diff options
Diffstat (limited to 'plugins/funind/Recdef.v')
-rw-r--r-- | plugins/funind/Recdef.v | 38 |
1 files changed, 20 insertions, 18 deletions
diff --git a/plugins/funind/Recdef.v b/plugins/funind/Recdef.v index 51ede26e..a63941f0 100644 --- a/plugins/funind/Recdef.v +++ b/plugins/funind/Recdef.v @@ -1,10 +1,13 @@ (************************************************************************) (* 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 *) (************************************************************************) + +Require Import PeanoNat. + Require Compare_dec. Require Wf_nat. @@ -19,30 +22,29 @@ Fixpoint iter (n : nat) : (A -> A) -> A -> A := end. End Iter. -Theorem SSplus_lt : forall p p' : nat, p < S (S (p + p')). - intro p; intro p'; change (S p <= S (S (p + p'))); - apply le_S; apply Gt.gt_le_S; change (p < S (p + p')); - apply Lt.le_lt_n_Sm; apply Plus.le_plus_l. +Theorem le_lt_SS x y : x <= y -> x < S (S y). +Proof. + intros. now apply Nat.lt_succ_r, Nat.le_le_succ_r. Qed. - -Theorem Splus_lt : forall p p' : nat, p' < S (p + p'). - intro p; intro p'; change (S p' <= S (p + p')); - apply Gt.gt_le_S; change (p' < S (p + p')); apply Lt.le_lt_n_Sm; - apply Plus.le_plus_r. +Theorem Splus_lt x y : y < S (x + y). +Proof. + apply Nat.lt_succ_r. rewrite Nat.add_comm. apply Nat.le_add_r. Qed. -Theorem le_lt_SS : forall x y, x <= y -> x < S (S y). -intro x; intro y; intro H; change (S x <= S (S y)); - apply le_S; apply Gt.gt_le_S; change (x < S y); - apply Lt.le_lt_n_Sm; exact H. +Theorem SSplus_lt x y : x < S (S (x + y)). +Proof. + apply le_lt_SS, Nat.le_add_r. Qed. Inductive max_type (m n:nat) : Set := cmt : forall v, m <= v -> n <= v -> max_type m n. -Definition max : forall m n:nat, max_type m n. -intros m n; case (Compare_dec.le_gt_dec m n). -intros h; exists n; [exact h | apply le_n]. -intros h; exists m; [apply le_n | apply Lt.lt_le_weak; exact h]. +Definition max m n : max_type m n. +Proof. + destruct (Compare_dec.le_gt_dec m n) as [h|h]. + - exists n; [exact h | apply le_n]. + - exists m; [apply le_n | apply Nat.lt_le_incl; exact h]. Defined. + +Definition Acc_intro_generator_function := fun A R => @Acc_intro_generator A R 100. |