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-(************************************************************************)
-(* 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: Lexicographic_Exponentiation.v,v 1.1.2.1 2004/07/16 19:31:41 herbelin Exp $ i*)
-
-(** Author: Cristina Cornes
-
- From : Constructing Recursion Operators in Type Theory
- L. Paulson JSC (1986) 2, 325-355 *)
-
-Require Eqdep.
-Require PolyList.
-Require PolyListSyntax.
-Require Relation_Operators.
-Require Transitive_Closure.
-
-Section Wf_Lexicographic_Exponentiation.
-Variable A:Set.
-Variable leA: A->A->Prop.
-
-Notation Power := (Pow A leA).
-Notation Lex_Exp := (lex_exp A leA).
-Notation ltl := (Ltl A leA).
-Notation Descl := (Desc A leA).
-
-Notation List := (list A).
-Notation Nil := (nil A).
-(* useless but symmetric *)
-Notation Cons := (cons 1!A).
-Notation "<< x , y >>" := (exist List Descl x y) (at level 0)
- V8only (at level 0, x,y at level 100).
-
-V7only[
-Syntax constr
- level 1:
- List [ (list A) ] -> ["List"]
- | Nil [ (nil A) ] -> ["Nil"]
- | Cons [ (cons A) ] -> ["Cons"]
- ;
- level 10:
- Cons2 [ (cons A $e $l) ] -> ["Cons " $e:L " " $l:L ].
-
-Syntax constr
- level 1:
- pair_sig [ (exist (list A) Desc $e $d) ] -> ["<<" $e:L "," $d:L ">>"].
-].
-Hints Resolve d_one d_nil t_step.
-
-Lemma left_prefix : (x,y,z:List)(ltl x^y z)-> (ltl x z).
-Proof.
- Induction x.
- Induction z.
- Simpl;Intros H.
- Inversion_clear H.
- Simpl;Intros;Apply (Lt_nil A leA).
- Intros a l HInd.
- Simpl.
- Intros.
- Inversion_clear H.
- Apply (Lt_hd A leA);Auto with sets.
- Apply (Lt_tl A leA).
- Apply (HInd y y0);Auto with sets.
-Qed.
-
-
-Lemma right_prefix :
- (x,y,z:List)(ltl x y^z)-> (ltl x y) \/ (EX y':List | x=(y^y') /\ (ltl y' z)).
-Proof.
- Intros x y;Generalize x.
- Elim y;Simpl.
- Right.
- Exists x0 ;Auto with sets.
- Intros.
- Inversion H0.
- Left;Apply (Lt_nil A leA).
- Left;Apply (Lt_hd A leA);Auto with sets.
- Generalize (H x1 z H3) .
- Induction 1.
- Left;Apply (Lt_tl A leA);Auto with sets.
- Induction 1.
- Induction 1;Intros.
- Rewrite -> H8.
- Right;Exists x2 ;Auto with sets.
-Qed.
-
-
-
-Lemma desc_prefix: (x:List)(a:A)(Descl x^(Cons a Nil))->(Descl x).
-Proof.
- Intros.
- Inversion H.
- Generalize (app_cons_not_nil H1); Induction 1.
- Cut (x^(Cons a Nil))=(Cons x0 Nil); Auto with sets.
- Intro.
- Generalize (app_eq_unit H0) .
- Induction 1; Induction 1; Intros.
- Rewrite -> H4; Auto with sets.
- Discriminate H5.
- Generalize (app_inj_tail H0) .
- Induction 1; Intros.
- Rewrite <- H4; Auto with sets.
-Qed.
-
-Lemma desc_tail: (x:List)(a,b:A)
- (Descl (Cons b (x^(Cons a Nil))))-> (clos_trans A leA a b).
-Proof.
- Intro.
- Apply rev_ind with A:=A
- P:=[x:List](a,b:A)
- (Descl (Cons b (x^(Cons a Nil))))-> (clos_trans A leA a b).
- Intros.
-
- Inversion H.
- Cut (Cons b (Cons a Nil))= ((Nil^(Cons b Nil))^ (Cons a Nil)); Auto with sets; Intro.
- Generalize H0.
- Intro.
- Generalize (app_inj_tail 2!(l^(Cons y Nil)) 3!(Nil^(Cons b Nil)) H4);
- Induction 1.
- Intros.
-
- Generalize (app_inj_tail H6); Induction 1; Intros.
- Generalize H1.
- Rewrite <- H10; Rewrite <- H7; Intro.
- Apply (t_step A leA); Auto with sets.
-
-
-
- Intros.
- Inversion H0.
- Generalize (app_cons_not_nil H3); Intro.
- Elim H1.
-
- Generalize H0.
- Generalize (app_comm_cons (l^(Cons x0 Nil)) (Cons a Nil) b); Induction 1.
- Intro.
- Generalize (desc_prefix (Cons b (l^(Cons x0 Nil))) a H5); Intro.
- Generalize (H x0 b H6).
- Intro.
- Apply t_trans with A:=A y:=x0; Auto with sets.
-
- Apply t_step.
- Generalize H1.
- Rewrite -> H4; Intro.
-
- Generalize (app_inj_tail H8); Induction 1.
- Intros.
- Generalize H2; Generalize (app_comm_cons l (Cons x0 Nil) b).
- Intro.
- Generalize H10.
- Rewrite ->H12; Intro.
- Generalize (app_inj_tail H13); Induction 1.
- Intros.
- Rewrite <- H11; Rewrite <- H16; Auto with sets.
-Qed.
-
-
-Lemma dist_aux : (z:List)(Descl z)->(x,y:List)z=(x^y)->(Descl x)/\ (Descl y).
-Proof.
- Intros z D.
- Elim D.
- Intros.
- Cut (x^y)=Nil;Auto with sets; Intro.
- Generalize (app_eq_nil H0) ; Induction 1.
- Intros.
- Rewrite -> H2;Rewrite -> H3; Split;Apply d_nil.
-
- Intros.
- Cut (x0^y)=(Cons x Nil); Auto with sets.
- Intros E.
- Generalize (app_eq_unit E); Induction 1.
- Induction 1;Intros.
- Rewrite -> H2;Rewrite -> H3; Split.
- Apply d_nil.
-
- Apply d_one.
-
- Induction 1; Intros.
- Rewrite -> H2;Rewrite -> H3; Split.
- Apply d_one.
-
- Apply d_nil.
-
- Do 5 Intro.
- Intros Hind.
- Do 2 Intro.
- Generalize x0 .
- Apply rev_ind with A:=A
- P:=[y0:List]
- (x0:List)
- ((l^(Cons y Nil))^(Cons x Nil))=(x0^y0)->(Descl x0)/\(Descl y0).
-
- Intro.
- Generalize (app_nil_end x1) ; Induction 1; Induction 1.
- Split. Apply d_conc; Auto with sets.
-
- Apply d_nil.
-
- Do 3 Intro.
- Generalize x1 .
- Apply rev_ind with
- A:=A
- P:=[l0:List]
- (x1:A)
- (x0:List)
- ((l^(Cons y Nil))^(Cons x Nil))=(x0^(l0^(Cons x1 Nil)))
- ->(Descl x0)/\(Descl (l0^(Cons x1 Nil))).
-
-
- Simpl.
- Split.
- Generalize (app_inj_tail H2) ;Induction 1.
- Induction 1;Auto with sets.
-
- Apply d_one.
- Do 5 Intro.
- Generalize (app_ass x4 (l1^(Cons x2 Nil)) (Cons x3 Nil)) .
- Induction 1.
- Generalize (app_ass x4 l1 (Cons x2 Nil)) ;Induction 1.
- Intro E.
- Generalize (app_inj_tail E) .
- Induction 1;Intros.
- Generalize (app_inj_tail H6) ;Induction 1;Intros.
- Rewrite <- H7; Rewrite <- H10; Generalize H6.
- Generalize (app_ass x4 l1 (Cons x2 Nil)); Intro E1.
- Rewrite -> E1.
- Intro.
- Generalize (Hind x4 (l1^(Cons x2 Nil)) H11) .
- Induction 1;Split.
- Auto with sets.
-
- Generalize H14.
- Rewrite <- H10; Intro.
- Apply d_conc;Auto with sets.
-Qed.
-
-
-
-Lemma dist_Desc_concat : (x,y:List)(Descl x^y)->(Descl x)/\(Descl y).
-Proof.
- Intros.
- Apply (dist_aux (x^y) H x y); Auto with sets.
-Qed.
-
-
-Lemma desc_end:(a,b:A)(x:List)
- (Descl x^(Cons a Nil)) /\ (ltl x^(Cons a Nil) (Cons b Nil))
- -> (clos_trans A leA a b).
-
-Proof.
- Intros a b x.
- Case x.
- Simpl.
- Induction 1.
- Intros.
- Inversion H1;Auto with sets.
- Inversion H3.
-
- Induction 1.
- Generalize (app_comm_cons l (Cons a Nil) a0).
- Intros E; Rewrite <- E; Intros.
- Generalize (desc_tail l a a0 H0); Intro.
- Inversion H1.
- Apply t_trans with y:=a0 ;Auto with sets.
-
- Inversion H4.
-Qed.
-
-
-
-
-Lemma ltl_unit: (x:List)(a,b:A)
- (Descl (x^(Cons a Nil))) -> (ltl x^(Cons a Nil) (Cons b Nil))
- -> (ltl x (Cons b Nil)).
-Proof.
- Intro.
- Case x.
- Intros;Apply (Lt_nil A leA).
-
- Simpl;Intros.
- Inversion_clear H0.
- Apply (Lt_hd A leA a b);Auto with sets.
-
- Inversion_clear H1.
-Qed.
-
-
-Lemma acc_app:
- (x1,x2:List)(y1:(Descl x1^x2))
- (Acc Power Lex_Exp (exist List Descl (x1^x2) y1))
- ->(x:List)
- (y:(Descl x))
- (ltl x (x1^x2))->(Acc Power Lex_Exp (exist List Descl x y)).
-Proof.
- Intros.
- Apply (Acc_inv Power Lex_Exp (exist List Descl (x1^x2) y1)).
- Auto with sets.
-
- Unfold lex_exp ;Simpl;Auto with sets.
-Qed.
-
-
-Theorem wf_lex_exp :
- (well_founded A leA)->(well_founded Power Lex_Exp).
-Proof.
- Unfold 2 well_founded .
- Induction a;Intros x y.
- Apply Acc_intro.
- Induction y0.
- Unfold 1 lex_exp ;Simpl.
- Apply rev_ind with A:=A P:=[x:List]
- (x0:List)
- (y:(Descl x0))
- (ltl x0 x)
- ->(Acc Power Lex_Exp (exist List Descl x0 y)) .
- Intros.
- Inversion_clear H0.
-
- Intro.
- Generalize (well_founded_ind A (clos_trans A leA) (wf_clos_trans A leA H)).
- Intros GR.
- Apply GR with P:=[x0:A]
- (l:List)
- ((x1:List)
- (y:(Descl x1))
- (ltl x1 l)
- ->(Acc Power Lex_Exp (exist List Descl x1 y)))
- ->(x1:List)
- (y:(Descl x1))
- (ltl x1 (l^(Cons x0 Nil)))
- ->(Acc Power Lex_Exp (exist List Descl x1 y)) .
- Intro;Intros HInd; Intros.
- Generalize (right_prefix x2 l (Cons x1 Nil) H1) .
- Induction 1.
- Intro; Apply (H0 x2 y1 H3).
-
- Induction 1.
- Intro;Induction 1.
- Clear H4 H2.
- Intro;Generalize y1 ;Clear y1.
- Rewrite -> H2.
- Apply rev_ind with A:=A P:=[x3:List]
- (y1:(Descl (l^x3)))
- (ltl x3 (Cons x1 Nil))
- ->(Acc Power Lex_Exp
- (exist List Descl (l^x3) y1)) .
- Intros.
- Generalize (app_nil_end l) ;Intros Heq.
- Generalize y1 .
- Clear y1.
- Rewrite <- Heq.
- Intro.
- Apply Acc_intro.
- Induction y2.
- Unfold 1 lex_exp .
- Simpl;Intros x4 y3. Intros.
- Apply (H0 x4 y3);Auto with sets.
-
- Intros.
- Generalize (dist_Desc_concat l (l0^(Cons x4 Nil)) y1) .
- Induction 1.
- Intros.
- Generalize (desc_end x4 x1 l0 (conj ? ? H8 H5)) ; Intros.
- Generalize y1 .
- Rewrite <- (app_ass l l0 (Cons x4 Nil)); Intro.
- Generalize (HInd x4 H9 (l^l0)) ; Intros HInd2.
- Generalize (ltl_unit l0 x4 x1 H8 H5); Intro.
- Generalize (dist_Desc_concat (l^l0) (Cons x4 Nil) y2) .
- Induction 1;Intros.
- Generalize (H4 H12 H10); Intro.
- Generalize (Acc_inv Power Lex_Exp (exist List Descl (l^l0) H12) H14) .
- Generalize (acc_app l l0 H12 H14).
- Intros f g.
- Generalize (HInd2 f);Intro.
- Apply Acc_intro.
- Induction y3.
- Unfold 1 lex_exp ;Simpl; Intros.
- Apply H15;Auto with sets.
-Qed.
-
-
-End Wf_Lexicographic_Exponentiation.
generated by cgit v1.2.3 (git 2.39.1) at 2025-02-09 06:13:41 +0000