https://github.com/EasyCrypt/easycrypt
Tip revision: fc07a369c06d62559c2e8cf00fd6ec6f2e8f4a3a authored by Pierre-Yves Strub on 29 March 2022, 08:01:08 UTC
License change: CeCILL B/C -> MIT
License change: CeCILL B/C -> MIT
Tip revision: fc07a36
Monoid.eca
require import Int.
(* -------------------------------------------------------------------- *)
type t.
op idm : t.
op (+) : t -> t -> t.
theory Axioms.
axiom nosmt addmA: associative Self.(+).
axiom nosmt addmC: commutative Self.(+).
axiom nosmt add0m: left_id idm Self.(+).
end Axioms.
(* -------------------------------------------------------------------- *)
lemma addmA: associative Self.(+).
proof. by apply/Axioms.addmA. qed.
lemma addmC: commutative Self.(+).
proof. by apply/Axioms.addmC. qed.
lemma add0m: left_id idm Self.(+).
proof. by apply/Axioms.add0m. qed.
lemma addm0: right_id idm Self.(+).
proof. by move=> x; rewrite addmC add0m. qed.
lemma addmCA: left_commutative Self.(+).
proof. by move=> x y z; rewrite !addmA (addmC x). qed.
lemma addmAC: right_commutative Self.(+).
proof. by move=> x y z; rewrite -!addmA (addmC y). qed.
lemma addmACA: interchange Self.(+) Self.(+).
proof. by move=> x y z t; rewrite -!addmA (addmCA y). qed.
lemma iteropE n x: iterop n Self.(+) x idm = iter n ((+) x) idm.
proof.
elim/natcase n => [n le0_n|n ge0_n].
+ by rewrite ?(iter0, iterop0).
+ by rewrite iterSr // addm0 iteropS.
qed.