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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.
|
