diff options
Diffstat (limited to 'plugins/setoid_ring/Ncring_initial.v')
| -rw-r--r-- | plugins/setoid_ring/Ncring_initial.v | 56 |
1 files changed, 22 insertions, 34 deletions
diff --git a/plugins/setoid_ring/Ncring_initial.v b/plugins/setoid_ring/Ncring_initial.v index 3c79f7d9..528ad4f1 100644 --- a/plugins/setoid_ring/Ncring_initial.v +++ b/plugins/setoid_ring/Ncring_initial.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -27,20 +27,17 @@ Definition NotConstant := false. (** Z is a ring and a setoid*) -Lemma Zsth : Setoid_Theory Z (@eq Z). -constructor;red;intros;subst;trivial. -Qed. +Lemma Zsth : Equivalence (@eq Z). +Proof. exact Z.eq_equiv. Qed. -Instance Zops:@Ring_ops Z 0%Z 1%Z Zplus Zmult Zminus Zopp (@eq Z). +Instance Zops:@Ring_ops Z 0%Z 1%Z Z.add Z.mul Z.sub Z.opp (@eq Z). Instance Zr: (@Ring _ _ _ _ _ _ _ _ Zops). -constructor; -try (try apply Zsth; - try (unfold respectful, Proper; unfold equality; unfold eq_notation in *; - intros; try rewrite H; try rewrite H0; reflexivity)). - exact Zplus_comm. exact Zplus_assoc. - exact Zmult_1_l. exact Zmult_1_r. exact Zmult_assoc. - exact Zmult_plus_distr_l. intros; apply Zmult_plus_distr_r. exact Zminus_diag. +Proof. +constructor; try apply Zsth; try solve_proper. + exact Z.add_comm. exact Z.add_assoc. + exact Z.mul_1_l. exact Z.mul_1_r. exact Z.mul_assoc. + exact Z.mul_add_distr_r. intros; apply Z.mul_add_distr_l. exact Z.sub_diag. Defined. (*Instance ZEquality: @Equality Z:= (@eq Z).*) @@ -102,7 +99,7 @@ Ltac rsimpl := simpl. Qed. Lemma ARgen_phiPOS_Psucc : forall x, - gen_phiPOS1 (Psucc x) == 1 + (gen_phiPOS1 x). + gen_phiPOS1 (Pos.succ x) == 1 + (gen_phiPOS1 x). Proof. induction x;rsimpl;norm. rewrite IHx. gen_rewrite. add_push 1. reflexivity. @@ -112,7 +109,7 @@ Ltac rsimpl := simpl. gen_phiPOS1 (x + y) == (gen_phiPOS1 x) + (gen_phiPOS1 y). Proof. induction x;destruct y;simpl;norm. - rewrite Pplus_carry_spec. + rewrite Pos.add_carry_spec. rewrite ARgen_phiPOS_Psucc. rewrite IHx;norm. add_push (gen_phiPOS1 y);add_push 1;reflexivity. @@ -152,20 +149,13 @@ Ltac rsimpl := simpl. == gen_phiPOS1 x + -gen_phiPOS1 y. Proof. intros x y. - rewrite Z.pos_sub_spec. - assert (HH0 := Pminus_mask_Gt x y). unfold Pos.gt in HH0. - assert (HH1 := Pminus_mask_Gt y x). unfold Pos.gt in HH1. - rewrite Pos.compare_antisym in HH1. - destruct (Pos.compare_spec x y) as [HH|HH|HH]. - subst. rewrite ring_opp_def;reflexivity. - destruct HH1 as [h [HHeq1 [HHeq2 HHor]]];trivial. - unfold Pminus; rewrite HHeq1;rewrite <- HHeq2. - rewrite ARgen_phiPOS_add;simpl;norm. - rewrite ring_opp_def;norm. - destruct HH0 as [h [HHeq1 [HHeq2 HHor]]];trivial. - unfold Pminus; rewrite HHeq1;rewrite <- HHeq2. - rewrite ARgen_phiPOS_add;simpl;norm. - add_push (gen_phiPOS1 h). rewrite ring_opp_def ; norm. + generalize (Z.pos_sub_discr x y). + destruct (Z.pos_sub x y) as [|p|p]; intros; subst. + - now rewrite ring_opp_def. + - rewrite ARgen_phiPOS_add;simpl;norm. + add_push (gen_phiPOS1 p). rewrite ring_opp_def;norm. + - rewrite ARgen_phiPOS_add;simpl;norm. + rewrite ring_opp_def;norm. Qed. Lemma match_compOpp : forall x (B:Type) (be bl bg:B), @@ -206,16 +196,14 @@ Lemma gen_phiZ_opp : forall x, [- x] == - [x]. Global Instance gen_phiZ_morph : (@Ring_morphism (Z:Type) R _ _ _ _ _ _ _ Zops Zr _ _ _ _ _ _ _ _ _ gen_phiZ) . (* beurk!*) apply Build_Ring_morphism; simpl;try reflexivity. - apply gen_phiZ_add. intros. rewrite ring_sub_def. -replace (Zminus x y) with (x + (-y))%Z. rewrite gen_phiZ_add. -rewrite gen_phiZ_opp. rewrite ring_sub_def. reflexivity. + apply gen_phiZ_add. intros. rewrite ring_sub_def. +replace (x-y)%Z with (x + (-y))%Z. +now rewrite gen_phiZ_add, gen_phiZ_opp, ring_sub_def. reflexivity. - apply gen_phiZ_mul. apply gen_phiZ_opp. apply gen_phiZ_ext. + apply gen_phiZ_mul. apply gen_phiZ_opp. apply gen_phiZ_ext. Defined. End ZMORPHISM. Instance multiplication_phi_ring{R:Type}`{Ring R} : Multiplication := {multiplication x y := (gen_phiZ x) * y}. - - |