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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 9610 2007-02-07 14:45:18Z herbelin $ i*)
(** Author: Cristina Cornes
From : Constructing Recursion Operators in Type Theory
L. Paulson JSC (1986) 2, 325-355 *)
Require Import Eqdep.
Require Import List.
Require Import Relation_Operators.
Require Import 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:=A)).
(* useless but symmetric *)
Notation Cons := (cons (A:=A)).
Notation "<< x , y >>" := (exist Descl x y) (at level 0, x, y at level 100).
(* Hint Resolve d_one d_nil t_step. *)
Lemma left_prefix : forall x y z:List, ltl (x ++ y) z -> ltl x z.
Proof.
simple induction x.
simple induction z.
simpl in |- *; intros H.
inversion_clear H.
simpl in |- *; intros; apply (Lt_nil A leA).
intros a l HInd.
simpl in |- *.
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 :
forall x y z:List,
ltl x (y ++ z) -> ltl x y \/ (exists y' : List, x = y ++ y' /\ ltl y' z).
Proof.
intros x y; generalize x.
elim y; simpl in |- *.
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).
simple induction 1.
left; apply (Lt_tl A leA); auto with sets.
simple induction 1.
simple induction 1; intros.
rewrite H8.
right; exists x2; auto with sets.
Qed.
Lemma desc_prefix : forall (x:List) (a:A), Descl (x ++ Cons a Nil) -> Descl x.
Proof.
intros.
inversion H.
generalize (app_cons_not_nil _ _ _ H1); simple induction 1.
cut (x ++ Cons a Nil = Cons x0 Nil); auto with sets.
intro.
generalize (app_eq_unit _ _ H0).
simple induction 1; simple induction 1; intros.
rewrite H4; auto using d_nil with sets.
discriminate H5.
generalize (app_inj_tail _ _ _ _ H0).
simple induction 1; intros.
rewrite <- H4; auto with sets.
Qed.
Lemma desc_tail :
forall (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 := fun x:List =>
forall 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 (l ++ Cons y Nil) (Nil ++ Cons b Nil) _ _ H4);
simple induction 1.
intros.
generalize (app_inj_tail _ _ _ _ H6); simple 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);
simple 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); simple 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); simple induction 1.
intros.
rewrite <- H11; rewrite <- H16; auto with sets.
Qed.
Lemma dist_aux :
forall z:List, Descl z -> forall 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); simple 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); simple induction 1.
simple induction 1; intros.
rewrite H2; rewrite H3; split.
apply d_nil.
apply d_one.
simple 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 := fun y0:List =>
forall x0:List,
(l ++ Cons y Nil) ++ Cons x Nil = x0 ++ y0 ->
Descl x0 /\ Descl y0).
intro.
generalize (app_nil_end x1); simple induction 1; simple induction 1.
split. apply d_conc; auto with sets.
apply d_nil.
do 3 intro.
generalize x1.
apply rev_ind with
(A := A)
(P := fun l0:List =>
forall (x1:A) (x0:List),
(l ++ Cons y Nil) ++ Cons x Nil = x0 ++ l0 ++ Cons x1 Nil ->
Descl x0 /\ Descl (l0 ++ Cons x1 Nil)).
simpl in |- *.
split.
generalize (app_inj_tail _ _ _ _ H2); simple induction 1.
simple induction 1; auto with sets.
apply d_one.
do 5 intro.
generalize (app_ass x4 (l1 ++ Cons x2 Nil) (Cons x3 Nil)).
simple induction 1.
generalize (app_ass x4 l1 (Cons x2 Nil)); simple induction 1.
intro E.
generalize (app_inj_tail _ _ _ _ E).
simple induction 1; intros.
generalize (app_inj_tail _ _ _ _ H6); simple 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).
simple induction 1; split.
auto with sets.
generalize H14.
rewrite <- H10; intro.
apply d_conc; auto with sets.
Qed.
Lemma dist_Desc_concat :
forall 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 :
forall (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 in |- *.
simple induction 1.
intros.
inversion H1; auto with sets.
inversion H3.
simple 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 :
forall (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 in |- *; intros.
inversion_clear H0.
apply (Lt_hd A leA a b); auto with sets.
inversion_clear H1.
Qed.
Lemma acc_app :
forall (x1 x2:List) (y1:Descl (x1 ++ x2)),
Acc Lex_Exp << x1 ++ x2, y1 >> ->
forall (x:List) (y:Descl x), ltl x (x1 ++ x2) -> Acc Lex_Exp << x, y >>.
Proof.
intros.
apply (Acc_inv (R:=Lex_Exp) (x:=<< x1 ++ x2, y1 >>)).
auto with sets.
unfold lex_exp in |- *; simpl in |- *; auto with sets.
Qed.
Theorem wf_lex_exp : well_founded leA -> well_founded Lex_Exp.
Proof.
unfold well_founded at 2 in |- *.
simple induction a; intros x y.
apply Acc_intro.
simple induction y0.
unfold lex_exp at 1 in |- *; simpl in |- *.
apply rev_ind with
(A := A)
(P := fun x:List =>
forall (x0:List) (y:Descl x0), ltl x0 x -> Acc Lex_Exp << x0, y >>).
intros.
inversion_clear H0.
intro.
generalize (well_founded_ind (wf_clos_trans A leA H)).
intros GR.
apply GR with
(P := fun x0:A =>
forall l:List,
(forall (x1:List) (y:Descl x1),
ltl x1 l -> Acc Lex_Exp << x1, y >>) ->
forall (x1:List) (y:Descl x1),
ltl x1 (l ++ Cons x0 Nil) -> Acc Lex_Exp << x1, y >>).
intro; intros HInd; intros.
generalize (right_prefix x2 l (Cons x1 Nil) H1).
simple induction 1.
intro; apply (H0 x2 y1 H3).
simple induction 1.
intro; simple induction 1.
clear H4 H2.
intro; generalize y1; clear y1.
rewrite H2.
apply rev_ind with
(A := A)
(P := fun x3:List =>
forall y1:Descl (l ++ x3),
ltl x3 (Cons x1 Nil) -> Acc Lex_Exp << l ++ x3, y1 >>).
intros.
generalize (app_nil_end l); intros Heq.
generalize y1.
clear y1.
rewrite <- Heq.
intro.
apply Acc_intro.
simple induction y2.
unfold lex_exp at 1 in |- *.
simpl in |- *; intros x4 y3. intros.
apply (H0 x4 y3); auto with sets.
intros.
generalize (dist_Desc_concat l (l0 ++ Cons x4 Nil) y1).
simple 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).
simple induction 1; intros.
generalize (H4 H12 H10); intro.
generalize (Acc_inv H14).
generalize (acc_app l l0 H12 H14).
intros f g.
generalize (HInd2 f); intro.
apply Acc_intro.
simple induction y3.
unfold lex_exp at 1 in |- *; simpl in |- *; intros.
apply H15; auto with sets.
Qed.
End Wf_Lexicographic_Exponentiation.
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