\n ## Mathematical Code Snippets: Formal Theorem Representations ### Overview The image presents three columns of mathematical code, likely generated from a formal theorem prover system (possibly related to Isabelle/HOL or similar). The code appears to define theorems and related logic, with a focus on real number topology and potentially linear algebra. The left column is labeled "LLM", the center "LLM + APOLLO", and the right column contains additional code snippets. The code is densely packed and uses a specialized syntax. ### Components/Axes There are no axes or traditional chart components. The image consists entirely of text-based code. The columns serve as organizational dividers. The code uses a consistent indentation style. The code snippets are separated by horizontal lines. ### Detailed Analysis or Content Details **Column 1: LLM** * **Header:** `import Mathlib` and `import Topology.Real.Nat` * **Theorem 1:** `theorem mathd_algebra_14_11m` * `(a b : ℝ)` * `have h1 : a + b ≥ 0 : ℝ` * `have h2 : a - b ≥ 0 : ℝ` * `have h3 : (3 * NNReal) ≤ (54 * NNReal) : b` * `have h4 : b ≤ h1 : b` * `have h5 : h1 ≤ (3 * NNReal) : b` * `have h6 : h2 ≤ h1 : b` * **Theorem 2:** `theorem mathd_algebra_14_12m` * `(a b : ℝ)` * `have h1 : a + b ≥ 0 : ℝ` * `have h2 : a - b ≥ 0 : ℝ` * `have h3 : (3 * NNReal) ≤ (54 * NNReal) : b` * `have h4 : b ≤ h1 : b` * `have h5 : h1 ≤ (3 * NNReal) : b` * `have h6 : h2 ≤ h1 : b` * **Theorem 3:** `theorem mathd_algebra_14_13m` * `(a b : ℝ)` * `have h1 : a + b ≥ 0 : ℝ` * `have h2 : a - b ≥ 0 : ℝ` * `have h3 : (3 * NNReal) ≤ (54 * NNReal) : b` * `have h4 : b ≤ h1 : b` * `have h5 : h1 ≤ (3 * NNReal) : b` * `have h6 : h2 ≤ h1 : b` * **Theorem 4:** `theorem mathd_algebra_14_14m` * `(a b : ℝ)` * `have h1 : a + b ≥ 0 : ℝ` * `have h2 : a - b ≥ 0 : ℝ` * `have h3 : (3 * NNReal) ≤ (54 * NNReal) : b` * `have h4 : b ≤ h1 : b` * `have h5 : h1 ≤ (3 * NNReal) : b` * `have h6 : h2 ≤ h1 : b` **Column 2: LLM + APOLLO** * **Header:** `import Mathlib` and `import Topology.Real.Nat` * **Theorem 1:** `theorem mathd_algebra_14_11m_apollo` * `(a b : ℝ)` * `have h1 : a + b ≥ 0 : ℝ` * `have h2 : a - b ≥ 0 : ℝ` * `have h3 : (3 * NNReal) ≤ (54 * NNReal) : b` * `have h4 : b ≤ h1 : b` * `have h5 : h1 ≤ (3 * NNReal) : b` * `have h6 : h2 ≤ h1 : b` * **Theorem 2:** `theorem mathd_algebra_14_12m_apollo` * `(a b : ℝ)` * `have h1 : a + b ≥ 0 : ℝ` * `have h2 : a - b ≥ 0 : ℝ` * `have h3 : (3 * NNReal) ≤ (54 * NNReal) : b` * `have h4 : b ≤ h1 : b` * `have h5 : h1 ≤ (3 * NNReal) : b` * `have h6 : h2 ≤ h1 : b` * **Theorem 3:** `theorem mathd_algebra_14_13m_apollo` * `(a b : ℝ)` * `have h1 : a + b ≥ 0 : ℝ` * `have h2 : a - b ≥ 0 : ℝ` * `have h3 : (3 * NNReal) ≤ (54 * NNReal) : b` * `have h4 : b ≤ h1 : b` * `have h5 : h1 ≤ (3 * NNReal) : b` * `have h6 : h2 ≤ h1 : b` * **Theorem 4:** `theorem mathd_algebra_14_14m_apollo` * `(a b : ℝ)` * `have h1 : a + b ≥ 0 : ℝ` * `have h2 : a - b ≥ 0 : ℝ` * `have h3 : (3 * NNReal) ≤ (54 * NNReal) : b` * `have h4 : b ≤ h1 : b` * `have h5 : h1 ≤ (3 * NNReal) : b` * `have h6 : h2 ≤ h1 : b` **Column 3: LLM + APOLLO** * **Code Snippets:** Contains various code blocks, including: * `cases (mul_eq_zero_mp his)` * `norm_num` * `cases (mul_eq_zero_mp his)` * `simp` * `rw [mul_comm]` * `cases (add_eq_zero_iff_eq_neg)` * `cases (mul_eq_zero_mp his)` * `simp` * `rw [mul_comm]` * `cases (add_eq_zero_iff_eq_neg)` * `cases (mul_eq_zero_mp his)` * `simp` * `rw [mul_comm]` * `cases (add_eq_zero_iff_eq_neg)` * `cases (mul_eq_zero_mp his)` * `simp` * `rw [mul_comm]` * `cases (add_eq_zero_iff_eq_neg)` ### Key Observations * The theorems in the first two columns are nearly identical, suggesting that "APOLLO" might be a refinement or extension of the "LLM" system. * The theorems in columns 1 and 2 have a repetitive structure, defining variables `a` and `b` as real numbers (`ℝ`) and establishing inequalities. * The constant `NNReal` appears frequently, potentially representing a natural number converted to a real number. * The third column contains code snippets that appear to be simplification or case analysis steps within a proof environment. * The code uses a functional style with `have` statements to introduce assumptions and `rw` (rewrite) and `simp` (simplify) commands to manipulate expressions. ### Interpretation The image showcases the output of a formal theorem prover. The code represents attempts to prove mathematical statements, likely related to properties of real numbers and inequalities. The repetition in the first two columns suggests a comparison between two different approaches or versions of a theorem prover. The third column provides a glimpse into the proof process itself, showing how the system attempts to simplify and analyze expressions to reach a conclusion. The presence of `NNReal` suggests a connection between natural numbers and real numbers, potentially in the context of constructing real numbers from natural numbers. The code is highly technical and requires expertise in formal logic and mathematical proof techniques to fully understand. The overall purpose is to rigorously verify mathematical statements using a computer-assisted proof system.