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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** * Int63 numbers defines Z/(2^63)Z, and can hence be equipped
with a ring structure and a ring tactic *)
Require Import Cyclic63 CyclicAxioms.
Local Open Scope int63_scope.
(** Detection of constants *)
Ltac isInt63cst t :=
match eval lazy delta [add] in (t + 1)%int63 with
| add _ _ => constr:(false)
| _ => constr:(true)
end.
Ltac Int63cst t :=
match eval lazy delta [add] in (t + 1)%int63 with
| add _ _ => constr:(NotConstant)
| _ => constr:(t)
end.
(** The generic ring structure inferred from the Cyclic structure *)
Module Int63ring := CyclicRing Int63Cyclic.
(** Unlike in the generic [CyclicRing], we can use Leibniz here. *)
Lemma Int63_canonic : forall x y, to_Z x = to_Z y -> x = y.
Proof to_Z_inj.
Lemma ring_theory_switch_eq :
forall A (R R':A->A->Prop) zero one add mul sub opp,
(forall x y : A, R x y -> R' x y) ->
ring_theory zero one add mul sub opp R ->
ring_theory zero one add mul sub opp R'.
Proof.
intros A R R' zero one add mul sub opp Impl Ring.
constructor; intros; apply Impl; apply Ring.
Qed.
Lemma Int63Ring : ring_theory 0 1 add mul sub opp Logic.eq.
Proof.
exact (ring_theory_switch_eq _ _ _ _ _ _ _ _ _ Int63_canonic Int63ring.CyclicRing).
Qed.
Lemma eq31_correct : forall x y, eqb x y = true -> x=y.
Proof. now apply eqb_spec. Qed.
Add Ring Int63Ring : Int63Ring
(decidable eq31_correct,
constants [Int63cst]).
Section TestRing.
Let test : forall x y, 1 + x*y + x*x + 1 = 1*1 + 1 + y*x + 1*x*x.
intros. ring.
Qed.
End TestRing.
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