Tactics.md

Brief documentation of the tactics in the Hahn library

Tactics to solve a goal

• done solves trivial goals (adapted from SSReflect).
• edone solves trivial goals with eassumption or done.
• by tac is a shorthand for tac; done.
• eby tac is a shorthand for tac; edone.
• vauto tries to solve the goal by up to two nested econstructor calls.
• basic_solver solves simple goals containing relational operators.

Generic simplification tactics

• simpls is shorthand for simpl in *; try done.
• ins is shorthand for intros; simpls.
• clarify simplifies the goal by simple injections/rewrites.
• clarsimp is similar, but also does a few further rewrites.
• clarassoc does further rewrites with associativity lemmas.
• desc destructs conjunctions and existentials in hypotheses.
• des also destructs disjunctions.
• desf also performs case splits on conditionals/matches.
• des_eqrefl simplifies weird-looking dependent matches generated by Program.

Specialized simplification tactics

• clarify_not simplifies negated hypotheses.
• in_simp simplifies In x list assumptions.
• rels simplifies goals with relational operators.
• relsf also does case splits.

Forward reasoning

• tertium_non_datur X [as Pattern] does a case split on X \/ ~X.
• forward eapply H [with ...] as Name generates subgoals for all the premises of H; the final subgoal names the conclusion of H as Name.
• exploit H is similar to forward eapply H (adapted from CompCert, deprecated).

Backward reasoning

• splits repeatedly applies split to conjunctions.
• hahn_frame/hahn_frame_l/hahn_frame_r try to apply the frame rule on goals with relational operators.

Rewriting support for relations

• hahn_rewrite H rewrites with a relational inclusion/equivalence lemma
• seq_rewrite H rewrites with an equivalence lemma up to associativity of seq
• sin_rewrite H rewrites with an inclusion lemma up to associativity of seq
• arewrite EQ generates a subgoal to prove EQ and seq/sin-rewrites with EQ
• arewrite_false term is shorthand for arewrite (term ≡ ∅₂)
• arewrite_id term is shorthand for arewrite (term ⊆ ⦗fun _ => True⦘)
• case_union x y case splits on relations of the form _ ⨾ (x ∪ y) ⨾ _