diff options
Diffstat (limited to 'theories/PArith/Pnat.v')
| -rw-r--r-- | theories/PArith/Pnat.v | 81 |
1 files changed, 41 insertions, 40 deletions
diff --git a/theories/PArith/Pnat.v b/theories/PArith/Pnat.v index f9df70bd..31e88a40 100644 --- a/theories/PArith/Pnat.v +++ b/theories/PArith/Pnat.v @@ -1,7 +1,7 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* 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 *) @@ -30,7 +30,7 @@ Qed. Theorem inj_add p q : to_nat (p + q) = to_nat p + to_nat q. Proof. - revert q. induction p using Pind; intros q. + revert q. induction p using peano_ind; intros q. now rewrite add_1_l, inj_succ. now rewrite add_succ_l, !inj_succ, IHp. Qed. @@ -378,55 +378,56 @@ End SuccNat2Pos. (** For compatibility, old names and old-style lemmas *) -Notation Psucc_S := Pos2Nat.inj_succ (only parsing). -Notation Pplus_plus := Pos2Nat.inj_add (only parsing). -Notation Pmult_mult := Pos2Nat.inj_mul (only parsing). -Notation Pcompare_nat_compare := Pos2Nat.inj_compare (only parsing). -Notation nat_of_P_xH := Pos2Nat.inj_1 (only parsing). -Notation nat_of_P_xO := Pos2Nat.inj_xO (only parsing). -Notation nat_of_P_xI := Pos2Nat.inj_xI (only parsing). -Notation nat_of_P_is_S := Pos2Nat.is_succ (only parsing). -Notation nat_of_P_pos := Pos2Nat.is_pos (only parsing). -Notation nat_of_P_inj_iff := Pos2Nat.inj_iff (only parsing). -Notation nat_of_P_inj := Pos2Nat.inj (only parsing). -Notation Plt_lt := Pos2Nat.inj_lt (only parsing). -Notation Pgt_gt := Pos2Nat.inj_gt (only parsing). -Notation Ple_le := Pos2Nat.inj_le (only parsing). -Notation Pge_ge := Pos2Nat.inj_ge (only parsing). -Notation Pminus_minus := Pos2Nat.inj_sub (only parsing). -Notation iter_nat_of_P := @Pos2Nat.inj_iter (only parsing). - -Notation nat_of_P_of_succ_nat := SuccNat2Pos.id_succ (only parsing). -Notation P_of_succ_nat_of_P := Pos2SuccNat.id_succ (only parsing). - -Notation nat_of_P_succ_morphism := Pos2Nat.inj_succ (only parsing). -Notation nat_of_P_plus_morphism := Pos2Nat.inj_add (only parsing). -Notation nat_of_P_mult_morphism := Pos2Nat.inj_mul (only parsing). -Notation nat_of_P_compare_morphism := Pos2Nat.inj_compare (only parsing). -Notation lt_O_nat_of_P := Pos2Nat.is_pos (only parsing). -Notation ZL4 := Pos2Nat.is_succ (only parsing). -Notation nat_of_P_o_P_of_succ_nat_eq_succ := SuccNat2Pos.id_succ (only parsing). -Notation P_of_succ_nat_o_nat_of_P_eq_succ := Pos2SuccNat.id_succ (only parsing). -Notation pred_o_P_of_succ_nat_o_nat_of_P_eq_id := Pos2SuccNat.pred_id (only parsing). +Notation Psucc_S := Pos2Nat.inj_succ (compat "8.3"). +Notation Pplus_plus := Pos2Nat.inj_add (compat "8.3"). +Notation Pmult_mult := Pos2Nat.inj_mul (compat "8.3"). +Notation Pcompare_nat_compare := Pos2Nat.inj_compare (compat "8.3"). +Notation nat_of_P_xH := Pos2Nat.inj_1 (compat "8.3"). +Notation nat_of_P_xO := Pos2Nat.inj_xO (compat "8.3"). +Notation nat_of_P_xI := Pos2Nat.inj_xI (compat "8.3"). +Notation nat_of_P_is_S := Pos2Nat.is_succ (compat "8.3"). +Notation nat_of_P_pos := Pos2Nat.is_pos (compat "8.3"). +Notation nat_of_P_inj_iff := Pos2Nat.inj_iff (compat "8.3"). +Notation nat_of_P_inj := Pos2Nat.inj (compat "8.3"). +Notation Plt_lt := Pos2Nat.inj_lt (compat "8.3"). +Notation Pgt_gt := Pos2Nat.inj_gt (compat "8.3"). +Notation Ple_le := Pos2Nat.inj_le (compat "8.3"). +Notation Pge_ge := Pos2Nat.inj_ge (compat "8.3"). +Notation Pminus_minus := Pos2Nat.inj_sub (compat "8.3"). +Notation iter_nat_of_P := @Pos2Nat.inj_iter (compat "8.3"). + +Notation nat_of_P_of_succ_nat := SuccNat2Pos.id_succ (compat "8.3"). +Notation P_of_succ_nat_of_P := Pos2SuccNat.id_succ (compat "8.3"). + +Notation nat_of_P_succ_morphism := Pos2Nat.inj_succ (compat "8.3"). +Notation nat_of_P_plus_morphism := Pos2Nat.inj_add (compat "8.3"). +Notation nat_of_P_mult_morphism := Pos2Nat.inj_mul (compat "8.3"). +Notation nat_of_P_compare_morphism := Pos2Nat.inj_compare (compat "8.3"). +Notation lt_O_nat_of_P := Pos2Nat.is_pos (compat "8.3"). +Notation ZL4 := Pos2Nat.is_succ (compat "8.3"). +Notation nat_of_P_o_P_of_succ_nat_eq_succ := SuccNat2Pos.id_succ (compat "8.3"). +Notation P_of_succ_nat_o_nat_of_P_eq_succ := Pos2SuccNat.id_succ (compat "8.3"). +Notation pred_o_P_of_succ_nat_o_nat_of_P_eq_id := Pos2SuccNat.pred_id (compat "8.3"). Lemma nat_of_P_minus_morphism p q : - Pcompare p q Eq = Gt -> Pos.to_nat (p - q) = Pos.to_nat p - Pos.to_nat q. -Proof (fun H => Pos2Nat.inj_sub p q (ZC1 _ _ H)). + Pos.compare_cont p q Eq = Gt -> + Pos.to_nat (p - q) = Pos.to_nat p - Pos.to_nat q. +Proof (fun H => Pos2Nat.inj_sub p q (Pos.gt_lt _ _ H)). Lemma nat_of_P_lt_Lt_compare_morphism p q : - Pcompare p q Eq = Lt -> Pos.to_nat p < Pos.to_nat q. + Pos.compare_cont p q Eq = Lt -> Pos.to_nat p < Pos.to_nat q. Proof (proj1 (Pos2Nat.inj_lt p q)). Lemma nat_of_P_gt_Gt_compare_morphism p q : - Pcompare p q Eq = Gt -> Pos.to_nat p > Pos.to_nat q. + Pos.compare_cont p q Eq = Gt -> Pos.to_nat p > Pos.to_nat q. Proof (proj1 (Pos2Nat.inj_gt p q)). Lemma nat_of_P_lt_Lt_compare_complement_morphism p q : - Pos.to_nat p < Pos.to_nat q -> Pcompare p q Eq = Lt. + Pos.to_nat p < Pos.to_nat q -> Pos.compare_cont p q Eq = Lt. Proof (proj2 (Pos2Nat.inj_lt p q)). Definition nat_of_P_gt_Gt_compare_complement_morphism p q : - Pos.to_nat p > Pos.to_nat q -> Pcompare p q Eq = Gt. + Pos.to_nat p > Pos.to_nat q -> Pos.compare_cont p q Eq = Gt. Proof (proj2 (Pos2Nat.inj_gt p q)). (** Old intermediate results about [Pmult_nat] *) @@ -445,7 +446,7 @@ Proof. Qed. Lemma Pmult_nat_succ_morphism : - forall p n, Pmult_nat (Psucc p) n = n + Pmult_nat p n. + forall p n, Pmult_nat (Pos.succ p) n = n + Pmult_nat p n. Proof. intros. now rewrite !Pmult_nat_mult, Pos2Nat.inj_succ. Qed. @@ -457,7 +458,7 @@ Proof. Qed. Theorem Pmult_nat_plus_carry_morphism : - forall p q n, Pmult_nat (Pplus_carry p q) n = n + Pmult_nat (p + q) n. + forall p q n, Pmult_nat (Pos.add_carry p q) n = n + Pmult_nat (p + q) n. Proof. intros. now rewrite Pos.add_carry_spec, Pmult_nat_succ_morphism. Qed. |