ajout de theories/Wellfounded

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+
+(* $Id$ *)
+
+(****************************************************************************)
+(*                         Bruno Barras  Cristina Cornes                    *)
+(*                                                                          *)
+(****************************************************************************)
+
+Require Eqdep.
+Require Relation_Operators.
+Require Transitive_Closure.
+
+(*  From : Constructing Recursion Operators in Type Theory            
+           L. Paulson  JSC (1986) 2, 325-355                                
+*)
+Section WfLexicographic_Product.
+Variable A:Set.
+Variable B:A->Set.
+Variable leA: A->A->Prop.
+Variable leB: (x:A)(B x)->(B x)->Prop.
+
+
+Syntactic Definition LexProd := (lexprod A B leA leB).
+
+Hints Resolve t_step Acc_clos_trans wf_clos_trans.
+
+Lemma acc_A_B_lexprod : (x:A)(Acc A leA x)
+        ->((x0:A)(clos_trans A leA x0 x)->(well_founded (B x0) (leB x0)))
+                ->(y:(B x))(Acc (B x) (leB x) y)
+                        ->(Acc (sigS A B) LexProd (existS A B x y)).
+Proof.
+ Induction 1.
+ Induction 4;Intros.
+ Apply Acc_intro.
+ Induction y0.
+ Intros.
+ Simple Inversion H6;Intros.
+ Cut (leA x2 x0);Intros.
+ Apply H1;Auto with sets.
+ Intros.
+ Apply H2.
+ Apply t_trans with x2 ;Auto with sets.
+
+ Red in H2.
+ Apply H2.
+ Auto with sets.
+
+ Injection H8. (*** BUG ***)
+ Induction 2.
+ Injection H9.
+ Induction 2;Auto with sets.
+
+ Elim H8.
+ Generalize y2 y' H9 H7 .
+ Replace x3 with x0.
+ Clear  H7 H8 H9 y2 y' x3 H6 y1 x2 y0.
+ Intros.
+ Apply H5.
+ Elim inj_pair2 with A B x0 y' x1 ;Auto with sets.
+
+ Injection H9;Auto with sets.
+Qed.
+
+Theorem wf_lexprod: 
+   (well_founded A leA) ->((x:A) (well_founded (B x) (leB x))) 
+             -> (well_founded (sigS A B) LexProd).
+Proof.
+ Intros wfA wfB;Unfold well_founded .
+ Induction a;Intros.
+ Apply acc_A_B_lexprod;Auto with sets;Intros.
+ Red in wfB.
+ Auto with sets.
+Save.
+
+
+End WfLexicographic_Product.
+
+
+Section Wf_Symmetric_Product.
+  Variable A:Set.
+  Variable B:Set.
+  Variable leA: A->A->Prop.
+  Variable leB: B->B->Prop.
+
+  Syntactic Definition Symprod := (symprod A B leA leB).
+
+(*
+  Local sig_prod:=
+         [x:A*B]<{_:A&B}>Case x of [a:A][b:B](existS A [_:A]B a b) end.
+
+Lemma incl_sym_lexprod: (included (A*B) Symprod  
+            (R_o_f (A*B) {_:A&B} sig_prod (lexprod A [_:A]B leA [_:A]leB))).
+Proof.
+ Red.
+ Induction x.
+ (Induction y1;Intros).
+ Red.
+ Unfold sig_prod .
+ Inversion_clear H.
+ (Apply left_lex;Auto with sets).
+
+ (Apply right_lex;Auto with sets).
+Save.
+*)
+
+  Lemma Acc_symprod: (x:A)(Acc A leA x)->(y:B)(Acc B leB y)
+                        ->(Acc (A*B) Symprod (x,y)).
+Proof.
+ Induction 1;Intros.
+ Elim H2;Intros.
+ Apply Acc_intro;Intros.
+ Inversion_clear H5;Auto with sets.
+ Apply H1;Auto with sets.
+ Apply Acc_intro;Auto with sets.
+Save.
+
+
+Lemma wf_symprod: (well_founded A leA)->(well_founded B leB)
+                        ->(well_founded (A*B) Symprod).
+Proof.
+ Red.
+ Induction a;Intros.
+ Apply Acc_symprod;Auto with sets.
+Save.
+
+End Wf_Symmetric_Product.
+
+
+Section Swap.
+
+  Variable A:Set.
+  Variable R:A->A->Prop.
+
+  Syntactic Definition SwapProd :=(swapprod A R).
+
+
+  Lemma swap_Acc: (x,y:A)(Acc A*A SwapProd (x,y))->(Acc A*A SwapProd (y,x)).
+Proof.
+ Intros.
+ Inversion_clear H.
+ Apply Acc_intro.
+ Destruct y0;Intros.
+ Inversion_clear H;Inversion_clear H1;Apply H0.
+ Apply sp_swap.
+ Apply right_sym;Auto with sets.
+
+ Apply sp_swap.
+ Apply left_sym;Auto with sets.
+
+ Apply sp_noswap.
+ Apply right_sym;Auto with sets.
+
+ Apply sp_noswap.
+ Apply left_sym;Auto with sets.
+Save.
+
+
+  Lemma Acc_swapprod: (x,y:A)(Acc A R x)->(Acc A R y)
+                                ->(Acc A*A SwapProd (x,y)).
+Proof.
+ Induction 1;Intros.
+ Cut (y0:A)(R y0 x0)->(Acc ? SwapProd (y0,y)).
+ Clear  H1.
+ Elim H2;Intros.
+ Cut (y:A)(R y x1)->(Acc ? SwapProd (x0,y)).
+ Clear  H3.
+ Intro.
+ Apply Acc_intro.
+ Induction y0;Intros.
+ Inversion_clear H5.
+ Inversion_clear H6;Auto with sets.
+
+ Apply swap_Acc.
+ Inversion_clear H6;Auto with sets.
+
+ Intros.
+ Apply H3;Auto with sets;Intros.
+ Apply Acc_inv with (y1,x1) ;Auto with sets.
+ Apply sp_noswap.
+ Apply right_sym;Auto with sets.
+
+ Auto with sets.
+Save.
+
+
+  Lemma wf_swapprod: (well_founded A R)->(well_founded A*A SwapProd).
+Proof.
+ Red.
+ Induction a;Intros.
+ Apply Acc_swapprod;Auto with sets.
+Save.
+
+End Swap.