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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(*i $Id$ i*)
(** Author: Cristina Cornes
From : Constructing Recursion Operators in Type Theory
L. Paulson JSC (1986) 2, 325-355 *)
Require Relation_Operators.
Section Wf_Disjoint_Union.
Variable A,B:Set.
Variable leA: A->A->Prop.
Variable leB: B->B->Prop.
Syntactic Definition Le_AsB := (le_AsB A B leA leB).
Lemma acc_A_sum: (x:A)(Acc A leA x)->(Acc A+B Le_AsB (inl A B x)).
Proof.
Induction 1;Intros.
Apply Acc_intro;Intros.
Inversion_clear H2.
Auto with sets.
Qed.
Lemma acc_B_sum: (well_founded A leA) ->(x:B)(Acc B leB x)
->(Acc A+B Le_AsB (inr A B x)).
Proof.
Induction 2;Intros.
Apply Acc_intro;Intros.
Inversion_clear H3;Auto with sets.
Apply acc_A_sum;Auto with sets.
Save.
Lemma wf_disjoint_sum:
(well_founded A leA)
-> (well_founded B leB) -> (well_founded A+B Le_AsB).
Proof.
Intros.
Unfold well_founded .
Induction a.
Intro a0.
Apply (acc_A_sum a0).
Apply (H a0).
Intro b.
Apply (acc_B_sum H b).
Apply (H0 b).
Qed.
End Wf_Disjoint_Union.
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