rweq-proofs

RwEq proofs establish rewrite equivalence by composing transitivity, congruence, and canonical lemmas within the ComputationalPaths library. Use rweq_trans to chain equivalences, rweq_tt for congruence on subterms, and canonical lemmas like rweq_cmpA_refl_left to simplify path equalities in Lean with the LND_EQ-TRS rule set.

Essential RwEq lemmas include rweq_refl for reflexivity, rweq_symm for symmetry, rweq_trans for transitivity, unit laws (rweq_cmpA_refl_left, rweq_cmpA_refl_right), inverse laws, congruence schemas (rweq_trans_congr_left, rweq_trans_congr_right), and rweq_transport_refl for transport-based reasoning in path equalities and quotient constructions.

Yes, RwEq proofs support direct transitivity chains, calc-chain notation, and nested congruence patterns. Path-tactics integrate with the LND_EQ-TRS rule set to enable efficient proof strategies, and rweq_symm_congr and rweq_congrArg_of_rweq handle symmetry and argument congruence within larger proof structures.

Use congruence lemmas like rweq_tt and rweq_congrArg_of_rweq when equality holds on subterms or function arguments rather than at the top level. Direct transitivity works for chaining established equivalences; congruence is necessary when you need to lift equality from components to composite terms.

Apply rweq_cmpA_refl_left and rweq_cmpA_refl_right to eliminate reflexive left and right compositions. For direct proof of RwEq (trans refl p) p, use path_simp or rweq_refl to discharge the goal immediately without manual transitivity chains.

RwEq proofs are executed within the ComputationalPaths library and use the LND_EQ-TRS rule set with path-tactics support. No external dependencies are listed; the Skill provides a self-contained lemma library for rewrite-equivalence reasoning in Lean formal proofs.