## Code Snippet: Theorem Definition ### Overview The image shows a code snippet defining a theorem named `amc12a_2002_p6`. The theorem takes a natural number `n` and a proof `h0` that `0 < n` as input. It asserts the existence of `m` and `p` such that `m > n`, `m >= p`, and `m * p <= m + p`. The proof is constructed using tactics like `lift`, `cases'`, and `exact`. ### Components/Axes * **Theorem Name:** `amc12a_2002_p6` * **Input Parameters:** * `n : N` (n is a natural number) * `h0 : 0 < n` (h0 is a proof that 0 is less than n) * **Goal:** `∃ m, (m > n ∧ ∃ p, m * p ≤ m + p)` (There exists an m such that m > n and there exists a p such that m * p <= m + p) * **Proof Strategy:** * `lift n to N+ using h0` (Lift n to the positive natural numbers using the proof h0) * `cases' n with n` (Perform case analysis on n) * `exact (_, lt_add_of_pos_right _ zero_lt_one, 1, by simp)` (Provide the exact proof term) ### Detailed Analysis or ### Content Details The code snippet defines a theorem and provides a proof using a tactic-based approach. The theorem states that for any natural number `n` greater than 0, there exist numbers `m` and `p` satisfying certain conditions. The proof involves lifting `n` to the positive natural numbers, performing case analysis, and providing an exact proof term. The proof term `(_, lt_add_of_pos_right _ zero_lt_one, 1, by simp)` suggests that the proof relies on the lemma `lt_add_of_pos_right` and the fact that `0 < 1`. The `by simp` tactic likely simplifies the goal to complete the proof. ### Key Observations * The theorem is named `amc12a_2002_p6`, which suggests it might be related to a problem from the AMC 12A competition in 2002. * The proof uses tactics like `lift`, `cases'`, and `exact`, which are common in interactive theorem provers. * The proof term involves lemmas related to inequalities and simplification. ### Interpretation The code snippet demonstrates a formal proof of a mathematical theorem. The theorem states that for any natural number `n` greater than 0, there exist numbers `m` and `p` such that `m > n` and `m * p <= m + p`. The proof is constructed using a combination of tactics and lemmas, showcasing a typical workflow in interactive theorem proving. The specific lemmas used in the proof term suggest that the proof relies on basic properties of inequalities and natural numbers. The theorem and its proof might be related to a problem from a mathematical competition, indicating its potential relevance to problem-solving in mathematics.

\n ## Screenshot: Theorem Statement and Proof ### Overview The image is a screenshot of a dark-themed text editor or terminal window displaying a theorem statement and a partial proof. The theorem appears to be related to natural numbers and inequalities. There are three colored dots in the top-left corner (red, orange, green). ### Components/Axes There are no axes or traditional chart components. The key elements are: * **Theorem Statement:** A formal mathematical statement. * **Proof Steps:** A sequence of commands or instructions likely used in a proof assistant. * **Colored Dots:** Three dots in the top-left corner, colored red, orange, and green. Their purpose is unclear without further context. ### Detailed Analysis or Content Details The text content is as follows: ``` theorem amc12a_2002_p6 (n : ℕ) (h₀ : 0 < n) : ∃ m, (m > n ∧ ∃ p, m * p ≤ m + p) := by lift n to ℕ+ using h₀ cases' n with n exact (_, lt_add_of_pos_right _ zero_lt_one, 1, by simp) ``` Breaking down the theorem statement: * `theorem amc12a_2002_p6 (n : ℕ) (h₀ : 0 < n)`: This declares a theorem named `amc12a_2002_p6` that takes a natural number `n` (denoted by `n : ℕ`) and a hypothesis `h₀` stating that `n` is greater than 0 (`h₀ : 0 < n`). * `: ∃ m, (m > n ∧ ∃ p, m * p ≤ m + p)`: This is the statement of the theorem. It asserts that there exists a natural number `m` such that `m` is greater than `n` (`m > n`) and there exists a natural number `p` such that `m * p` is less than or equal to `m + p` (`m * p ≤ m + p`). The proof steps are: * `lift n to ℕ+ using h₀`: This step likely lifts the natural number `n` to a positive natural number `ℕ+` using the hypothesis `h₀`. * `cases' n with n`: This step performs case analysis on `n`. * `exact (_, lt_add_of_pos_right _ zero_lt_one, 1, by simp)`: This step completes the proof using a combination of tactics. `exact` indicates a direct proof. The arguments likely refer to lemmas or properties used in the proof. `lt_add_of_pos_right` suggests a lemma related to inequalities. `zero_lt_one` is a known inequality. `simp` is a simplification tactic. ### Key Observations The theorem statement involves existential quantifiers (`∃`) and inequalities. The proof appears to be written in a formal proof language, likely Lean or Coq, given the syntax. The use of `lift` and `cases'` suggests a proof by induction or case analysis. ### Interpretation The theorem states that for any natural number `n` greater than 0, there exists a natural number `m` greater than `n` such that `m * p ≤ m + p` for some natural number `p`. This inequality can be rearranged to `m * p - m - p ≤ 0`, or `m * p - m - p + 1 ≤ 1`, which factors to `(m - 1) * (p - 1) ≤ 1`. This suggests that the theorem is related to finding `m` and `p` that satisfy this inequality. The proof steps indicate a formal verification of this statement using a proof assistant. The colored dots in the top-left corner are likely indicators of the status of the proof (e.g., red for error, orange for warning, green for success), but this is speculative without more context.

\n ## [Code Screenshot]: Formal Proof of a Mathematical Theorem ### Overview The image is a screenshot of a code editor window (dark theme) displaying a formal mathematical theorem and its proof written in a dependently-typed proof assistant language, most likely Lean 4. The code defines a theorem named `amc12a_2002_p6` and provides a concise proof script. ### Components/Axes * **Window Frame**: A dark gray rectangular window with rounded corners, centered on a light gray background. The top-left corner contains three colored circles (red, yellow, green), mimicking macOS window controls. * **Code Content**: The primary content is a block of monospaced text with syntax highlighting. The text is left-aligned within the window. * **Syntax Highlighting Colors**: * **Yellow/Gold**: Keywords (`theorem`, `lift`, `to`, `using`, `cases'`, `with`, `exact`, `by`, `simp`). * **Green**: Type identifiers and some logical symbols (`ℕ`, `ℕ+`, `∃`, `∧`, `≤`). * **White/Light Gray**: Theorem name, variable names, hypothesis names, proof terms, and most operators (`amc12a_2002_p6`, `n`, `h₀`, `m`, `p`, `*`, `+`, `_`, `lt_add_of_pos_right`, `zero_lt_one`, `1`). ### Detailed Analysis / Content Details **Transcription of the Code:** ```lean theorem amc12a_2002_p6 (n : ℕ) (h₀ : 0 < n) : ∃ m, (m > n ∧ ∃ p, m * p ≤ m + p) := by lift n to ℕ+ using h₀ cases' n with n exact (_, lt_add_of_pos_right _ zero_lt_one, 1, by simp) ``` **Line-by-Line Breakdown:** 1. **Theorem Declaration**: * `theorem amc12a_2002_p6`: Declares a theorem with the name `amc12a_2002_p6`, indicating it is likely the solution to Problem 6 from the 2002 AMC 12A competition. * `(n : ℕ)`: The theorem is parameterized by a variable `n` of type `ℕ` (natural numbers). * `(h₀ : 0 < n)`: A hypothesis `h₀` stating that `n` is greater than 0 (i.e., `n` is positive). * `: ∃ m, (m > n ∧ ∃ p, m * p ≤ m + p)`: The proposition to be proven. It states: "There exists a natural number `m` such that `m` is greater than `n`, AND there exists a natural number `p` such that `m * p` is less than or equal to `m + p`." * `:= by`: Begins the proof script. 2. **Proof Script**: * `lift n to ℕ+ using h₀`: This tactic uses the hypothesis `h₀` (0 < n) to "lift" `n` from the type `ℕ` (natural numbers) to `ℕ+` (positive natural numbers). This is a type refinement step. * `cases' n with n`: This tactic performs case analysis on the positive natural number `n`. The `with n` part suggests it might be destructing it into its underlying natural number component, though the exact behavior depends on the definition of `ℕ+`. * `exact (_, lt_add_of_pos_right _ zero_lt_one, 1, by simp)`: This provides the exact proof term to close the goal. It constructs a tuple (likely a dependent pair) containing: * `_`: A placeholder for the existential witness `m`. The underscore indicates it will be inferred by the type checker. * `lt_add_of_pos_right _ zero_lt_one`: A proof of the first conjunct `m > n`. It applies a lemma `lt_add_of_pos_right` with arguments `_` (likely `n`) and `zero_lt_one` (a proof that 0 < 1). * `1`: The witness for the existential `p`. The proof asserts that `p = 1` works. * `by simp`: A proof of the second conjunct `m * 1 ≤ m + 1`. The `simp` tactic simplifies this inequality, which holds for all natural numbers `m`. ### Key Observations 1. **Conciseness**: The entire proof is completed in three tactic steps after the theorem declaration, demonstrating the power of the proof assistant's automation and library. 2. **Proof Strategy**: The proof does not explicitly construct the witness `m`. Instead, it relies on type inference (`_`) and a general lemma (`lt_add_of_pos_right`) that likely states `n < n + k` for any positive `k`. The choice of `p = 1` is explicit and simplifies the inequality to a trivial truth. 3. **Syntax**: The code uses specific syntax (`:= by`, `cases'`, `ℕ+`) characteristic of the Lean 4 proof assistant and its mathematical library (Mathlib). 4. **Visual Layout**: The code is presented in a clean, syntax-highlighted block within a stylized window, suggesting it might be from a documentation page, a tutorial, or a social media post showcasing a formal proof. ### Interpretation This image captures a formal verification of a number theory problem from a mathematics competition. The theorem states a property about positive integers: for any positive integer `n`, you can always find a larger integer `m` and another integer `p` such that the product `m*p` is no greater than the sum `m+p`. The proof reveals the underlying mathematical insight: the property holds trivially if you choose `p = 1`. In that case, the inequality becomes `m * 1 ≤ m + 1`, which simplifies to `m ≤ m + 1`, always true. The only non-trivial part is ensuring `m > n`, which the proof handles by invoking a general lemma about addition increasing order. The existence of such a formal proof demonstrates how competition mathematics problems can be translated into and rigorously verified within a computer-based formal system. The code is not just a description but a machine-checkable certificate of the theorem's truth. The use of placeholders (`_`) and automation (`simp`) highlights the balance between human-guided strategy and computer-driven verification in modern interactive theorem proving.

## 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.