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(*         *   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          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Require Import Relation_Operators.

Section Wf_Disjoint_Union.
  Variables A B : Type.
  Variable leA : A -> A -> Prop.
  Variable leB : B -> B -> Prop.

  Notation Le_AsB := (le_AsB A B leA leB).

  Lemma acc_A_sum : forall x:A, Acc leA x -> Acc Le_AsB (inl B x).
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
forall x : A, Acc leA x -> Acc Le_AsB (inl x)
  Proof.
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
forall x : A, Acc leA x -> Acc Le_AsB (inl x)
    induction 1.
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
x:A
H:forall y : A, leA y x -> Acc leA y
H0:forall y : A, leA y x -> Acc Le_AsB (inl y)
Acc Le_AsB (inl x)
    apply Acc_intro; intros y H2.
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
x:A
H:forall y0 : A, leA y0 x -> Acc leA y0
H0:forall y0 : A, leA y0 x -> Acc Le_AsB (inl y0)
y:(A + B)%type
H2:Le_AsB y (inl x)
Acc Le_AsB y
    inversion_clear H2.
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
x:A
H:forall y0 : A, leA y0 x -> Acc leA y0
H0:forall y0 : A, leA y0 x -> Acc Le_AsB (inl y0)
y:(A + B)%type
x0:A
H1:leA x0 x
Acc Le_AsB (inl x0)
    auto with sets.
  Qed.

  Lemma acc_B_sum :
    well_founded leA -> forall x:B, Acc leB x -> Acc Le_AsB (inr A x).
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
well_founded leA ->
forall x : B, Acc leB x -> Acc Le_AsB (inr x)
  Proof.
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
well_founded leA ->
forall x : B, Acc leB x -> Acc Le_AsB (inr x)
    induction 2.
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
H:well_founded leA
x:B
H0:forall y : B, leB y x -> Acc leB y
H1:forall y : B, leB y x -> Acc Le_AsB (inr y)
Acc Le_AsB (inr x)
    apply Acc_intro; intros y H3.
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
H:well_founded leA
x:B
H0:forall y0 : B, leB y0 x -> Acc leB y0
H1:forall y0 : B, leB y0 x -> Acc Le_AsB (inr y0)
y:(A + B)%type
H3:Le_AsB y (inr x)
Acc Le_AsB y
    inversion_clear H3; auto with sets.
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
H:well_founded leA
x:B
H0:forall y0 : B, leB y0 x -> Acc leB y0
H1:forall y0 : B, leB y0 x -> Acc Le_AsB (inr y0)
y:(A + B)%type
x0:A
Acc Le_AsB (inl x0)
    apply acc_A_sum; auto with sets.
  Qed.


  Lemma wf_disjoint_sum :
    well_founded leA -> well_founded leB -> well_founded Le_AsB.
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
well_founded leA ->
well_founded leB -> well_founded Le_AsB
  Proof.
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
well_founded leA ->
well_founded leB -> well_founded Le_AsB
    intros.
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
H:well_founded leA
H0:well_founded leB
well_founded Le_AsB
    unfold well_founded.
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
H:well_founded leA
H0:well_founded leB
forall a : A + B, Acc Le_AsB a
    destruct a as [a| b].
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
H:well_founded leA
H0:well_founded leB
a:A
Acc Le_AsB (inl a)
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
H:well_founded leA
H0:well_founded leB
b:B
Acc Le_AsB (inr b)
    -
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
H:well_founded leA
H0:well_founded leB
a:A
Acc Le_AsB (inl a) apply (acc_A_sum a).
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
H:well_founded leA
H0:well_founded leB
a:A
Acc leA a
      apply (H a).

    -
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
H:well_founded leA
H0:well_founded leB
b:B
Acc Le_AsB (inr b) apply (acc_B_sum H b).
A, B:Type
leA:A -> A -> Prop
leB:B -> B -> Prop
H:well_founded leA
H0:well_founded leB
b:B
Acc leB b
      apply (H0 b).
  Qed.

End Wf_Disjoint_Union.