diff options
Diffstat (limited to 'theories/Numbers/NatInt/NZDomain.v')
| -rw-r--r-- | theories/Numbers/NatInt/NZDomain.v | 40 |
1 files changed, 23 insertions, 17 deletions
diff --git a/theories/Numbers/NatInt/NZDomain.v b/theories/Numbers/NatInt/NZDomain.v index c9001db2..ffb04f08 100644 --- a/theories/Numbers/NatInt/NZDomain.v +++ b/theories/Numbers/NatInt/NZDomain.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -14,14 +14,12 @@ Require Import NZBase NZOrder NZAddOrder Plus Minus. translation from Peano numbers [nat] into NZ. *) -(** First, some complements about [nat_iter] *) +Local Notation "f ^ n" := (fun x => nat_rect _ x (fun _ => f) n). -Local Notation "f ^ n" := (nat_iter n f). - -Instance nat_iter_wd n {A} (R:relation A) : - Proper ((R==>R)==>R==>R) (nat_iter n). +Instance nat_rect_wd n {A} (R:relation A) : + Proper (R==>(R==>R)==>R) (fun x f => nat_rect (fun _ => _) x (fun _ => f) n). Proof. -intros f f' Hf. induction n; simpl; red; auto. +intros x y eq_xy f g eq_fg; induction n; [assumption | now apply eq_fg]. Qed. Module NZDomainProp (Import NZ:NZDomainSig'). @@ -33,17 +31,24 @@ Include NZBaseProp NZ. Lemma itersucc_or_itersucc n m : exists k, n == (S^k) m \/ m == (S^k) n. Proof. -nzinduct n m. +revert n. +apply central_induction with (z:=m). + { intros x y eq_xy; apply ex_iff_morphism. + intros n; apply or_iff_morphism. + + split; intros; etransitivity; try eassumption; now symmetry. + + split; intros; (etransitivity; [eassumption|]); [|symmetry]; + (eapply nat_rect_wd; [eassumption|apply succ_wd]). + } exists 0%nat. now left. intros n. split; intros [k [L|R]]. exists (Datatypes.S k). left. now apply succ_wd. destruct k as [|k]. simpl in R. exists 1%nat. left. now apply succ_wd. -rewrite nat_iter_succ_r in R. exists k. now right. +rewrite nat_rect_succ_r in R. exists k. now right. destruct k as [|k]; simpl in L. exists 1%nat. now right. apply succ_inj in L. exists k. now left. -exists (Datatypes.S k). right. now rewrite nat_iter_succ_r. +exists (Datatypes.S k). right. now rewrite nat_rect_succ_r. Qed. (** Generalized version of [pred_succ] when iterating *) @@ -53,7 +58,7 @@ Proof. induction k. simpl; auto with *. simpl; intros. apply pred_wd in H. rewrite pred_succ in H. apply IHk in H; auto. -rewrite <- nat_iter_succ_r in H; auto. +rewrite <- nat_rect_succ_r in H; auto. Qed. (** From a given point, all others are iterated successors @@ -319,7 +324,7 @@ Lemma ofnat_add : forall n m, [n+m] == [n]+[m]. Proof. intros. rewrite ofnat_add_l. induction n; simpl. reflexivity. - rewrite ofnat_succ. now f_equiv. + now f_equiv. Qed. Lemma ofnat_mul : forall n m, [n*m] == [n]*[m]. @@ -327,15 +332,15 @@ Proof. induction n; simpl; intros. symmetry. apply mul_0_l. rewrite plus_comm. - rewrite ofnat_succ, ofnat_add, mul_succ_l. + rewrite ofnat_add, mul_succ_l. now f_equiv. Qed. Lemma ofnat_sub_r : forall n m, n-[m] == (P^m) n. Proof. induction m; simpl; intros. - rewrite ofnat_zero. apply sub_0_r. - rewrite ofnat_succ, sub_succ_r. now f_equiv. + apply sub_0_r. + rewrite sub_succ_r. now f_equiv. Qed. Lemma ofnat_sub : forall n m, m<=n -> [n-m] == [n]-[m]. @@ -346,9 +351,10 @@ Proof. intros. destruct n. inversion H. - rewrite nat_iter_succ_r. + rewrite nat_rect_succ_r. simpl. - rewrite ofnat_succ, pred_succ; auto with arith. + etransitivity. apply IHm. auto with arith. + eapply nat_rect_wd; [symmetry;apply pred_succ|apply pred_wd]. Qed. End NZOfNatOps. |