## Code Comparison: LLM vs. LLM + APOLLO ### Overview The image presents a side-by-side comparison of code snippets labeled "LLM" and "LLM + APOLLO". Both snippets appear to be related to mathematical proofs or symbolic computations, likely within a formal verification or automated reasoning context. The code includes commands, definitions, and theorems, with annotations and comments. The comparison highlights differences in the approaches or optimizations used in each version. ### Components/Axes * **Headers:** * Left: "LLM" (Red background) * Right: "LLM + APOLLO" (Green background) * **Code Snippets:** Each section contains a block of code, including imports, options, theorem definitions, and proof steps. * **Annotations:** Numbered annotations "#1", "#2", "#3" appear in both code snippets, possibly indicating key steps or differences. ### Detailed Analysis or ### Content Details **LLM (Left Side - Red Background)** * **Imports:** * `import Mathlib` * `import Aesop` * **Options:** * `set_option maxHeartbeats 0` * `set_option pp.numericTypes true` * `set_option pp.coercions.types true` * **Open:** * `open BigOperators Real Nat Topology Rat` * **Theorem:** * `theorem algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7_llm (f z : C)` * `(h₀ : f + (3 : C) * z = (11 : C))` * `(h₁ : (3 : C) * (f - 1) - (5 : C) * z = (-68 : C))` * `f = (-10 : C) ∧ z = (7 : C) := by` * **Proof Steps:** * `-- Simplify the second equation using mul_sub` * `have h₂ : 3 * f - 3 - 5 * z = -68 := by` * `simpa [mul_sub] using h₁` * `-- Combine like terms to isolate 3*f - 5*z` * `have h₃ : 3 * f - 5 * z = -65 := by #1` * `linarith [h₂]` * `-- Use the two equations to solve for z` * `have z_eq : z = 7 := by #2` * `linarith [h₀, h₃]` * `-- Substitute z back to solve for f` * `have f_eq : f = -10 := by #3` * `linarith [h₀, z_eq]` * `-- Conclude both equalities` * `exact And.intro f_eq z_eq` **LLM + APOLLO (Right Side - Green Background)** * **Imports:** * `import Mathlib` * `import Aesop` * **Options:** * `set_option maxHeartbeats 0` * `set_option pp.numericTypes true` * `set_option pp.coercions.types true` * **Open:** * `open BigOperators Real Nat Topology Rat` * **Theorem:** * `theorem algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7_apollo (f z : C)` * `(h₀ : f + (3 : C) * z = (11 : C))` * `(h₁ : (3 : C) * (f - 1) - (5 : C) * z = (-68 : C))` * `f = (-10 : C) ∧ z = (7 : C) := by` * **Proof Steps:** * `have h₂ : 3 * f - 3 - 5 * z = -68 := by` * `simpa [mul_sub] using h₁` * `have h₃ : 3 * f - 5 * z = -65 := by #1` * `try norm_cast; try norm_num; try simp_all;` * `try ring_nf at *; try native_decide; try` * `linarith; try nlinarith` * `have h_expanded : (3 : C) * f - (3 : C) - (5 : C) * z = (-68 : C) := by` * `gcongr` * `have h_added : (3 : C) * f - (5 : C) * z = ((3 : C) * f - (3 : C) - (5 : C) * z) + (3 : C) := by` * `try norm_cast; try norm_num; try` * `simp_all; try ring_nf at *; try` * `native_decide; try linarith; try nlinarith` * `have h_add : f * (3 : C) - z * (5 : C) = (-68 : C) + (3 : C) := by` * `have lhs : (-3 : C) + (f * (3 : C) - z * (5 : C)) = (f * (3 : C) - z * (5 : C)) := by` * `exact neg_add_cancel_comm (3 : C) (f * (3 : C) - z * (5 : C))` * `have rhs : (-68 : C) + (3 : C) = (-68 : C) + (3 : C) := by` * `exact rfl` * `rw [← lhs] at h₁` * `simp_all only` * `have h_sum : (-68 : C) + (3 : C) = (-65 : C) := by` * `try norm_cast; try norm_num; try` * `simp_all; try ring_nf at *; try native_decide;` * `try linarith; try nlinarith` * `rw [h_add, h_sum]` * `have h_subst : (3 : C) * f - (5 : C) * z = (-68 : C) + (3 : C) := by` * `rw [h_expanded] at h_added` * `exact h_added` * `have h_simpl : (-68 : C) + (3 : C) = (-65 : C) := by` * `norm_num` * `exact Eq.trans h_subst h_simpl` * `have z_eq : z = 7 := by #2` * `try norm_cast; try norm_num; try simp_all;` * `try ring_nf at *; try native_decide;` * `try linarith; try nlinarith` * `have f_eq_def : f = (11 : C) - (3 : C) * z := by` * `exact eq_sub_of_add_eq h₀` * `have h3_sub : (3 : C) * ((11 : C) - (3 : C) * z) - (5 : C) * z = (-65 : C) := by` * `rw [f_eq_def] at h₃` * `exact h₃` * `have h3_simplified : (33 : C) - (14 : C) * z = (-65 : C) := by` * `norm_num` * `try norm_cast; try norm_num; try` * `simp_all; try ring_nf at *; try native_decide;` * `try linarith; try nlinarith` * `gcongr` * `have isolate_z : (14 : C) * z = 98 := by` * `try norm_cast; try norm_num; try` * `simp_all; try ring_nf at *; try native_decide;` * `try linarith; try nlinarith` * `have eq1 : (30 : C) = ((-68 : C) + z * (14 : C)) := by` * `rw [sub_eq_iff_eq_add] at h₁` * `gcongr` * `have eq2 : (30 : C) + (68 : C) = ((-68 : C) + (68 : C)) + z * (14 : C) := by` * `simp_all only [add_sub_cancel_right, neg_add_cancel_comm, neg_add_cancel, zero_add]` * `have eq3 : (98 : C) = (0 : C) + z * (14 : C) := by` * `norm_num at eq2` * `simp_all only [add_sub_cancel_right, zero_add]` * `have eq4 : (0 : C) + z * (14 : C) = z * (14 : C) := by` * `simp` * `have eq_final : (98 : C) = z * (14 : C) := by` * `simp_all only [add_sub_cancel_right, neg_add_cancel_comm, neg_add_cancel, zero_add]` * `gcongr` * `have z_sol : z = (7 : C) := by` * `try norm_cast; try norm_num; try simp_all;` * `try ring_nf at *; try native_decide; try` * `linarith; try nlinarith` * `have h_nonzero : (14 : C) ≠ 0 := by` * `norm_num` * `have z_inv : z = 98 * ((14 : C)⁻¹) := by` * `exact (eq_mul_inv_iff_mul_eq₀ h_nonzero).mpr isolate_z` * `have z_div : z = 98 / (14 : C) := by` * `exact z_inv` * `have z_simpl : 98 / (14 : C) = (7 : C) := by` * `norm_num` * `rw [z_div, z_simpl]` * `exact z_sol` * `have f_eq : f = -10 := by #3` * `try norm_cast; try norm_num; try simp_all;` * `try ring_nf at *; try native_decide; try` * `linarith; try nlinarith` * `have h_f : f = (11 : C) - (21 : C) := by` * `exact eq_sub_of_add_eq' h₀` * `have h_num : (11 : C) - (21 : C) = (-10 : C) := by` * `norm_num` * `rw [h_f, h_num]` * `exact And.intro f_eq z_eq` ### Key Observations * **Similarities:** Both code snippets address the same theorem and initial setup (imports, options, theorem statement). They both aim to prove `f = -10` and `z = 7`. * **Differences:** The "LLM + APOLLO" version includes more explicit and detailed proof steps, using tactics like `try norm_cast`, `try norm_num`, `try simp_all`, `try ring_nf at *`, `try native_decide`, `try linarith`, and `try nlinarith`. It also introduces intermediate lemmas (e.g., `h_expanded`, `h_added`, `h_add`, `h_subst`, `isolate_z`, `eq1`, `eq2`, `eq3`, `eq4`, `eq_final`, `z_sol`, `z_inv`, `z_div`, `z_simpl`, `h_f`, `h_num`) to break down the proof into smaller, more manageable steps. The LLM version uses `linarith` more directly. * **Annotations:** The annotations `#1`, `#2`, and `#3` likely mark corresponding steps in both proofs where significant transformations or deductions occur. ### Interpretation The "LLM + APOLLO" version likely represents an enhanced or more explicit proof strategy compared to the "LLM" version. The additional proof steps and tactics in "LLM + APOLLO" suggest a more robust or automated approach, possibly designed to be more easily verified or adapted to similar problems. The "LLM" version might be a more concise or human-readable proof, while "LLM + APOLLO" prioritizes rigor and automation. The use of tactics like `try norm_cast` and `try linarith` indicates an attempt to handle potential type casting and linear arithmetic automatically. The APOLLO addition seems to be adding more explicit steps to guide the proof assistant.