## Screenshot: Coq Theorem Proof Code
### Overview
The image shows a code snippet from a Coq proof assistant environment. The code defines a theorem named `amc12a_2002_p6` with parameters and a proof structure. The syntax is color-coded, with keywords in yellow, variables/parameters in green, and other text in white. The UI includes three circular buttons (red, yellow, green) in the top-left corner, typical of macOS window controls.
### Components/Axes
- **UI Elements**:
- Top-left corner: Three circular buttons (red, yellow, green) for window controls.
- **Code Structure**:
- **Theorem Declaration**: `theorem amc12a_2002_p6 (n : N) (h0 : 0 < n) : ∃ m, (m > n ∧ ∃ p, m * p ≤ m + p) := by`
- **Proof Steps**:
1. `lift n to N+ using h0`
2. `cases' n with n`
3. `exact (_, lt_add_of_pos_right _ zero_lt_one, 1, by simp)`
### Detailed Analysis
- **Theorem Name**: `amc12a_2002_p6` (likely referencing a specific problem or lemma).
- **Parameters**:
- `n : N` (natural number `n`).
- `h0 : 0 < n` (proof that `n` is positive).
- **Proof Goal**:
- `∃ m, (m > n ∧ ∃ p, m * p ≤ m + p)` (there exists an `m` greater than `n` such that there exists a `p` where `m * p ≤ m + p`).
- **Proof Steps**:
1. **`lift n to N+ using h0`**: Inductively lifts `n` to a positive natural number using the hypothesis `h0`.
2. **`cases' n with n`**: Case analysis on `n` (likely splitting into base and inductive cases).
3. **`exact (_, lt_add_of_pos_right _ zero_lt_one, 1, by simp)`**: Applies a lemma (`lt_add_of_pos_right`) with `zero_lt_one` (proof that `0 < 1`) and simplifies the result.
### Key Observations
- The code uses Coq's proof-by-induction and case analysis syntax.
- The theorem's goal involves existential quantification (`∃`) and arithmetic inequalities.
- The `exact` command suggests a direct application of a pre-proven lemma, followed by simplification (`simp`).
### Interpretation
This code represents a formal proof in Coq, a proof assistant used to verify mathematical theorems. The theorem `amc12a_2002_p6` asserts the existence of a number `m` greater than `n` (a positive natural number) such that there exists a `p` satisfying `m * p ≤ m + p`. The proof leverages induction on `n` and applies a lemma (`lt_add_of_pos_right`) to establish the inequality. The use of `simp` indicates automated simplification of the proof steps. This demonstrates Coq's role in formalizing and verifying mathematical reasoning, ensuring correctness through machine-checked proofs.
