, 9, 35, 8, 8]


Time Statistics:

  Min: 22.00 sec, Max: 100.00 sec, Average: 38.20 sec, Standard Deviation: 30.91 sec

Call Statistics:

  Min: 8 calls, Max: 35 calls, Average: 13.80 calls, Standard Deviation: 10.61 calls

Trial  6
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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'].
  - (* n = 0 *)
    reflexivity.
File "./ex.v", line 16, characters 4-15:
Error:
 The relation lt is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
["\n  induction n as [|n' IHn'].\n  - (* n = 0 *)\n    reflexivity."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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'].
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  apply lia.
File "./ex.v", line 15, characters 8-11:
Error: The variable lia was not found in the current environment.


SCORE
None
['  apply lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  apply lia.
Qed.
File "./ex.v", line 14, characters 8-11:
Error: The variable lia was not found in the current environment.


SCORE
-1.0
['\nQed.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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 *)
    reflexivity.
File "./ex.v", line 16, characters 4-15:
Error:
 The relation lt is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
["\n  induction n as [|n' IHn'].\n  - (* base case: n = 0 *)\n    reflexivity."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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'].
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.
SCORE
1.0
['\nQed.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn']. apply (@helper0 ).
  simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['  simpl.']
CHOSEN SOLUTION

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn']. apply (@helper0 ).
  simpl.```

STATS

number of nodes: 21
number of gen nodes: 11 (including leaves: 2)
number of back nodes: 10 (including leaves: 9)

expansion count: 11 (including failed: 1)

cache stats {'hit': 0, 'miss': 11}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22, 22, 31]

Trial calls: [9, 9, 35, 8, 8, 11]


Time Statistics:

  Min: 22.00 sec, Max: 100.00 sec, Average: 37.00 sec, Standard Deviation: 28.34 sec

Call Statistics:

  Min: 8 calls, Max: 35 calls, Average: 13.33 calls, Standard Deviation: 9.74 calls

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

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22, 22, 31]

Trial calls: [9, 9, 35, 8, 8, 11]


Time Statistics:

  Min: 22.00 sec, Max: 100.00 sec, Average: 37.00 sec, Standard Deviation: 28.34 sec

Call Statistics:

  Min: 8 calls, Max: 35 calls, Average: 13.33 calls, Standard Deviation: 9.74 calls

Trial  7
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

Fixpoint factorial (n:nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\nFixpoint factorial (n:nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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'].
  - (* case n = 0 *)
    reflexivity.
File "./ex.v", line 16, characters 4-15:
Error:
 The relation lt is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
["\n  induction n as [|n' IHn'].\n  - (* case n = 0 *)\n    reflexivity."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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'].
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.
SCORE
1.0
['\nQed.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn']. apply (@helper0 ).
  rewrite IHn'.
  lia.
File "./ex.v", line 22, characters 2-5:
Error: Tactic failure:  Cannot find witness.


SCORE
-1.0
["  rewrite IHn'.\n  lia."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn']. apply (@helper0 ).
  rewrite IHn'.
</CODE>
Created Lemma helper1.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n' : nat,
0 < factorial n' ->
factorial n' < factorial n' + n' * factorial n'

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.


Lemma helper1: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  assert (H : n' * factorial n' = factorial n' + (n' - 1) * factorial (n' - 1)).
  {
    induction n' as [| n' IHn''].
    - reflexivity.
File "./ex.v", line 25, characters 6-17:
Error: In environment
IHn' : 0 < factorial 0
Unable to unify "factorial 0 + (0 - 1) * factorial (0 - 1)" with
 "0 * factorial 0".


SCORE
-1.0
["  intros n' IHn'.\n  assert (H : n' * factorial n' = factorial n' + (n' - 1) * factorial (n' - 1)).\n  {\n    induction n' as [| n' IHn''].\n    - reflexivity."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn']. apply (@helper0 ).
  simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['  simpl.']
CHOSEN SOLUTION

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn']. apply (@helper0 ).
  simpl.```

STATS

number of nodes: 19
number of gen nodes: 10 (including leaves: 2)
number of back nodes: 9 (including leaves: 8)

expansion count: 10 (including failed: 1)

cache stats {'hit': 0, 'miss': 10}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22, 22, 31, 31]

Trial calls: [9, 9, 35, 8, 8, 11, 10]


Time Statistics:

  Min: 22.00 sec, Max: 100.00 sec, Average: 36.14 sec, Standard Deviation: 26.32 sec

Call Statistics:

  Min: 8 calls, Max: 35 calls, Average: 12.86 calls, Standard Deviation: 9.09 calls

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

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22, 22, 31, 31]

Trial calls: [9, 9, 35, 8, 8, 11, 10]


Time Statistics:

  Min: 22.00 sec, Max: 100.00 sec, Average: 36.14 sec, Standard Deviation: 26.32 sec

Call Statistics:

  Min: 8 calls, Max: 35 calls, Average: 12.86 calls, Standard Deviation: 9.09 calls

Trial  8
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import ListNotations.
File "./ex.v", line 4, characters 7-20:
Error: Cannot find module ListNotations


SCORE
-1.0
['Import ListNotations.']
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Lia. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.
SCORE
1.0
['Import List.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

(* Define the factorial function *)
Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\n\n(* Define the factorial function *)\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n\n(* Lemma: factorial n is always positive *)\nLemma factorial_positive : forall n, 0 < factorial n.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  (* Base case: n = 0 *)
  intros n.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n  (* Base case: n = 0 *)\n  intros n.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  (* Base case: n = 0 *)
  intros n.
  simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n  simpl.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  (* Base case: n = 0 *)
  intros n.
  simpl.
  reflexivity.
File "./ex.v", line 19, characters 2-13:
Error:
 The relation lt is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
['\n  reflexivity.']
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  (* Base case: n = 0 *)
  intros n.
  simpl.
</CODE>
['\n  reflexivity.']
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  (* Base case: n = 0 *)
  intros n.
  simpl.
</CODE>
['\n  simpl.']
['\n  reflexivity.']
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  (* Base case: n = 0 *)
  intros n.
  simpl.
</CODE>
['\n  reflexivity.']
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  (* Base case: n = 0 *)
  intros n.
  simpl.
</CODE>
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: factorial 0 = 1, which is positive *)
    reflexivity.
File "./ex.v", line 19, characters 4-15:
Error:
 The relation lt is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
["\n  induction n as [|n' IHn'].\n  - (* base case: factorial 0 = 1, which is positive *)\n    reflexivity."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: 0 < 1.
Proof.
  apply lia.
File "./ex.v", line 18, characters 8-11:
Error: The variable lia was not found in the current environment.


SCORE
None
['  apply lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: 0 < 1.
Proof.
  apply lia.
Qed.
File "./ex.v", line 17, characters 8-11:
Error: The variable lia was not found in the current environment.


SCORE
-1.0
['\nQed.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* n = 0 *)
    reflexivity.
File "./ex.v", line 19, characters 4-15:
Error:
 The relation lt is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
["\n  induction n as [|n' IHn'].\n  - (* n = 0 *)\n    reflexivity."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: 0 < 1.
Proof.
  apply lt_succ_self.
File "./ex.v", line 18, characters 8-20:
Error: The variable lt_succ_self was not found in the current environment.


SCORE
None
['  apply lt_succ_self.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: 0 < 1.
Proof.
  apply lt_succ_self.
Qed.
File "./ex.v", line 17, characters 8-20:
Error: The variable lt_succ_self was not found in the current environment.


SCORE
-1.0
['\nQed.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: 0 < 1.
Proof.
  now.
File "./ex.v", line 18, characters 5-6:
Error: Syntax error: [tactic] expected after 'now' (in [simple_tactic]).


SCORE
None
['  now.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: 0 < 1.
Proof.
  now.
Qed.
File "./ex.v", line 17, characters 5-6:
Error: Syntax error: [tactic] expected after 'now' (in [simple_tactic]).


SCORE
-1.0
['\nQed.']
['\n  reflexivity.']
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  (* Base case: n = 0 *)
  intros n.
  simpl.
</CODE>
['\nProof.']
['\n  intros n.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
["\n  induction n as [|n' IHn'].\n  - (* base case: n = 0 *)\n    simpl."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    apply lt_1_0.
File "./ex.v", line 20, characters 10-16:
Error: The variable lt_1_0 was not found in the current environment.


SCORE
None
['\n    apply lt_1_0.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    apply lt_1_0.
  - (* inductive case: n = S n' *)
    simpl.
File "./ex.v", line 20, characters 10-16:
Error: The variable lt_1_0 was not found in the current environment.


SCORE
-1.0
["\n  - (* inductive case: n = S n' *)\n    simpl."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: 0 < 1.
Proof.
  apply (lt_le_0 0 1).
  auto.
File "./ex.v", line 18, characters 9-16:
Error: The variable lt_le_0 was not found in the current environment.


SCORE
-1.0
['  apply (lt_le_0 0 1).\n  auto.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['\n    lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
["\n  - (* inductive case: n = S n' *)\n    simpl."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n' + n' * factorial n'

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
    rewrite IHn'.
    lia.
File "./ex.v", line 24, characters 4-7:
Error: Tactic failure:  Cannot find witness.


SCORE
-1.0
["\n    rewrite IHn'.\n    lia."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
    rewrite IHn'.
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n' : nat,
0 < factorial n' ->
factorial n' < factorial n' + n' * factorial n'

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  induction n' as [|n' IHn'].
  - simpl.
File "./ex.v", line 19, characters 2-28:
Error: IHn' is already used.


SCORE
-1.0
["  intros n' IHn'.\n  induction n' as [|n' IHn'].\n  - simpl."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n' + n' * factorial n'

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
    rewrite IHn'.
    (* The goal is now 0 < S (n' * factorial n') *)
    lia.
File "./ex.v", line 25, characters 4-7:
Error: Tactic failure:  Cannot find witness.


SCORE
-1.0
["\n    rewrite IHn'.\n    (* The goal is now 0 < S (n' * factorial n') *)\n    lia."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* inductive case: n = S n' *)
    simpl.
    rewrite IHn'.
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n' : nat,
0 < factorial n' ->
factorial n' < factorial n' + n' * factorial n'

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  induction n' as [|n' IHn'].
  - (* base case: n' = 0 *)
    simpl.
File "./ex.v", line 19, characters 2-28:
Error: IHn' is already used.


SCORE
-1.0
["  intros n' IHn'.\n  induction n' as [|n' IHn'].\n  - (* base case: n' = 0 *)\n    simpl."]
["  intros n' IHn'.\n  induction n' as [|n' IHn'].\n  - simpl."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* induction step: n = S n' *)
    simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
["\n  - (* induction step: n = S n' *)\n    simpl."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n' + n' * factorial n'

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* induction step: n = S n' *)
    simpl.
    rewrite IHn'.
    lia.
File "./ex.v", line 24, characters 4-7:
Error: Tactic failure:  Cannot find witness.


SCORE
-1.0
["\n    rewrite IHn'.\n    lia."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* induction step: n = S n' *)
    simpl.
    rewrite IHn'.
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n' : nat,
0 < factorial n' ->
factorial n' < factorial n' + n' * factorial n'

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  rewrite <- (mul_comm 1 (factorial n')).
  rewrite mul_1_r.
File "./ex.v", line 19, characters 14-22:
Error: The variable mul_comm was not found in the current environment.


SCORE
-1.0
["  intros n' IHn'.\n  rewrite <- (mul_comm 1 (factorial n')).\n  rewrite mul_1_r."]
["\n    rewrite IHn'.\n    lia."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
    lia.
  - (* induction step: n = S n' *)
    simpl.
    rewrite IHn'.
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n' : nat,
0 < factorial n' ->
factorial n' < factorial n' + n' * factorial n'

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: forall n' : nat, forall IHn' : 0 < factorial n', factorial n' < factorial n' + n' * factorial n'.
Proof.
  intros n' IHn'.
  simpl in IHn'.
  lia.
File "./ex.v", line 20, characters 2-5:
Error: Tactic failure:  Cannot find witness.


SCORE
-1.0
["  intros n' IHn'.\n  simpl in IHn'.\n  lia."]
["\n  - (* inductive case: n = S n' *)\n    simpl."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)
Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl.
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: 0 < 1.
Proof.
  lia.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.
SCORE
1.0
['\nQed.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl. apply (@helper0 ).
  - (* inductive case: n = S n' *)
    simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
["  - (* inductive case: n = S n' *)\n    simpl."]
CHOSEN SOLUTION

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import List.

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

(* Lemma: factorial n is always positive *)

Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Lemma factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn'].
  - (* base case: n = 0 *)
    simpl. apply (@helper0 ).
  - (* inductive case: n = S n' *)
    simpl.```

STATS

number of nodes: 59
number of gen nodes: 30 (including leaves: 11)
number of back nodes: 29 (including leaves: 19)

expansion count: 44 (including failed: 15)

cache stats {'hit': 10, 'miss': 34}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22, 22, 31, 31, 106]

Trial calls: [9, 9, 35, 8, 8, 11, 10, 44]


Time Statistics:

  Min: 22.00 sec, Max: 106.00 sec, Average: 44.88 sec, Standard Deviation: 33.77 sec

Call Statistics:

  Min: 8 calls, Max: 44 calls, Average: 16.75 calls, Standard Deviation: 13.36 calls

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

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22, 22, 31, 31, 106]

Trial calls: [9, 9, 35, 8, 8, 11, 10, 44]


Time Statistics:

  Min: 22.00 sec, Max: 106.00 sec, Average: 44.88 sec, Standard Deviation: 33.77 sec

Call Statistics:

  Min: 8 calls, Max: 44 calls, Average: 16.75 calls, Standard Deviation: 13.36 calls

Trial  9
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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'].
  - reflexivity.
File "./ex.v", line 15, characters 4-15:
Error:
 The relation lt is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
["\n  induction n as [|n' IHn'].\n  - reflexivity."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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'].
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.
SCORE
1.0
['\nQed.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn']. apply (@helper0 ).
  simpl.
Error: There are pending proofs in file ./ex.v: factorial_positive.

SCORE
1.0
['  simpl.']
CHOSEN SOLUTION

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Theorem factorial_positive : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [|n' IHn']. apply (@helper0 ).
  simpl.```

STATS

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

expansion count: 8 (including failed: 0)

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

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22, 22, 31, 31, 106, 22]

Trial calls: [9, 9, 35, 8, 8, 11, 10, 44, 8]


Time Statistics:

  Min: 22.00 sec, Max: 106.00 sec, Average: 42.33 sec, Standard Deviation: 32.64 sec

Call Statistics:

  Min: 8 calls, Max: 44 calls, Average: 15.78 calls, Standard Deviation: 12.89 calls

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

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22, 22, 31, 31, 106, 22]

Trial calls: [9, 9, 35, 8, 8, 11, 10, 44, 8]


Time Statistics:

  Min: 22.00 sec, Max: 106.00 sec, Average: 42.33 sec, Standard Deviation: 32.64 sec

Call Statistics:

  Min: 8 calls, Max: 44 calls, Average: 15.78 calls, Standard Deviation: 12.89 calls

Trial  10
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.
Import ListNotations.
File "./ex.v", line 4, characters 7-20:
Error: Cannot find module ListNotations


SCORE
-1.0
['Import ListNotations.']
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.
</CODE>
err !! (ERROR/3) Coq raised an exception:
     > Syntax error: illegal begin of vernac.
   The offending chunk is delimited by >>>…<<< below:
     >
     > From Coq Require Import Arith.Arith.
     > Require Import Lia. >>>simpl<<<.
   Results past this point may be unreliable.

TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
["\nFixpoint factorial (n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end."]
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context


[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial n

### given assumptions

n as follows.

n : nat.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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'].
  - reflexivity.
File "./ex.v", line 14, characters 4-15:
Error:
 The relation lt is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library


SCORE
-1.0
[" induction n as [|n' IHn'].\n  - reflexivity."]
<CODE> 
From Coq Require Import Arith.Arith.
Require Import Lia.

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'].
</CODE>
Created Lemma helper0.
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq

From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Error: There are pending proofs in file ./ex.v: helper0.

SCORE
1.0
['  lia.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < 1

### given assumptions

(none).

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.
SCORE
1.0
['\nQed.']
TEXT

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n. induction n as [|n' IHn']. apply (@helper0 ).
  simpl.
Error: There are pending proofs in file ./ex.v: factorial_pos.

SCORE
1.0
['  simpl.']
CHOSEN SOLUTION

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



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

## Instructions
### Spec: In Coq, write a factorial function and prove that the factorial is always strictly positive.

## Out


## Context
### Need to show

0 < factorial (S n')

### given assumptions

n', IHn' as follows.

n' : nat
IHn' : 0 < factorial n'.

[/INST]
## Code

```Coq
From Coq Require Import Arith.Arith.
Require Import Lia.

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


Lemma helper0: 0 < 1.
Proof.
  lia.
Qed.

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n. induction n as [|n' IHn']. apply (@helper0 ).
  simpl.```

STATS

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

expansion count: 9 (including failed: 1)

cache stats {'hit': 0, 'miss': 9}
=====SUMMARY STATISTICS FOR ALL TRIALS=====

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22, 22, 31, 31, 106, 22, 24]

Trial calls: [9, 9, 35, 8, 8, 11, 10, 44, 8, 9]


Time Statistics:

  Min: 22.00 sec, Max: 106.00 sec, Average: 40.50 sec, Standard Deviation: 31.45 sec

Call Statistics:

  Min: 8 calls, Max: 44 calls, Average: 15.10 calls, Standard Deviation: 12.40 calls

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

Experiment: run_meta.py for 10 trials

Trial times: [23, 24, 100, 22, 22, 31, 31, 106, 22, 24]

Trial calls: [9, 9, 35, 8, 8, 11, 10, 44, 8, 9]


Time Statistics:

  Min: 22.00 sec, Max: 106.00 sec, Average: 40.50 sec, Standard Deviation: 31.45 sec

Call Statistics:

  Min: 8 calls, Max: 44 calls, Average: 15.10 calls, Standard Deviation: 12.40 calls

To complete all experiments, 7 minutes and 26 seconds elapsed.