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author | Samuel Mimram <samuel.mimram@ens-lyon.org> | 2004-07-28 21:54:47 +0000 |
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committer | Samuel Mimram <samuel.mimram@ens-lyon.org> | 2004-07-28 21:54:47 +0000 |
commit | 6b649aba925b6f7462da07599fe67ebb12a3460e (patch) | |
tree | 43656bcaa51164548f3fa14e5b10de5ef1088574 /theories/Wellfounded/Inverse_Image.v |
Imported Upstream version 8.0pl1upstream/8.0pl1
Diffstat (limited to 'theories/Wellfounded/Inverse_Image.v')
-rw-r--r-- | theories/Wellfounded/Inverse_Image.v | 55 |
1 files changed, 55 insertions, 0 deletions
diff --git a/theories/Wellfounded/Inverse_Image.v b/theories/Wellfounded/Inverse_Image.v new file mode 100644 index 00000000..f2cf1d2e --- /dev/null +++ b/theories/Wellfounded/Inverse_Image.v @@ -0,0 +1,55 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Inverse_Image.v,v 1.10.2.1 2004/07/16 19:31:19 herbelin Exp $ i*) + +(** Author: Bruno Barras *) + +Section Inverse_Image. + + Variables A B : Set. + Variable R : B -> B -> Prop. + Variable f : A -> B. + + Let Rof (x y:A) : Prop := R (f x) (f y). + + Remark Acc_lemma : forall y:B, Acc R y -> forall x:A, y = f x -> Acc Rof x. + induction 1 as [y _ IHAcc]; intros x H. + apply Acc_intro; intros y0 H1. + apply (IHAcc (f y0)); try trivial. + rewrite H; trivial. + Qed. + + Lemma Acc_inverse_image : forall x:A, Acc R (f x) -> Acc Rof x. + intros; apply (Acc_lemma (f x)); trivial. + Qed. + + Theorem wf_inverse_image : well_founded R -> well_founded Rof. + red in |- *; intros; apply Acc_inverse_image; auto. + Qed. + + Variable F : A -> B -> Prop. + Let RoF (x y:A) : Prop := + exists2 b : B, F x b & (forall c:B, F y c -> R b c). + +Lemma Acc_inverse_rel : forall b:B, Acc R b -> forall x:A, F x b -> Acc RoF x. +induction 1 as [x _ IHAcc]; intros x0 H2. +constructor; intros y H3. +destruct H3. +apply (IHAcc x1); auto. +Qed. + + +Theorem wf_inverse_rel : well_founded R -> well_founded RoF. + red in |- *; constructor; intros. + case H0; intros. + apply (Acc_inverse_rel x); auto. +Qed. + +End Inverse_Image. + |