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+(************************************************************************)
+(* v * The Coq Proof Assistant / The Coq Development Team *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* \VV/ **************************************************************)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(************************************************************************)
+
+(** * Int63 numbers defines Z/(2^63)Z, and can hence be equipped
+ with a ring structure and a ring tactic *)
+
+Require Import Int63Lib Cyclic63 CyclicAxioms.
+
+Local Open Scope int63_scope.
+
+(** Detection of constants *)
+
+Local Open Scope list_scope.
+
+Ltac isInt63cst_lst l :=
+ match l with
+ | nil => constr:true
+ | ?t::?l => match t with
+ | D1 => isInt63cst_lst l
+ | D0 => isInt63cst_lst l
+ | _ => constr:false
+ end
+ | _ => constr:false
+ end.
+
+Ltac isInt63cst t :=
+ match t with
+ | I63 ?i0 ?i1 ?i2 ?i3 ?i4 ?i5 ?i6 ?i7 ?i8 ?i9 ?i10
+ ?i11 ?i12 ?i13 ?i14 ?i15 ?i16 ?i17 ?i18 ?i19 ?i20
+ ?i21 ?i22 ?i23 ?i24 ?i25 ?i26 ?i27 ?i28 ?i29 ?i30
+ ?i31 ?i32 ?i33 ?i34 ?i35 ?i36 ?i37 ?i38 ?i39 ?i40
+ ?i41 ?i42 ?i43 ?i44 ?i45 ?i46 ?i47 ?i48 ?i49 ?i50
+ ?i51 ?i52 ?i53 ?i54 ?i55 ?i56 ?i57 ?i58 ?i59 ?i60
+ ?i61 ?i62 =>
+ let l :=
+ constr:(i0::i1::i2::i3::i4::i5::i6::i7::i8::i9::i10
+ ::i11::i12::i13::i14::i15::i16::i17::i18::i19::i20
+ ::i21::i22::i23::i24::i25::i26::i27::i28::i29::i30
+ ::i31::i32::i33::i34::i35::i36::i37::i38::i39::i40
+ ::i41::i42::i43::i44::i45::i46::i47::i48::i49::i50
+ ::i51::i52::i53::i54::i55::i56::i57::i58::i59::i60
+ ::i61::i62::nil)
+ in isInt63cst_lst l
+ | Int63Lib.On => constr:true
+ | Int63Lib.In => constr:true
+ | Int63Lib.Tn => constr:true
+ | Int63Lib.Twon => constr:true
+ | _ => constr:false
+ end.
+
+Ltac Int63cst t :=
+ match isInt63cst t with
+ | true => constr:t
+ | false => constr:NotConstant
+ 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, phi x = phi y -> x = y.
+Proof.
+ intros x y EQ.
+ now rewrite <- (phi_inv_phi x), <- (phi_inv_phi y), EQ.
+Qed.
+
+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 add63 mul63 sub63 opp63 Logic.eq.
+Proof.
+exact (ring_theory_switch_eq _ _ _ _ _ _ _ _ _ Int63_canonic Int63ring.CyclicRing).
+Qed.
+
+Lemma eqb63_eq : forall x y, eqb63 x y = true <-> x=y.
+Proof.
+unfold eqb63. intros x y.
+rewrite Cyclic63.spec_compare. case Z.compare_spec.
+intuition. apply Int63_canonic; auto.
+intuition; subst; auto with zarith; try discriminate.
+intuition; subst; auto with zarith; try discriminate.
+Qed.
+
+Lemma eqb63_correct : forall x y, eqb63 x y = true -> x=y.
+Proof. now apply eqb63_eq. Qed.
+
+Add Ring Int63Ring : Int63Ring
+ (decidable eqb63_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.
+
+Let test2 : forall x, (x - 1) + 1 = x.
+intros. ring.
+Qed.
+End TestRing.
+