## Diagram: Comparison of Original and Synthetic Proofs in Formal Verification ### Overview The diagram contrasts an original proof structure (left) with a synthetic proof (right), illustrating steps in formal verification. It uses labeled components (G1–G5, H3–H4) to represent mathematical assertions and proof tactics. ### Components/Axes - **Original Proof (Left Side)**: - **Trimmed Sub-Tree**: Contains components G1–G4 with mathematical assertions. - **G1**: `∀a b c : nat, (a + b) + c = a + (b + c)` (associativity of addition). - **G2**: Same as G1, labeled as "induction a as [[a']]". - **G3**: `(0 + b) + c = 0 + (b + c)` (trivial case for `a = 0`). - **G4**: `(a' + b) + c = a' + (b + c)` (rewritten induction hypothesis `IHa'`). - **G5**: Trivial assertion `S (a' + (b + c)) = S (a' + (b + c))`. - **Synthetic Proof (Right Side)**: - **H3**: `(0 + b) + c = 0 + (b + c)` (trivial case). - **H4**: `∀a' : nat, (a' + b) + c = a' + (b + c) → (S a' + b) + c = S a' + (b + c)` (inductive step). - **Proof Steps**: - `induction a as [[a']].` - `auto.` (automated proof tactic). - `Qed.` (conclusion of proof). ### Detailed Analysis - **Original Proof**: - **G1–G4**: Establish associativity of addition via induction. G3 handles the base case (`a = 0`), while G4 represents the inductive hypothesis. - **G5**: Trivial equality after simplification. - **Synthetic Proof**: - **H3**: Directly asserts the base case. - **H4**: Combines the inductive hypothesis with the successor function `S` to prove the inductive step. - **Proof Tactics**: Uses `induction` to decompose `a` into cases and `auto` to discharge subgoals automatically. ### Key Observations 1. **Simplification**: The synthetic proof reduces redundancy by leveraging `auto` to handle trivial subgoals. 2. **Inductive Structure**: Both proofs rely on induction over `a`, but the synthetic version explicitly separates base and inductive cases. 3. **Notation**: The use of `S` (successor) and `→` (implication) aligns with dependent type theory conventions. ### Interpretation The diagram demonstrates how formal proofs can be streamlined using automated tactics (`auto`) and structured induction. The synthetic proof abstracts away low-level case analysis, focusing on high-level logical structure. This reflects a common pattern in proof assistants like Coq, where automation reduces manual case handling. The equivalence `(a + b) + c = a + (b + c)` is foundational in arithmetic, and the diagram highlights its role in both manual and automated verification workflows.