## Code Snippet: Mathematical Theorems ### Overview The image presents a code snippet containing three mathematical theorems, labeled `mathd_algebra_141`, `mathd_algebra_329`, and `mathd_algebra_547`. Each theorem defines variables, provides hypotheses, and states a conclusion, followed by a proof strategy. ### Components/Axes The code snippet is structured as follows: - **Theorem Declaration:** `theorem mathd_algebra_XXX` (where XXX is a number). - **Variable Declaration:** `(a b : R)` or `(x y : R)`, indicating variables `a` and `b` or `x` and `y` are real numbers. - **Hypotheses:** `(h0: ...)` and `(h1: ...)` and `(h2: ...)` define assumptions. - **Conclusion:** An equation to be proven. - **Proof Strategy:** `:= by ...` indicates the method used to prove the conclusion. ### Detailed Analysis or ### Content Details **Theorem mathd_algebra_141:** - Variables: `a`, `b` are real numbers. - Hypotheses: - `h1: (a * b) = 180` - `h2: 2 * (a + b) = 54` - Conclusion: `(a^2 + b^2) = 369` - Proof Strategy: `nlinarith` **Theorem mathd_algebra_329:** - Variables: `x`, `y` are real numbers. - Hypotheses: - `h0: 3 * y = x` - `h1: 2 * x + 5 * y = 11` - Conclusion: `x + y = 4` - Proof Strategy: `linarith` **Theorem mathd_algebra_547:** - Variables: `x`, `y` are real numbers. - Hypotheses: - `h0: x = 5` - `h1: y = 2` - Conclusion: `Real.sqrt (x^3 - 2^y) = 11` - Proof Strategy: - `simp [h0, h1, sq]` - `rw [Real.sqrt_eq_iff_sq_eq] <;> norm_num` ### Key Observations - Each theorem involves real number variables and algebraic equations. - The proof strategies vary, including `nlinarith`, `linarith`, `simp`, `rw`, and `norm_num`. - The theorems appear to be designed for automated theorem proving, given the specific syntax and proof strategies. ### Interpretation The code snippet demonstrates the formalization of mathematical theorems in a language suitable for automated reasoning. The theorems are relatively simple algebraic problems, likely used as examples or test cases for a theorem prover. The use of tactics like `nlinarith` and `linarith` suggests the system is capable of automatically solving linear and non-linear arithmetic problems. The `simp` and `rw` tactics indicate simplification and rewriting capabilities, while `norm_num` likely normalizes numerical expressions. The overall structure suggests a system designed for verifying mathematical proofs.