## Code Snippet: Number Theory Theorems ### Overview The image presents a code snippet, likely from a formal verification system or proof assistant, defining two theorems related to number theory. The code uses a specific syntax for defining theorems, assumptions, and proofs. ### Components/Axes The image contains the following elements: * **Window Decorations:** Standard window controls (red, yellow, green circles) in the top-left corner. * **Theorem Definitions:** Two theorem definitions, `mathd_numbertheory_293` and `mathd_numbertheory_233`. * **Variables and Types:** Declarations of variables with their corresponding types (e.g., `n : N`, `b : ZMod (11^2)`). * **Assumptions:** Hypotheses or preconditions for the theorems (e.g., `h0 : n ≤ 9`, `h1 : 11|20 * 100 + 10 * n + 7`, `h0 : b = 24⁻¹`). * **Statements:** Assertions to be proven (e.g., `n = 5`, `b = 116`). * **Proof Strategies:** Tactics used to prove the statements (e.g., `:= by omega`, `:= by exact h0`). ### Detailed Analysis or ### Content Details **Theorem `mathd_numbertheory_293`:** * **Name:** `theorem mathd_numbertheory_293` * **Variable:** `n` of type `N` (likely natural numbers). * **Assumption `h0`:** `n ≤ 9` (n is less than or equal to 9). * **Assumption `h1`:** `11|20 * 100 + 10 * n + 7` (11 divides 20 * 100 + 10 * n + 7). * **Statement:** `n = 5` * **Proof:** `:= by omega` (likely an automated proof tactic). **Theorem `mathd_numbertheory_233`:** * **Name:** `theorem mathd_numbertheory_233` * **Variable:** `b` of type `ZMod (11^2)` (integers modulo 11 squared, i.e., modulo 121). * **Assumption `h0`:** `b = 24⁻¹` (b is the modular inverse of 24 modulo 121). * **Statement:** `b = 116` * **Proof:** `:= by exact h0` (the statement follows directly from assumption h0). ### Key Observations * The code uses a formal syntax, suggesting it's part of a proof assistant or formal verification tool. * The theorems involve basic number theory concepts like divisibility, modular arithmetic, and modular inverses. * The proofs are concise, relying on automated tactics or direct application of assumptions. ### Interpretation The code snippet demonstrates the formalization of number theory theorems within a proof assistant. The theorems are precisely stated with explicit assumptions and types. The proofs, while brief, indicate the logical steps required to establish the truth of the statements. The use of tactics like `omega` suggests the system can automatically handle certain types of proofs. The second theorem shows a direct application of an assumption to prove the statement.

\n ## Text Block: Mathematical Theorems ### Overview The image displays a screenshot of a code editor or similar environment containing two mathematical theorems written in a formal, likely proof-assistant language. The theorems relate to number theory and modular arithmetic. ### Components/Axes There are no axes or charts in this image. The content is purely textual. The image contains three colored circles at the top-left corner: red, orange, and green. ### Detailed Analysis or Content Details **Theorem 1: mathd_numbertheory_293** * `(n : N)`: Declares a variable 'n' of type 'N' (presumably natural numbers). * `(h₀ : n ≤ 9)`: Defines a hypothesis 'h₀' stating that 'n' is less than or equal to 9. * `(h₁ : 11 | 20 * 100 + 10 * n + 7)`: Defines a hypothesis 'h₁' stating that 11 divides (20 * 100 + 10 * n + 7). The symbol '|' likely represents divisibility. * `n = 5 := by omega`: States that 'n' is equal to 5, and this is proven 'by omega' (likely a proof tactic or command within the system). **Theorem 2: mathd_numbertheory_233** * `(b : ZMod (11^2))` : Declares a variable 'b' of type 'ZMod (11^2)' (integers modulo 11 squared, or 121). * `(h₀ : b = 24⁻¹)`: Defines a hypothesis 'h₀' stating that 'b' is the modular inverse of 24 modulo 121. * `b = 116 := by exact h₀`: States that 'b' is equal to 116, and this is proven 'by exact h₀' (meaning it directly follows from the hypothesis h₀). ### Key Observations The theorems are presented in a highly structured format, typical of formal mathematical proofs. The use of hypotheses (h₀, h₁) and proof tactics (omega, exact) suggests this is code intended for a proof assistant like Lean, Coq, or Isabelle. The theorems themselves involve basic number theory concepts like divisibility and modular inverses. ### Interpretation The image demonstrates a snippet of formal mathematical reasoning. The theorems are likely part of a larger effort to rigorously prove properties of numbers. The use of a proof assistant ensures that the proofs are logically sound and free from errors. The theorems themselves are relatively simple, but they illustrate the power of formal methods in mathematics. The 'omega' tactic in the first theorem suggests an automated proof search, while 'exact h₀' in the second theorem indicates a direct application of a previously stated hypothesis. The theorems are named with a specific naming convention, `mathd_numbertheory_XXX`, suggesting they are part of a larger library or collection of number theory results.

## Screenshot: Mathematical Theorems in a Code Editor ### Overview The image is a screenshot of a dark-themed code editor or terminal window displaying two formal mathematical theorems and their proofs, written in a syntax consistent with a proof assistant language like Lean. The window has a standard macOS-style title bar with three colored control buttons (red, yellow, green) in the top-left corner. The text is syntax-highlighted in various colors (pink, blue, yellow, white) against a dark gray background. ### Components/Axes * **Window Frame:** A dark gray rectangular window with rounded corners, centered on a light gray background. * **Window Controls:** Three circular buttons in the top-left corner of the window: red (left), yellow (center), green (right). * **Text Content:** Two distinct blocks of code, each defining a theorem and its proof. The text uses a monospaced font. * **Syntax Highlighting:** * Keywords (`theorem`, `by`, `exact`): Pink/Salmon * Theorem Names (`mathd_numbertheory_293`, `mathd_numbertheory_233`): Light Blue * Variables and Types (`n`, `N`, `h₀`, `h₁`, `b`, `ZMod`): White/Light Gray * Numerical Values (`9`, `11`, `20`, `100`, `7`, `5`, `24`, `116`): Yellow/Orange * Mathematical Operators and Symbols (`≤`, `|`, `*`, `+`, `:=`, `^`, `-`): White/Light Gray ### Detailed Analysis / Content Details **Theorem 1 (Top Block):** * **Theorem Name:** `theorem mathd_numbertheory_293` * **Hypotheses:** * `(n : N)` - Declares `n` as a natural number. * `(h₀ : n ≤ 9)` - Hypothesis `h₀`: `n` is less than or equal to 9. * `(h₁ : 11|20 * 100 + 10 * n + 7)` - Hypothesis `h₁`: 11 divides the expression `20 * 100 + 10 * n + 7`. The vertical bar `|` denotes divisibility. * **Conclusion:** `n = 5` * **Proof Tactic:** `:= by omega` - The proof is completed by the `omega` tactic, which is often used for solving linear integer arithmetic problems. **Theorem 2 (Bottom Block):** * **Theorem Name:** `theorem mathd_numbertheory_233` * **Hypotheses:** * `(b : ZMod (11^2))` - Declares `b` as an element of the ring of integers modulo `11^2` (which is 121). * `(h₀ : b = 24⁻¹)` - Hypothesis `h₀`: `b` is the multiplicative inverse of 24 in the ring `ZMod 121`. * **Conclusion:** `b = 116` * **Proof Tactic:** `:= by exact h₀` - The proof is completed by the `exact` tactic, directly using hypothesis `h₀`. This suggests the statement `b = 116` is a direct computation or simplification of `b = 24⁻¹` within `ZMod 121`. ### Key Observations 1. **Formal Verification Context:** The code is not standard mathematical notation but formal statements meant for a proof assistant, where every logical step must be justified by a tactic (`omega`, `exact`). 2. **Number Theory Focus:** Both theorems are from number theory, dealing with divisibility (`11|...`) and modular arithmetic (`ZMod`). 3. **Specific Numerical Results:** The theorems assert specific integer solutions: `n=5` for the first, and the inverse of 24 modulo 121 being `116` for the second. 4. **Proof Strategy:** The first proof uses an automated tactic (`omega`) suitable for linear constraints, while the second uses a direct appeal to a hypothesis (`exact`), indicating the conclusion is essentially a restatement or immediate consequence of the definition given. ### Interpretation This image captures a snippet of work in formal mathematics or computer-aided theorem proving. The theorems are likely exercises or lemmas from a collection (indicated by names like `mathd_numbertheory_293`). * **Theorem 293** solves a constrained divisibility puzzle: Find the single-digit `n` (0-9) such that the number formed by the digits `2, 0, n, 7` (interpreted as `2000 + 10*n + 7`? Wait, the expression is `20 * 100 + 10 * n + 7`, which equals `2000 + 10n + 7 = 2007 + 10n`) is divisible by 11. The proof by `omega` confirms the unique solution is `n=5`, making the number 2057 (2007 + 50). 2057 / 11 = 187. * **Theorem 233** computes a modular inverse. It states that in the ring of integers modulo 121, the multiplicative inverse of 24 is 116. This can be verified: `24 * 116 = 2784`. `2784 mod 121`: 121 * 23 = 2783, so `2784 - 2783 = 1`. Therefore, `24 * 116 ≡ 1 (mod 121)`, confirming `116` is indeed the inverse. The image demonstrates the process of encoding classical number theory problems into a formal language for machine verification, highlighting the intersection of pure mathematics and computer science. The use of specific tactics reveals the underlying logical structure required for the computer to accept the proofs.

## Screenshot: Code Snippet from Number Theory Theorems ### Overview The image displays a code snippet with two mathematical theorems (`mathd_numbertheory_293` and `mathd_numbertheory_233`) written in a programming language with syntax highlighting. The code defines variables, hypotheses, and conclusions using formal logic and modular arithmetic. ### Components/Axes - **Theorems**: - `theorem mathd_numbertheory_293` - `theorem mathd_numbertheory_233` - **Variables**: - `n : ℕ` (natural number) - `h₀ : n ≤ 9` (hypothesis 0) - `h₁ : 11/20 * 100 + 10 * n + 7` (hypothesis 1) - `b : ZMod(11^2)` (modular integer) - `h₀ : b = 24⁻¹` (hypothesis 0 for theorem 233) - **Operators/Syntax**: - `:=` (assignment) - `*` (multiplication) - `^` (exponentiation) - `ZMod` (modular arithmetic) - `exact` (proof keyword) ### Detailed Analysis #### Theorem `mathd_numbertheory_293` - **Variables**: - `n` is constrained to natural numbers (`ℕ`). - `h₀` restricts `n` to values ≤ 9. - `h₁` computes `11/20 * 100 + 10 * n + 7`. - **Conclusion**: - `n = 5` is derived via `omega` (likely a proof tactic for termination). #### Theorem `mathd_numbertheory_233` - **Variables**: - `b` is defined as `ZMod(11^2)`, equivalent to integers modulo 121. - `h₀` asserts `b = 24⁻¹` (modular inverse of 24 modulo 121). - **Conclusion**: - `b = 116` is proven using `exact h₀`, confirming the modular inverse relationship. ### Key Observations 1. **Modular Arithmetic**: - `ZMod(11^2)` implies operations are performed modulo 121. - `24⁻¹ ≡ 116 (mod 121)` because `24 * 116 = 2784 ≡ 1 (mod 121)`. 2. **Hypothesis Constraints**: - `h₀` in theorem 293 limits `n` to a small range (0–9), simplifying the search for solutions. 3. **Syntax Highlighting**: - Keywords (`theorem`, `h₀`, `exact`) are in red. - Variables (`n`, `b`) and values (`5`, `116`) are in blue/orange. ### Interpretation This code demonstrates formal verification of number theory results using a proof assistant (e.g., Lean or Coq). The theorems: 1. **Theorem 293**: Solves for `n` under constraints, showing `n = 5` satisfies the equation `11/20 * 100 + 10n + 7` within the hypothesis bounds. 2. **Theorem 233**: Computes the modular inverse of 24 modulo 121, yielding `116`, verified via `exact h₀`. The use of `ZMod` and modular inverses highlights applications in cryptography or cyclic group theory. The structured hypotheses and conclusions suggest a focus on automated theorem proving, where constraints (`h₀`, `h₁`) guide the proof search.