diff options
Diffstat (limited to 'theories/Arith/PeanoNat.v')
| -rw-r--r-- | theories/Arith/PeanoNat.v | 40 |
1 files changed, 31 insertions, 9 deletions
diff --git a/theories/Arith/PeanoNat.v b/theories/Arith/PeanoNat.v index e3240bb7..bc58995f 100644 --- a/theories/Arith/PeanoNat.v +++ b/theories/Arith/PeanoNat.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) @@ -313,7 +315,7 @@ Import Private_Parity. Lemma even_spec : forall n, even n = true <-> Even n. Proof. - fix 1. + fix even_spec 1. destruct n as [|[|n]]; simpl. - split; [ now exists 0 | trivial ]. - split; [ discriminate | intro H; elim (Even_1 H) ]. @@ -323,7 +325,7 @@ Qed. Lemma odd_spec : forall n, odd n = true <-> Odd n. Proof. unfold odd. - fix 1. + fix odd_spec 1. destruct n as [|[|n]]; simpl. - split; [ discriminate | intro H; elim (Odd_0 H) ]. - split; [ now exists 0 | trivial ]. @@ -471,7 +473,7 @@ Notation "( x | y )" := (divide x y) (at level 0) : nat_scope. Lemma gcd_divide : forall a b, (gcd a b | a) /\ (gcd a b | b). Proof. - fix 1. + fix gcd_divide 1. intros [|a] b; simpl. split. now exists 0. @@ -500,7 +502,7 @@ Qed. Lemma gcd_greatest : forall a b c, (c|a) -> (c|b) -> (c|gcd a b). Proof. - fix 1. + fix gcd_greatest 1. intros [|a] b; simpl; auto. fold (b mod (S a)). intros c H H'. apply gcd_greatest; auto. @@ -534,7 +536,7 @@ Qed. Lemma le_div2 n : div2 (S n) <= n. Proof. revert n. - fix 1. + fix le_div2 1. destruct n; simpl; trivial. apply lt_succ_r. destruct n; [simpl|]; trivial. now constructor. Qed. @@ -724,6 +726,26 @@ Definition shiftr_spec a n m (_:0<=m) := shiftr_specif a n m. Include NExtraProp. +(** Properties of tail-recursive addition and multiplication *) + +Lemma tail_add_spec n m : tail_add n m = n + m. +Proof. + revert m. induction n as [|n IH]; simpl; trivial. + intros. now rewrite IH, add_succ_r. +Qed. + +Lemma tail_addmul_spec r n m : tail_addmul r n m = r + n * m. +Proof. + revert m r. induction n as [| n IH]; simpl; trivial. + intros. rewrite IH, tail_add_spec. + rewrite add_assoc. f_equal. apply add_comm. +Qed. + +Lemma tail_mul_spec n m : tail_mul n m = n * m. +Proof. + unfold tail_mul. now rewrite tail_addmul_spec. +Qed. + End Nat. (** Re-export notations that should be available even when |