diff options
Diffstat (limited to 'theories/PArith/Pnat.v')
| -rw-r--r-- | theories/PArith/Pnat.v | 19 |
1 files changed, 10 insertions, 9 deletions
diff --git a/theories/PArith/Pnat.v b/theories/PArith/Pnat.v index 0fa6972b7..e9c77b05d 100644 --- a/theories/PArith/Pnat.v +++ b/theories/PArith/Pnat.v @@ -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. @@ -410,23 +410,24 @@ 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. |