coq_fact_run_focus_diversity.txt

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

Loading checkpoint shards:   0%|          | 0/7 [00:00<?, ?it/s]
Loading checkpoint shards:  14%|█▍        | 1/7 [00:06<00:41,  6.96s/it]
Loading checkpoint shards:  29%|██▊       | 2/7 [00:13<00:34,  6.86s/it]
Loading checkpoint shards:  43%|████▎     | 3/7 [00:20<00:26,  6.63s/it]
Loading checkpoint shards:  57%|█████▋    | 4/7 [00:26<00:19,  6.54s/it]
Loading checkpoint shards:  71%|███████▏  | 5/7 [00:32<00:12,  6.49s/it]
Loading checkpoint shards:  86%|████████▌ | 6/7 [00:39<00:06,  6.46s/it]
Loading checkpoint shards: 100%|██████████| 7/7 [00:45<00:00,  6.32s/it]
Loading checkpoint shards: 100%|██████████| 7/7 [00:45<00:00,  6.48s/it]
/home/namin/mambaforge/envs/trl/lib/python3.10/site-packages/transformers/utils/hub.py:374: FutureWarning: The `use_auth_token` argument is deprecated and will be removed in v5 of Transformers. Please use `token` instead.
  warnings.warn(
You are using the default legacy behaviour of the <class 'transformers.models.llama.tokenization_llama.LlamaTokenizer'>. This is expected, and simply means that the `legacy` (previous) behavior will be used so nothing changes for you. If you want to use the new behaviour, set `legacy=False`. This should only be set if you understand what it means, and thouroughly read the reason why this was added as explained in https://github.com/huggingface/transformers/pull/24565
["n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end.", "n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end.", "n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end.", "n : nat) : nat :=\n  match n with\n  | 0 => 1\n  | S n' => n * factorial n'\n  end.", "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. Do not repeat the previous mistakes.

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.

### Hint: Use some lemmas provided in Out. You cannot solve the base case by `reflexivity` since `<` is not reflexive. You also need a lemma in the inductive case. Use `apply` to use a lemma.

## Out
Nat.lt_0_1: 0 < 1
Nat.lt_lt_add_l: forall n m p : nat, n < m -> n < p + m
Nat.lt_lt_add_r: forall n m p : nat, n < m -> n < m + p
Nat.add_pos_l: forall n m : nat, 0 < n -> 0 < n + m
Nat.lt_add_pos_l: forall n m : nat, 0 < n -> m < n + m
Nat.lt_add_pos_r: forall n m : nat, 0 < n -> m < m + n
Nat.add_pos_r: forall n m : nat, 0 < m -> 0 < n + m
Nat.add_pos_pos: forall n m : nat, 0 < n -> 0 < m -> 0 < n + m
Nat.lt_sub_lt_add_l: forall n m p : nat, n - m < p -> n < m + p
Nat.lt_sub_lt_add_r: forall n m p : nat, n - p < m -> n < m + p
Nat.add_nonneg_pos: forall n m : nat, 0 <= n -> 0 < m -> 0 < n + m
Nat.add_lt_mono: forall n m p q : nat, n < m -> p < q -> n + p < m + q
Nat.add_le_lt_mono: forall n m p q : nat, n <= m -> p < q -> n + p < m + q


## Context




[/INST]
## Code

```Coq
Require Import Arith.
Search (0 < 1) inside Nat.
Search (_ < _ -> _ < _ + _) inside Nat.

Fixpoint factorial (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n' => n * factorial n'
  end.
SCORE
1.0
['Lemma factorial_pos : forall n, 0 < factorial n.', 'Theorem factorial_pos : forall n, 0 < factorial n.', 'Theorem factorial_pos : forall n, 0 < factorial n.', 'Lemma factorial_pos : forall n, 0 < factorial n.', 'Theorem 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. Do not repeat the previous mistakes.

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.

### Hint: Use some lemmas provided in Out. You cannot solve the base case by `reflexivity` since `<` is not reflexive. You also need a lemma in the inductive case. Use `apply` to use a lemma.

## Out
Nat.lt_0_1: 0 < 1
Nat.lt_lt_add_l: forall n m p : nat, n < m -> n < p + m
Nat.lt_lt_add_r: forall n m p : nat, n < m -> n < m + p
Nat.add_pos_l: forall n m : nat, 0 < n -> 0 < n + m
Nat.lt_add_pos_l: forall n m : nat, 0 < n -> m < n + m
Nat.lt_add_pos_r: forall n m : nat, 0 < n -> m < m + n
Nat.add_pos_r: forall n m : nat, 0 < m -> 0 < n + m
Nat.add_pos_pos: forall n m : nat, 0 < n -> 0 < m -> 0 < n + m
Nat.lt_sub_lt_add_l: forall n m p : nat, n - m < p -> n < m + p
Nat.lt_sub_lt_add_r: forall n m p : nat, n - p < m -> n < m + p
Nat.add_nonneg_pos: forall n m : nat, 0 <= n -> 0 < m -> 0 < n + m
Nat.add_lt_mono: forall n m p q : nat, n < m -> p < q -> n + p < m + q
Nat.add_le_lt_mono: forall n m p q : nat, n <= m -> p < q -> n + p < m + q


## Context




[/INST]
## Code

```Coq
Require Import Arith.
Search (0 < 1) inside Nat.
Search (_ < _ -> _ < _ + _) inside Nat.

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

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

SCORE
1.0
['roof.', 'roof.', 'roof.', 'roof.', 'roof.']
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. Do not repeat the previous mistakes.

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.

### Hint: Use some lemmas provided in Out. You cannot solve the base case by `reflexivity` since `<` is not reflexive. You also need a lemma in the inductive case. Use `apply` to use a lemma.

## Out
Nat.lt_0_1: 0 < 1
Nat.lt_lt_add_l: forall n m p : nat, n < m -> n < p + m
Nat.lt_lt_add_r: forall n m p : nat, n < m -> n < m + p
Nat.add_pos_l: forall n m : nat, 0 < n -> 0 < n + m
Nat.lt_add_pos_l: forall n m : nat, 0 < n -> m < n + m
Nat.lt_add_pos_r: forall n m : nat, 0 < n -> m < m + n
Nat.add_pos_r: forall n m : nat, 0 < m -> 0 < n + m
Nat.add_pos_pos: forall n m : nat, 0 < n -> 0 < m -> 0 < n + m
Nat.lt_sub_lt_add_l: forall n m p : nat, n - m < p -> n < m + p
Nat.lt_sub_lt_add_r: forall n m p : nat, n - p < m -> n < m + p
Nat.add_nonneg_pos: forall n m : nat, 0 <= n -> 0 < m -> 0 < n + m
Nat.add_lt_mono: forall n m p q : nat, n < m -> p < q -> n + p < m + q
Nat.add_le_lt_mono: forall n m p q : nat, n <= m -> p < q -> n + p < m + q


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).



[/INST]
## Code

```Coq
Require Import Arith.
Search (0 < 1) inside Nat.
Search (_ < _ -> _ < _ + _) inside Nat.

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

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

SCORE
1.0
[' intros n.', ' intros n.', ' intros n.', ' intros 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. Do not repeat the previous mistakes.

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.

### Hint: Use some lemmas provided in Out. You cannot solve the base case by `reflexivity` since `<` is not reflexive. You also need a lemma in the inductive case. Use `apply` to use a lemma.

## Out
Nat.lt_0_1: 0 < 1
Nat.lt_lt_add_l: forall n m p : nat, n < m -> n < p + m
Nat.lt_lt_add_r: forall n m p : nat, n < m -> n < m + p
Nat.add_pos_l: forall n m : nat, 0 < n -> 0 < n + m
Nat.lt_add_pos_l: forall n m : nat, 0 < n -> m < n + m
Nat.lt_add_pos_r: forall n m : nat, 0 < n -> m < m + n
Nat.add_pos_r: forall n m : nat, 0 < m -> 0 < n + m
Nat.add_pos_pos: forall n m : nat, 0 < n -> 0 < m -> 0 < n + m
Nat.lt_sub_lt_add_l: forall n m p : nat, n - m < p -> n < m + p
Nat.lt_sub_lt_add_r: forall n m p : nat, n - p < m -> n < m + p
Nat.add_nonneg_pos: forall n m : nat, 0 <= n -> 0 < m -> 0 < n + m
Nat.add_lt_mono: forall n m p q : nat, n < m -> p < q -> n + p < m + q
Nat.add_le_lt_mono: forall n m p q : nat, n <= m -> p < q -> n + p < m + q


## Context
### Need to show

forall n : nat, 0 < factorial n

### given assumptions

(none).



[/INST]
## Code

```Coq
Require Import Arith.
Search (0 < 1) inside Nat.
Search (_ < _ -> _ < _ + _) inside Nat.

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

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

SCORE
1.0
[" induction n as [|n' IHn'].\n  - apply Nat.", " induction n as [| n' IHn'].\n  - (* Base case *)\n    apply Nat.", " induction n as [|n' IHn'].\n  - apply Nat.", " induction n as [|n' IHn'].\n  - (* base case *)\n    apply Nat.", " induction n as [|n' IHn'].\n  - apply 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. Do not repeat the previous mistakes.

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.

### Hint: Use some lemmas provided in Out. You cannot solve the base case by `reflexivity` since `<` is not reflexive. You also need a lemma in the inductive case. Use `apply` to use a lemma.

## Out
Nat.lt_0_1: 0 < 1
Nat.lt_lt_add_l: forall n m p : nat, n < m -> n < p + m
Nat.lt_lt_add_r: forall n m p : nat, n < m -> n < m + p
Nat.add_pos_l: forall n m : nat, 0 < n -> 0 < n + m
Nat.lt_add_pos_l: forall n m : nat, 0 < n -> m < n + m
Nat.lt_add_pos_r: forall n m : nat, 0 < n -> m < m + n
Nat.add_pos_r: forall n m : nat, 0 < m -> 0 < n + m
Nat.add_pos_pos: forall n m : nat, 0 < n -> 0 < m -> 0 < n + m
Nat.lt_sub_lt_add_l: forall n m p : nat, n - m < p -> n < m + p
Nat.lt_sub_lt_add_r: forall n m p : nat, n - p < m -> n < m + p
Nat.add_nonneg_pos: forall n m : nat, 0 <= n -> 0 < m -> 0 < n + m
Nat.add_lt_mono: forall n m p q : nat, n < m -> p < q -> n + p < m + q
Nat.add_le_lt_mono: forall n m p q : nat, n <= m -> p < q -> n + p < m + q


## Context
### Need to show

0 < factorial n

### given assumptions

n : nat.



[/INST]
## Code

```Coq
Require Import Arith.
Search (0 < 1) inside Nat.
Search (_ < _ -> _ < _ + _) inside Nat.

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

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


SCORE
None
['_0_1.', '_0_1.', '_0_1.', '_0_1.', '_0_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. Do not repeat the previous mistakes.

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.

### Hint: Use some lemmas provided in Out. You cannot solve the base case by `reflexivity` since `<` is not reflexive. You also need a lemma in the inductive case. Use `apply` to use a lemma.

## Out
Nat.lt_0_1: 0 < 1
Nat.lt_lt_add_l: forall n m p : nat, n < m -> n < p + m
Nat.lt_lt_add_r: forall n m p : nat, n < m -> n < m + p
Nat.add_pos_l: forall n m : nat, 0 < n -> 0 < n + m
Nat.lt_add_pos_l: forall n m : nat, 0 < n -> m < n + m
Nat.lt_add_pos_r: forall n m : nat, 0 < n -> m < m + n
Nat.add_pos_r: forall n m : nat, 0 < m -> 0 < n + m
Nat.add_pos_pos: forall n m : nat, 0 < n -> 0 < m -> 0 < n + m
Nat.lt_sub_lt_add_l: forall n m p : nat, n - m < p -> n < m + p
Nat.lt_sub_lt_add_r: forall n m p : nat, n - p < m -> n < m + p
Nat.add_nonneg_pos: forall n m : nat, 0 <= n -> 0 < m -> 0 < n + m
Nat.add_lt_mono: forall n m p q : nat, n < m -> p < q -> n + p < m + q
Nat.add_le_lt_mono: forall n m p q : nat, n <= m -> p < q -> n + p < m + q


## Context
### Need to show

0 < factorial 0

### given assumptions

(none).



[/INST]
## Code

```Coq
Require Import Arith.
Search (0 < 1) inside Nat.
Search (_ < _ -> _ < _ + _) inside Nat.

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

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

SCORE
1.0
[' - (* Inductive case *)\n    apply Nat.', ' - (* Inductive case *)\n    apply Nat.', ' - (* Inductive case *)\n    apply Nat.', ' - (* Inductive case *)\n    apply Nat.', ' - (* Inductive case *)\n    apply 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. Do not repeat the previous mistakes.

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.

### Hint: Use some lemmas provided in Out. You cannot solve the base case by `reflexivity` since `<` is not reflexive. You also need a lemma in the inductive case. Use `apply` to use a lemma.

## Out
Nat.lt_0_1: 0 < 1
Nat.lt_lt_add_l: forall n m p : nat, n < m -> n < p + m
Nat.lt_lt_add_r: forall n m p : nat, n < m -> n < m + p
Nat.add_pos_l: forall n m : nat, 0 < n -> 0 < n + m
Nat.lt_add_pos_l: forall n m : nat, 0 < n -> m < n + m
Nat.lt_add_pos_r: forall n m : nat, 0 < n -> m < m + n
Nat.add_pos_r: forall n m : nat, 0 < m -> 0 < n + m
Nat.add_pos_pos: forall n m : nat, 0 < n -> 0 < m -> 0 < n + m
Nat.lt_sub_lt_add_l: forall n m p : nat, n - m < p -> n < m + p
Nat.lt_sub_lt_add_r: forall n m p : nat, n - p < m -> n < m + p
Nat.add_nonneg_pos: forall n m : nat, 0 <= n -> 0 < m -> 0 < n + m
Nat.add_lt_mono: forall n m p q : nat, n < m -> p < q -> n + p < m + q
Nat.add_le_lt_mono: forall n m p q : nat, n <= m -> p < q -> n + p < m + q


## Context
### Need to show

0 < factorial 0

### given assumptions

(none).



[/INST]
## Code

```Coq
Require Import Arith.
Search (0 < 1) inside Nat.
Search (_ < _ -> _ < _ + _) inside Nat.

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

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


SCORE
None
['_mul_pos_r.', '_lt_add_l.', '_lt_add_l.', "_mul_pos_l in IHn'.\n    (* Use the lemma Nat.", '_lt_add_r.']
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. Do not repeat the previous mistakes.

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.

### Hint: Use some lemmas provided in Out. You cannot solve the base case by `reflexivity` since `<` is not reflexive. You also need a lemma in the inductive case. Use `apply` to use a lemma.

## Out
Nat.lt_0_1: 0 < 1
Nat.lt_lt_add_l: forall n m p : nat, n < m -> n < p + m
Nat.lt_lt_add_r: forall n m p : nat, n < m -> n < m + p
Nat.add_pos_l: forall n m : nat, 0 < n -> 0 < n + m
Nat.lt_add_pos_l: forall n m : nat, 0 < n -> m < n + m
Nat.lt_add_pos_r: forall n m : nat, 0 < n -> m < m + n
Nat.add_pos_r: forall n m : nat, 0 < m -> 0 < n + m
Nat.add_pos_pos: forall n m : nat, 0 < n -> 0 < m -> 0 < n + m
Nat.lt_sub_lt_add_l: forall n m p : nat, n - m < p -> n < m + p
Nat.lt_sub_lt_add_r: forall n m p : nat, n - p < m -> n < m + p
Nat.add_nonneg_pos: forall n m : nat, 0 <= n -> 0 < m -> 0 < n + m
Nat.add_lt_mono: forall n m p q : nat, n < m -> p < q -> n + p < m + q
Nat.add_le_lt_mono: forall n m p q : nat, n <= m -> p < q -> n + p < m + q


## Context
### Need to show

0 < factorial (S n')

### given assumptions

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



[/INST]
## Code

```Coq
Require Import Arith.
Search (0 < 1) inside Nat.
Search (_ < _ -> _ < _ + _) inside Nat.

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

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

SCORE
1.0
["   - (* IHn' *)\n      apply IHn'.\n    - (* 0 < n *)\n      apply Nat.", "   apply IHn'.\n    apply Nat.", "   apply IHn'.\n    apply Nat.", "   exact IHn'.\nQed.", "   apply IHn'.\n    apply 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. Do not repeat the previous mistakes.

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.

### Hint: Use some lemmas provided in Out. You cannot solve the base case by `reflexivity` since `<` is not reflexive. You also need a lemma in the inductive case. Use `apply` to use a lemma.

## Out
Nat.lt_0_1: 0 < 1
Nat.lt_lt_add_l: forall n m p : nat, n < m -> n < p + m
Nat.lt_lt_add_r: forall n m p : nat, n < m -> n < m + p
Nat.add_pos_l: forall n m : nat, 0 < n -> 0 < n + m
Nat.lt_add_pos_l: forall n m : nat, 0 < n -> m < n + m
Nat.lt_add_pos_r: forall n m : nat, 0 < n -> m < m + n
Nat.add_pos_r: forall n m : nat, 0 < m -> 0 < n + m
Nat.add_pos_pos: forall n m : nat, 0 < n -> 0 < m -> 0 < n + m
Nat.lt_sub_lt_add_l: forall n m p : nat, n - m < p -> n < m + p
Nat.lt_sub_lt_add_r: forall n m p : nat, n - p < m -> n < m + p
Nat.add_nonneg_pos: forall n m : nat, 0 <= n -> 0 < m -> 0 < n + m
Nat.add_lt_mono: forall n m p q : nat, n < m -> p < q -> n + p < m + q
Nat.add_le_lt_mono: forall n m p q : nat, n <= m -> p < q -> n + p < m + q


## Context
### Need to show

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

### given assumptions

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



[/INST]
## Code

```Coq
Require Import Arith.
Search (0 < 1) inside Nat.
Search (_ < _ -> _ < _ + _) inside Nat.

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

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - (* Base case *)
    apply Nat.lt_0_1.
  - (* Inductive case *)
    apply Nat.lt_lt_add_r.
    exact IHn'.
Qed.
SCORE
1.0
['``\n\n## Context\n\nThe `factorial_pos` theorem states that the factorial function is always strictly positive.', '``\n\n## Explanation\nIn the above Coq script, we first import the `Arith` library to have access to the natural numbers and basic arithmetic lemmas.', "``\n\n## Explanation\n\nThe factorial function is defined by recursion on `n`. The base case is when `n` is `0`, in which case we return `1`. For the inductive case, we have `n` is `S n'`, so we compute the factorial of `S n'` and multiply it by `n`.\n\nThe proof of `factorial_pos` is by induction on `n`.\n\n- In the", '``\n\n## Explanation\n\nIn the code above, we define a factorial function on natural numbers, and then prove that the factorial of any natural number is strictly positive.', '``\n\n## Explanation\nIn this proof, we first import the Arithmetic library to use its lemmas about natural numbers.']
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. Do not repeat the previous mistakes.

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.

### Hint: Use some lemmas provided in Out. You cannot solve the base case by `reflexivity` since `<` is not reflexive. You also need a lemma in the inductive case. Use `apply` to use a lemma.

## Out
Nat.lt_0_1: 0 < 1
Nat.lt_lt_add_l: forall n m p : nat, n < m -> n < p + m
Nat.lt_lt_add_r: forall n m p : nat, n < m -> n < m + p
Nat.add_pos_l: forall n m : nat, 0 < n -> 0 < n + m
Nat.lt_add_pos_l: forall n m : nat, 0 < n -> m < n + m
Nat.lt_add_pos_r: forall n m : nat, 0 < n -> m < m + n
Nat.add_pos_r: forall n m : nat, 0 < m -> 0 < n + m
Nat.add_pos_pos: forall n m : nat, 0 < n -> 0 < m -> 0 < n + m
Nat.lt_sub_lt_add_l: forall n m p : nat, n - m < p -> n < m + p
Nat.lt_sub_lt_add_r: forall n m p : nat, n - p < m -> n < m + p
Nat.add_nonneg_pos: forall n m : nat, 0 <= n -> 0 < m -> 0 < n + m
Nat.add_lt_mono: forall n m p q : nat, n < m -> p < q -> n + p < m + q
Nat.add_le_lt_mono: forall n m p q : nat, n <= m -> p < q -> n + p < m + q


## Context




[/INST]
## Code

```Coq
Require Import Arith.
Search (0 < 1) inside Nat.
Search (_ < _ -> _ < _ + _) inside Nat.

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

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - (* Base case *)
    apply Nat.lt_0_1.
  - (* Inductive case *)
    apply Nat.lt_lt_add_r.
    exact IHn'.
Qed.
```

## Explanation

The factorial function is defined by recursion on `n`. The base case is when `n` is `0`, in which case we return `1`. For the inductive case, we have `n` is `S n'`, so we compute the factorial of `S n'` and multiply it by `n`.

The proof of `factorial_pos` is by induction on `n`.

- In the
SCORE
1.0
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. Do not repeat the previous mistakes.

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.

### Hint: Use some lemmas provided in Out. You cannot solve the base case by `reflexivity` since `<` is not reflexive. You also need a lemma in the inductive case. Use `apply` to use a lemma.

## Out
Nat.lt_0_1: 0 < 1
Nat.lt_lt_add_l: forall n m p : nat, n < m -> n < p + m
Nat.lt_lt_add_r: forall n m p : nat, n < m -> n < m + p
Nat.add_pos_l: forall n m : nat, 0 < n -> 0 < n + m
Nat.lt_add_pos_l: forall n m : nat, 0 < n -> m < n + m
Nat.lt_add_pos_r: forall n m : nat, 0 < n -> m < m + n
Nat.add_pos_r: forall n m : nat, 0 < m -> 0 < n + m
Nat.add_pos_pos: forall n m : nat, 0 < n -> 0 < m -> 0 < n + m
Nat.lt_sub_lt_add_l: forall n m p : nat, n - m < p -> n < m + p
Nat.lt_sub_lt_add_r: forall n m p : nat, n - p < m -> n < m + p
Nat.add_nonneg_pos: forall n m : nat, 0 <= n -> 0 < m -> 0 < n + m
Nat.add_lt_mono: forall n m p q : nat, n < m -> p < q -> n + p < m + q
Nat.add_le_lt_mono: forall n m p q : nat, n <= m -> p < q -> n + p < m + q


## Context




[/INST]
## Code

```Coq
Require Import Arith.
Search (0 < 1) inside Nat.
Search (_ < _ -> _ < _ + _) inside Nat.

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

Theorem factorial_pos : forall n, 0 < factorial n.
Proof.
  intros n.
  induction n as [| n' IHn'].
  - (* Base case *)
    apply Nat.lt_0_1.
  - (* Inductive case *)
    apply Nat.lt_lt_add_r.
    exact IHn'.
Qed.
```

## Explanation

The factorial function is defined by recursion on `n`. The base case is when `n` is `0`, in which case we return `1`. For the inductive case, we have `n` is `S n'`, so we compute the factorial of `S n'` and multiply it by `n`.

The proof of `factorial_pos` is by induction on `n`.

- In the