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Diffstat (limited to 'src/versions/standard/Int63/Ring63_standard.v')
-rw-r--r-- | src/versions/standard/Int63/Ring63_standard.v | 114 |
1 files changed, 114 insertions, 0 deletions
diff --git a/src/versions/standard/Int63/Ring63_standard.v b/src/versions/standard/Int63/Ring63_standard.v new file mode 100644 index 0000000..7b9fc1f --- /dev/null +++ b/src/versions/standard/Int63/Ring63_standard.v @@ -0,0 +1,114 @@ +(************************************************************************) +(* 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. + |