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diff --git a/theories/Wellfounded/Lexicographic_Exponentiation.v b/theories/Wellfounded/Lexicographic_Exponentiation.v new file mode 100644 index 00000000..d8a4d37c --- /dev/null +++ b/theories/Wellfounded/Lexicographic_Exponentiation.v @@ -0,0 +1,374 @@ +(************************************************************************) +(* 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 1.10.2.1 2004/07/16 19:31:19 herbelin Exp $ 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 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. |