## Screenshot: Lean Code for Algebraic Theorems ### Overview The image shows a code editor displaying three formalized mathematical theorems written in the Lean theorem prover. The code defines algebraic relationships, hypotheses, and proofs using Lean's syntax. ### Components/Axes - **Theorem Definitions**: - `theorem mathd_algebra_141` - `theorem mathd_algebra_329` - `theorem mathd_algebra_547` - **Variables**: - `a, b, x, y` (declared as real numbers: `a b : ℝ`, `x y : ℝ`) - **Equations/Hypotheses**: - `h1 : (a * b) = 180` - `h2 : 2 * (a + b) = 54` - `h0 : 3 * y = x` - `h1 : 2 * x + 5 * y = 11` - `h0 : x = 5` - `h1 : y = 2` - **Proof Tactics**: - `by linarith` (linear arithmetic solver) - `by Real.sqrt_eq_iff_sq_eq` (square root equivalence) - `simp [h0, h1, sq]` (simplification with hypotheses) - `rw [Real.sqrt_eq_iff_sq_eq]` (rewrite rule) ### Content Details 1. **Theorem `mathd_algebra_141`**: - **Variables**: `a, b ∈ ℝ` - **Equations**: - `a * b = 180` - `2 * (a + b) = 54` - **Proof**: `by linarith` - **Result**: `a² + b² = 369` (derived via `by` keyword). 2. **Theorem `mathd_algebra_329`**: - **Variables**: `x, y ∈ ℝ` - **Equations**: - `3 * y = x` - `2 * x + 5 * y = 11` - **Proof**: `by linarith` - **Result**: `x + y = 4` (derived via `by` keyword). 3. **Theorem `mathd_algebra_547`**: - **Variables**: `x, y ∈ ℝ` - **Equations**: - `x = 5` - `y = 2` - **Proof**: - `Real.sqrt (x³ - 2ᵧ) = 11` - Simplified using `simp [h0, h1, sq]` - Rewritten with `rw [Real.sqrt_eq_iff_sq_eq]` - **Result**: `norm_num` (normalizes numerical expressions). ### Key Observations - All theorems use **linear arithmetic** (`linarith`) or **simplification** (`simp`) tactics to prove equations. - Theorems involve **real-number arithmetic** with explicit variable declarations. - The final theorem (`mathd_algebra_547`) combines **exponentiation** (`x³`, `2ᵧ`) and **square roots**, requiring rewriting rules for equivalence. ### Interpretation This code demonstrates formal verification of algebraic identities in Lean. The theorems: 1. Solve systems of linear equations (e.g., `a * b = 180` and `2(a + b) = 54`). 2. Validate relationships between variables (e.g., `x = 3y` and `2x + 5y = 11`). 3. Prove equivalence of expressions involving roots and exponents. The use of `linarith` and `simp` highlights Lean's automated reasoning capabilities for arithmetic and algebraic proofs. The `norm_num` tactic ensures numerical expressions are simplified to their canonical form, confirming the correctness of derived results.