Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: Generalised_Binomial_Theorem.thy Sprache: Unknown

Original von: Coq^©

(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

(** Author: Cristina Cornes

    From : Constructing Recursion Operators in Type Theory
           L. Paulson  JSC (1986) 2, 325-355  *)

Require Import List.
Require Import Relation_Operators.
Require Import Operators_Properties.
Require Import Transitive_Closure.

Import ListNotations.

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 "<< 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; 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 :
    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.
    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 ++ [a]) -> Descl x.
  Proof.
    intros.
    inversion H.
    - apply app_cons_not_nil in H1 as [].
    - assert (x ++ [a] = [x0]) by auto with sets.
      apply app_eq_unit in H0 as [(->, _)| (_, [=])].
      auto using d_nil.
    - apply app_inj_tail in H0 as (<-, _).
      assumption.
  Qed.

  Lemma desc_tail :
    forall (x : List) (a b : A),
      Descl (b :: x ++ [a]) -> clos_refl_trans A leA a b.
  Proof.
    intro.
    apply rev_ind with
      (P :=
        fun x : List =>
        forall a b : A, Descl (b :: x ++ [a]) -> clos_refl_trans A leA a b);
     intros.
    - inversion H.
      assert ([b; a] = ([] ++ [b]) ++ [a]) by auto with sets.
      destruct (app_inj_tail (l ++ [y]) ([] ++ [b]) _ _ H0) as ((?, <-)%app_inj_tail, <-).
      inversion H1; subst; [ apply rt_step; assumption | apply rt_refl ].
    - inversion H0.
      + apply app_cons_not_nil in H3 as [].
      + rewrite app_comm_cons in H0, H1. apply desc_prefix in H0.
        pose proof (H x0 b H0).
        apply rt_trans with (y := x0); auto with sets.
        enough (H5 : clos_refl A leA a x0)
         by (inversion H5; subst; [ apply rt_step | apply rt_refl ];
              assumption).
        apply app_inj_tail in H1 as (H1, ->).
        rewrite app_comm_cons in H1.
        apply app_inj_tail in H1 as (_, <-).
        assumption.
  Qed.

  Lemma dist_aux :
    forall z : List,
      Descl z -> forall x y : List, z = x ++ y -> Descl x /\ Descl y.
  Proof.
    intros z D.
    induction D as [| | * H D Hind]; intros.
    - assert (H0 : x ++ y = []) by auto with sets.
      apply app_eq_nil in H0 as (->, ->).
      split; apply d_nil.
    - assert (E : x0 ++ y = [x]) by auto with sets.
      apply app_eq_unit in E as [(->, ->)| (->, ->)].
      + split.
        apply d_nil.
        apply d_one.
      + split.
        apply d_one.
        apply d_nil.
    - induction y0 using rev_ind in x0, H0 |- *.
      + rewrite <- app_nil_end in H0.
        rewrite <- H0.
        split.
        apply d_conc; auto with sets.
        apply d_nil.
      + induction y0 using rev_ind in x1, x0, H0 |- *.
        * simpl.
          split.
          apply app_inj_tail in H0 as (<-, _). assumption.
          apply d_one.
        * rewrite <- 2!app_assoc_reverse in H0.
          apply app_inj_tail in H0 as (H0, <-).
          pose proof H0 as H0'.
          apply app_inj_tail in H0' as (_, ->).
          rewrite app_assoc_reverse in H0.
          apply Hind in H0 as [].
          split.
          assumption.
          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 ++ [a]) /\ ltl (x ++ [a]) [b] -> clos_trans A leA a b.
  Proof.
    intros a b x.
    case x.
    simpl.
    simple induction 1.
    intros.
    inversion H1; auto with sets.
    inversion H3.

    simple induction 1.
    generalize (app_comm_cons l [a] a0).
    intros E; rewrite <- E; intros.
    generalize (desc_tail l a a0 H0); intro.
    inversion H1.
    eapply clos_rt_t; [ eassumption | apply t_step; assumption ].

    inversion H4.
  Qed.

  Lemma ltl_unit :
    forall (x : List) (a b : A),
      Descl (x ++ [a]) -> ltl (x ++ [a]) [b] -> ltl x [b].
  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 :
    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; simpl; auto with sets.
  Qed.

  Theorem wf_lex_exp : well_founded leA -> well_founded Lex_Exp.
  Proof.
    unfold well_founded at 2.
    simple induction a; intros x y.
    apply Acc_intro.
    simple induction y0.
    unfold lex_exp at 1; simpl.
    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 ++ [x0]) -> Acc Lex_Exp << x1, y >>).
    intro; intros HInd; intros.
    generalize (right_prefix x2 l [x1] 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 [x1] -> 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.
    simpl; intros x4 y3. intros.
    apply (H0 x4 y3); auto with sets.

    intros.
    generalize (dist_Desc_concat l (l0 ++ [x4]) y1).
    simple induction 1.
    intros.
    generalize (desc_end x4 x1 l0 (conj H8 H5)); intros.
    generalize y1.
    rewrite <- (app_assoc_reverse l l0 [x4]); intro.
    generalize (HInd x4 H9 (l ++ l0)); intros HInd2.
    generalize (ltl_unit l0 x4 x1 H8 H5); intro.
    generalize (dist_Desc_concat (l ++ l0) [x4] 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; simpl; intros.
    apply H15; auto with sets.
  Qed.

End Wf_Lexicographic_Exponentiation.

¤ Dauer der Verarbeitung: 0.23 Sekunden (vorverarbeitet) ¤

vermutete Sprache: Sekunden
vermutete Sprache: sprechenden Kalenders
Eigene Datei ansehen

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.