prove-plus-comm

Inductive proofs for addition break into base and inductive cases. Use lemmas like plus_n_O (addition by zero) and plus_n_Sm (addition by successor) with tactics such as simpl and rewrite to discharge goals step by step, then verify the complete proof compiles in Coq.

plus_n_O and plus_n_Sm are standard arithmetic lemmas in Coq's Arith library. plus_n_O proves n + 0 = n; plus_n_Sm proves n + S m = S (n + m). They simplify addition goals during inductive proof steps.

Prove addition commutativity by induction on one argument. Apply plus_n_O in the base case and the inductive hypothesis with plus_n_Sm in the inductive case. Use rewrite and simpl tactics to reduce goals, then verify compilation succeeds.

Yes. This workflow applies to natural-number arithmetic proofs in any Coq project using the Arith library. It guides tactic selection and lemma application for base and inductive steps on recursive numeric properties.

Essential tactics include simpl (simplification), rewrite (lemma substitution), and IH (inductive hypothesis application). These tactics work together to reduce proof goals after lemma selection and support compilation verification.

Verification occurs through Coq compilation. After applying lemmas and tactics to complete base and inductive cases, compile the project. Successful compilation confirms the proof is valid and syntactically sound.