diff options
Diffstat (limited to 'theories/Numbers')
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v | 2 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v | 3 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/Int31/Cyclic31.v | 43 | ||||
| -rw-r--r-- | theories/Numbers/NatInt/NZDomain.v | 38 | ||||
| -rw-r--r-- | theories/Numbers/Natural/BigN/Nbasic.v | 2 |
5 files changed, 46 insertions, 42 deletions
diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v index ed69a8f57..fe77bf5c7 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v @@ -287,7 +287,7 @@ Section DoubleBase. Lemma double_wB_wwB : forall n, double_wB n * double_wB n = double_wB (S n). Proof. intros n;unfold double_wB;simpl. - unfold base. rewrite Pshiftl_nat_S, (Pos2Z.inj_xO (_ << _)). + unfold base. rewrite (Pos2Z.inj_xO (_ << _)). replace (2 * Zpos (w_digits << n)) with (Zpos (w_digits << n) + Zpos (w_digits << n)) by ring. symmetry; apply Zpower_exp;intro;discriminate. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v index 5cb7405a6..23cbd1e8c 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v @@ -160,7 +160,7 @@ Section GENDIVN1. Lemma p_lt_double_digits : forall n, [|p|] <= Zpos (w_digits << n). Proof. induction n;simpl. trivial. - case (spec_to_Z p); rewrite Pshiftl_nat_S, Pos2Z.inj_xO;auto with zarith. + case (spec_to_Z p); rewrite Pos2Z.inj_xO;auto with zarith. Qed. Lemma spec_double_divn1_p : forall n r h l, @@ -305,7 +305,6 @@ Section GENDIVN1. Lemma spec_double_digits:forall n, Zpos w_digits <= Zpos (w_digits << n). Proof. induction n;simpl;auto with zarith. - rewrite Pshiftl_nat_S. change (Zpos (xO (w_digits << n))) with (2*Zpos (w_digits << n)). assert (0 < Zpos w_digits) by reflexivity. diff --git a/theories/Numbers/Cyclic/Int31/Cyclic31.v b/theories/Numbers/Cyclic/Int31/Cyclic31.v index 0284af7aa..5aa31d7bd 100644 --- a/theories/Numbers/Cyclic/Int31/Cyclic31.v +++ b/theories/Numbers/Cyclic/Int31/Cyclic31.v @@ -86,14 +86,14 @@ Section Basics. Lemma nshiftr_S_tail : forall n x, nshiftr (S n) x = nshiftr n (shiftr x). Proof. - induction n; simpl; auto. - intros; rewrite nshiftr_S, IHn, nshiftr_S; auto. + intros n; elim n; simpl; auto. + intros; now f_equal. Qed. Lemma nshiftr_n_0 : forall n, nshiftr n 0 = 0. Proof. induction n; simpl; auto. - rewrite nshiftr_S, IHn; auto. + rewrite IHn; auto. Qed. Lemma nshiftr_size : forall x, nshiftr size x = 0. @@ -108,7 +108,7 @@ Section Basics. replace k with ((k-size)+size)%nat by omega. induction (k-size)%nat; auto. rewrite nshiftr_size; auto. - simpl; rewrite nshiftr_S, IHn; auto. + simpl; rewrite IHn; auto. Qed. (** * Iterated shift to the left *) @@ -124,14 +124,13 @@ Section Basics. Lemma nshiftl_S_tail : forall n x, nshiftl (S n) x = nshiftl n (shiftl x). Proof. - induction n; simpl; auto. - intros; rewrite nshiftl_S, IHn, nshiftl_S; auto. + intros n; elim n; simpl; intros; now f_equal. Qed. Lemma nshiftl_n_0 : forall n, nshiftl n 0 = 0. Proof. induction n; simpl; auto. - rewrite nshiftl_S, IHn; auto. + rewrite IHn; auto. Qed. Lemma nshiftl_size : forall x, nshiftl size x = 0. @@ -146,7 +145,7 @@ Section Basics. replace k with ((k-size)+size)%nat by omega. induction (k-size)%nat; auto. rewrite nshiftl_size; auto. - simpl; rewrite nshiftl_S, IHn; auto. + simpl; rewrite IHn; auto. Qed. Lemma firstr_firstl : @@ -176,7 +175,7 @@ Section Basics. replace p with ((p-n)+n)%nat by omega. induction (p-n)%nat. simpl; auto. - simpl; rewrite nshiftr_S; rewrite IHn0; auto. + simpl; rewrite IHn0; auto. Qed. Lemma nshiftr_0_firstl : forall n x, n < size -> @@ -240,7 +239,7 @@ Section Basics. recr_aux p A case0 caserec (nshiftr (size - n) x). Proof. induction n. - simpl; intros. + simpl minus; intros. rewrite nshiftr_size; destruct p; simpl; auto. intros. destruct p. @@ -439,7 +438,7 @@ Section Basics. (phibis_aux n (nshiftr (size-n) x) < 2 ^ (Z.of_nat n))%Z. Proof. induction n. - simpl; unfold phibis_aux; simpl; auto with zarith. + simpl minus; unfold phibis_aux; simpl; auto with zarith. intros. unfold phibis_aux, recrbis_aux; fold recrbis_aux; fold (phibis_aux n (shiftr (nshiftr (size - S n) x))). @@ -529,7 +528,7 @@ Section Basics. remember (k'-k)%nat as n. clear Heqn H k'. induction n; simpl; auto. - rewrite 2 nshiftl_S; f_equal; auto. + f_equal; auto. Qed. Lemma EqShiftL_firstr : forall k x y, k < size -> @@ -823,7 +822,7 @@ Section Basics. nshiftr (size-n) x. Proof. induction n. - intros; simpl. + intros; simpl minus. rewrite nshiftr_size; auto. intros. unfold phibis_aux, recrbis_aux; fold recrbis_aux; @@ -879,12 +878,12 @@ Section Basics. Proof. induction n. simpl; intros; auto. - simpl; intros. - destruct p; simpl. + simpl p2ibis; intros. + destruct p; simpl snd. specialize IHn with p. - destruct (p2ibis n p); simpl in *. - rewrite nshiftr_S_tail. + destruct (p2ibis n p). simpl snd in *. +rewrite nshiftr_S_tail. destruct (le_lt_dec size n). rewrite nshiftr_above_size; auto. assert (H:=nshiftr_0_firstl _ _ l IHn). @@ -892,7 +891,7 @@ Section Basics. destruct i; simpl in *; rewrite H; auto. specialize IHn with p. - destruct (p2ibis n p); simpl in *. + destruct (p2ibis n p); simpl snd in *. rewrite nshiftr_S_tail. destruct (le_lt_dec size n). rewrite nshiftr_above_size; auto. @@ -1525,14 +1524,14 @@ Section Int31_Specs. unfold phibis_aux; simpl; rewrite H; fold (phibis_aux n (shiftr i)); generalize (phibis_aux_pos n (shiftr i)); intros; set (z := phibis_aux n (shiftr i)) in *; clearbody z; - rewrite <- iter_nat_plus. + rewrite <- nat_rect_plus. f_equal. rewrite Z.double_spec, <- Z.add_diag. symmetry; apply Zabs2Nat.inj_add; auto with zarith. - change (iter_nat (S (Z.abs_nat z + Z.abs_nat z)) A f a = - iter_nat (Z.abs_nat (Z.succ_double z)) A f a); f_equal. + change (iter_nat (S (Z.abs_nat z) + (Z.abs_nat z))%nat A f a = + iter_nat (Z.abs_nat (Z.succ_double z)) A f a); f_equal. rewrite Z.succ_double_spec, <- Z.add_diag. rewrite Zabs2Nat.inj_add; auto with zarith. rewrite Zabs2Nat.inj_add; auto with zarith. @@ -1557,7 +1556,7 @@ Section Int31_Specs. intros. simpl addmuldiv31_alt. replace (S n) with (n+1)%nat by (rewrite plus_comm; auto). - rewrite iter_nat_plus; simpl; auto. + rewrite nat_rect_plus; simpl; auto. Qed. Lemma spec_add_mul_div : forall x y p, [|p|] <= Zpos 31 -> diff --git a/theories/Numbers/NatInt/NZDomain.v b/theories/Numbers/NatInt/NZDomain.v index 4b71d5390..cee2e3210 100644 --- a/theories/Numbers/NatInt/NZDomain.v +++ b/theories/Numbers/NatInt/NZDomain.v @@ -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. diff --git a/theories/Numbers/Natural/BigN/Nbasic.v b/theories/Numbers/Natural/BigN/Nbasic.v index 5bde10087..161f523ca 100644 --- a/theories/Numbers/Natural/BigN/Nbasic.v +++ b/theories/Numbers/Natural/BigN/Nbasic.v @@ -371,7 +371,7 @@ Section CompareRec. intros n (H0, H); split; auto. apply Z.lt_le_trans with (1:= H). unfold double_wB, DoubleBase.double_wB; simpl. - rewrite Pshiftl_nat_S, base_xO. + rewrite base_xO. set (u := base (Pos.shiftl_nat wm_base n)). assert (0 < u). unfold u, base; auto with zarith. |