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(*
Example proof script for testing Coq Proof General.
$Id$
Maintainer: Proof General maintainer <proofgen@dcs.ed.ac.uk>
Please let us know if you could maintain this example file!
*)
(****************************************************************************)
(* The Calculus of Inductive Constructions *)
(* *)
(* Projet Coq *)
(* *)
(* INRIA ENS-CNRS *)
(* Rocquencourt Lyon *)
(* *)
(* Coq V6.1 *)
(* Oct 1st 1996 *)
(* *)
(****************************************************************************)
(* List.v *)
(****************************************************************************)
Require Le.
(* Some programs and results about lists *)
Parameter A:Set.
Inductive list : Set := nil : list | cons : A -> list -> list.
(*
Fixpoint app [l:list] : list -> list
:= [m:list]Case l of
(* nil *) m
(* cons a l1 *) [a:A][l1:list](cons a (app l1 m)) end.
*)
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.
Hint app_nil_end.
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.
Induction 1; Auto.
Qed.
Hint app_ass.
Lemma ass_app : (l,m,n : list)(app l (app m n))=(app (app l m) n).
Proof.
Auto.
Qed.
Hint ass_app.
(*
Definition tail := [l:list]Case l of
(* nil *) nil
(* cons a m *) [a:A][m:list]m end : list->list.
*)
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.
(****************************************)
(* Length of lists *)
(****************************************)
(*
Fixpoint length [l:list] : nat
:= Case l of (* nil *) O
(* cons a m *) [a:A][m:list](S (length m)) end.
*)
Fixpoint length [l:list] : nat
:= <nat>Cases l of (cons _ m) => (S (length m)) | _ => O end.
(******************************)
(* Length order of lists *)
(******************************)
Section length_order.
Definition lel := [l,m:list](le (length l) (length m)).
Variables a,b:A.
Variables l,m,n:list.
Lemma lel_refl : (lel l l).
Proof.
Unfold lel ; Auto.
Qed.
Lemma lel_trans : (lel l m)->(lel m n)->(lel l n).
Proof.
Unfold lel ; Intros.
Apply le_trans with (length m) ; Auto.
Qed.
Lemma lel_cons_cons : (lel l m)->(lel (cons a l) (cons b m)).
Proof.
Unfold lel ; Simpl ; Auto.
Qed.
Lemma lel_cons : (lel l m)->(lel l (cons b m)).
Proof.
Unfold lel ; Simpl ; Auto.
Qed.
Lemma lel_tail : (lel (cons a l) (cons b m)) -> (lel l m).
Proof.
Unfold lel ; Simpl ; Auto.
Qed.
Lemma lel_nil : (l':list)(lel l' nil)->(nil=l').
Proof.
Intro l' ; Elim l' ; Auto.
Intros a' y H H0.
(* <list>nil=(cons a' y)
============================
H0 : (lel (cons a' y) nil)
H : (lel y nil)->(<list>nil=y)
y : list
a' : A
l' : list *)
Absurd (le (S (length y)) O); Auto.
Qed.
End length_order.
Hint lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons.
Fixpoint In [a:A;l:list] : Prop :=
(Case l of (* nil *) False
(* cons b m *) [b:A][m:list](b=a)\/(In a m) end).
Lemma in_eq : (a:A)(l:list)(In a (cons a l)).
Proof.
Simpl ; Auto.
Qed.
Hint in_eq.
Lemma in_cons : (a,b:A)(l:list)(In b l)->(In b (cons a l)).
Proof.
Simpl ; Auto.
Qed.
Hint in_cons.
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.
Intros a0 y H H0.
(* ((<A>a0=a)\/(In a y))\/(In a m)
============================
H0 : (<A>a0=a)\/(In a (app y m))
H : (In a (app y m))->((In a y)\/(In a m))
y : list
a0 : A
a : A
m : list
l : list *)
Elim H0 ; Auto.
Intro H1.
(* ((<A>a0=a)\/(In a y))\/(In a m)
============================
H1 : (In a (app y m)) *)
Elim (H H1) ; Auto.
Qed.
Immediate in_app_or.
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.
(* 1 (In a m)
============================
H : False\/(In a m)
a : A
m : list
l : list *)
Elim H ; Auto ; Intro H0.
(* (In a m)
============================
H0 : False *)
Elim H0. (* subProof completed *)
Intros y H0 H1.
(* 2 (<A>H=a)\/(In a (app y m))
============================
H1 : ((<A>H=a)\/(In a y))\/(In a m)
H0 : ((In a y)\/(In a m))->(In a (app y m))
y : list *)
Elim H1 ; Auto 4.
Intro H2.
(* (<A>H=a)\/(In a (app y m))
============================
H2 : (<A>H=a)\/(In a y) *)
Elim H2 ; Auto.
Qed.
Hint in_or_app.
Definition incl := [l,m:list](a:A)(In a l)->(In a m).
Hint Unfold incl.
Lemma incl_refl : (l:list)(incl l l).
Proof.
Auto.
Qed.
Hint incl_refl.
Lemma incl_tl : (a:A)(l,m:list)(incl l m)->(incl l (cons a m)).
Proof.
Auto.
Qed.
Immediate incl_tl.
Lemma incl_tran : (l,m,n:list)(incl l m)->(incl m n)->(incl l n).
Proof.
Auto.
Qed.
Lemma incl_appl : (l,m,n:list)(incl l n)->(incl l (app n m)).
Proof.
Auto.
Qed.
Immediate incl_appl.
Lemma incl_appr : (l,m,n:list)(incl l n)->(incl l (app m n)).
Proof.
Auto.
Qed.
Immediate incl_appr.
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.
(* (In a0 m)
============================
H1 : (<A>a=a0)\/(In a0 l)
a0 : A
H0 : (a:A)(In a l)->(In a m)
H : (In a m)
m : list
l : list
a : A *)
Elim H1.
(* 1 (<A>a=a0)->(In a0 m) *)
Elim H1 ; Auto ; Intro H2.
(* (<A>a=a0)->(In a0 m)
============================
H2 : <A>a=a0 *)
Elim H2 ; Auto. (* solves subgoal *)
(* 2 (In a0 l)->(In a0 m) *)
Auto.
Qed.
Hint incl_cons.
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.
(* (In a n)
============================
H1 : (In a (app l m))
a : A
H0 : (a:A)(In a m)->(In a n)
H : (a:A)(In a l)->(In a n)
n : list
m : list
l : list *)
Elim (in_app_or l m a) ; Auto.
Qed.
Hint incl_app.
(* Id: List.v,v 1.4 1996/10/07 07:50:36 ccornes Exp *)
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