Module Coq.Lists.List

THIS IS A OLD CONTRIB. IT IS NO LONGER MAINTAINED **

Require Le.

Parameter List_Dom:Set.
Definition A := List_Dom.

Inductive list : Set := nil : list | cons : A -> list -> list.

Fixpoint app [l:list] : list -> list
      := [m:list]<list>Cases l of
                           nil => m
                       | (cons a l1) => (cons a (app l1 m))
                   end.

Lemma app_nil_end : (l:list)(l=(app l nil)).
Proof.
        Intro l ; Elim l ; Simpl ; Auto.
        Induction 1; Auto.
Qed.
Hints Resolve app_nil_end : list v62.

Lemma app_ass : (l,m,n : list)(app (app l m) n)=(app l (app m n)).
Proof.
        Intros l m n ; Elim l ; Simpl ; Auto with list.
        Induction 1; Auto with list.
Qed.
Hints Resolve app_ass : list v62.

Lemma ass_app : (l,m,n : list)(app l (app m n))=(app (app l m) n).
Proof.
        Auto with list.
Qed.
Hints Resolve ass_app : list v62.

Definition tail :=
    [l:list] <list>Cases l of (cons _ m) => m | _ => nil end : list->list.
                   

Lemma nil_cons : (a:A)(m:list)~nil=(cons a m).
  Intros; Discriminate.
Qed.

Fixpoint length [l:list] : nat
   := <nat>Cases l of (cons _ m) => (S (length m)) | _ => O end.

Section length_order.
Definition lel := [l,m:list](le (length l) (length m)).

Hints Unfold lel : list.

Variables a,b:A.
Variables l,m,n:list.

Lemma lel_refl : (lel l l).
Proof.
        Unfold lel ; Auto with list.
Qed.

Lemma lel_trans : (lel l m)->(lel m n)->(lel l n).
Proof.
        Unfold lel ; Intros.
        Apply le_trans with (length m) ; Auto with list.
Qed.

Lemma lel_cons_cons : (lel l m)->(lel (cons a l) (cons b m)).
Proof.
        Unfold lel ; Simpl ; Auto with list arith.
Qed.

Lemma lel_cons : (lel l m)->(lel l (cons b m)).
Proof.
        Unfold lel ; Simpl ; Auto with list arith.
Qed.

Lemma lel_tail : (lel (cons a l) (cons b m)) -> (lel l m).
Proof.
        Unfold lel ; Simpl ; Auto with list arith.
Qed.

Lemma lel_nil : (l':list)(lel l' nil)->(nil=l').
Proof.
        Intro l' ; Elim l' ; Auto with list arith.
        Intros a' y H H0.
        Absurd (le (S (length y)) O); Auto with list arith.
Qed.
End length_order.

Hints Resolve lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons : list v62.

Fixpoint In [a:A;l:list] : Prop :=
      Cases l of
         nil => False
      | (cons b m) => (b=a)\/(In a m)
      end.

Lemma in_eq : (a:A)(l:list)(In a (cons a l)).
Proof.
        Simpl ; Auto with list.
Qed.
Hints Resolve in_eq : list v62.

Lemma in_cons : (a,b:A)(l:list)(In b l)->(In b (cons a l)).
Proof.
        Simpl ; Auto with list.
Qed.
Hints Resolve in_cons : list v62.

Lemma in_app_or : (l,m:list)(a:A)(In a (app l m))->((In a l)\/(In a m)).
Proof.
        Intros l m a.
        Elim l ; Simpl ; Auto with list.
        Intros a0 y H H0.
        Elim H0 ; Auto with list.
        Intro H1.
        Elim (H H1) ; Auto with list.
Qed.
Hints Immediate in_app_or : list v62.

Lemma in_or_app : (l,m:list)(a:A)((In a l)\/(In a m))->(In a (app l m)).
Proof.
        Intros l m a.
        Elim l ; Simpl ; Intro H.
        Elim H ; Auto with list ; Intro H0.
        Elim H0.
        Intros y H0 H1.
        Elim H1 ; Auto 4 with list.
        Intro H2.
        Elim H2 ; Auto with list.
Qed.
Hints Resolve in_or_app : list v62.

Definition incl := [l,m:list](a:A)(In a l)->(In a m).

Hints Unfold incl : list v62.

Lemma incl_refl : (l:list)(incl l l).
Proof.
        Auto with list.
Qed.
Hints Resolve incl_refl : list v62.

Lemma incl_tl : (a:A)(l,m:list)(incl l m)->(incl l (cons a m)).
Proof.
        Auto with list.
Qed.
Hints Immediate incl_tl : list v62.

Lemma incl_tran : (l,m,n:list)(incl l m)->(incl m n)->(incl l n).
Proof.
        Auto with list.
Qed.

Lemma incl_appl : (l,m,n:list)(incl l n)->(incl l (app n m)).
Proof.
        Auto with list.
Qed.
Hints Immediate incl_appl : list v62.

Lemma incl_appr : (l,m,n:list)(incl l n)->(incl l (app m n)).
Proof.
        Auto with list.
Qed.
Hints Immediate incl_appr : list v62.

Lemma incl_cons : (a:A)(l,m:list)(In a m)->(incl l m)->(incl (cons a l) m).
Proof.
        Unfold incl ; Simpl ; Intros a l m H H0 a0 H1.
        Elim H1.
        Elim H1 ; Auto with list ; Intro H2.
        Elim H2 ; Auto with list.
        Auto with list.
Qed.
Hints Resolve incl_cons : list v62.

Lemma incl_app : (l,m,n:list)(incl l n)->(incl m n)->(incl (app l m) n).
Proof.
        Unfold incl ; Simpl ; Intros l m n H H0 a H1.
        Elim (in_app_or l m a) ; Auto with list.
Qed.
Hints Resolve incl_app : list v62.


Index