test branch

[helm.git] / helm / matita / library / list / list.ma

diff --git a/helm/matita/library/list/list.ma b/helm/matita/library/list/list.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/list/".
+include "logic/equality.ma".
+include "higher_order_defs/functions.ma".
+
+inductive list (A:Set) : Set :=
+  | nil: list A
+  | cons: A -> list A -> list A.
+
+notation "hvbox(hd break :: tl)"
+  right associative with precedence 46
+  for @{'cons $hd $tl}.
+
+notation "[ list0 x sep ; ]"
+  non associative with precedence 90
+  for ${fold right @'nil rec acc @{'cons $x $acc}}.
+
+notation "hvbox(l1 break @ l2)"
+  right associative with precedence 47
+  for @{'append $l1 $l2 }.
+
+interpretation "nil" 'nil = (cic:/matita/list/list.ind#xpointer(1/1/1) _).
+interpretation "cons" 'cons hd tl =
+  (cic:/matita/list/list.ind#xpointer(1/1/2) _ hd tl).
+
+(* theorem test_notation: [O; S O; S (S O)] = O :: S O :: S (S O) :: []. *)
+
+theorem nil_cons:
+  \forall A:Set.\forall l:list A.\forall a:A.
+    a::l <> [].
+  intros;
+  unfold Not;
+  intros;
+  discriminate H.
+qed.
+
+let rec id_list A (l: list A) on l :=
+  match l with
+  [ nil => []
+  | (cons hd tl) => hd :: id_list A tl ].
+
+let rec append A (l1: list A) l2 on l1 :=
+  match l1 with
+  [ nil => l2
+  | (cons hd tl) => hd :: append A tl l2 ].
+
+definition tail := \lambda A:Set. \lambda l: list A.
+  match l with
+  [ nil => []
+  | (cons hd tl) => tl].
+
+interpretation "append" 'append l1 l2 = (cic:/matita/list/append.con _ l1 l2).
+
+theorem append_nil: \forall A:Set.\forall l:list A.l @ [] = l.
+  intros;
+  elim l;
+  [ reflexivity;
+  | simplify;
+    rewrite > H;
+    reflexivity;
+  ]
+qed.
+
+theorem associative_append: \forall A:Set.associative (list A) (append A).
+  intros; unfold; intros;
+  elim x;
+  [ simplify;
+    reflexivity;
+  | simplify;
+    rewrite > H;
+    reflexivity;
+  ]
+qed.
+
+theorem cons_append_commute:
+  \forall A:Set.\forall l1,l2:list A.\forall a:A.
+    a :: (l1 @ l2) = (a :: l1) @ l2.
+  intros;
+  reflexivity;
+qed.
+
+(*
+theorem nil_append_nil_both:
+  \forall A:Set.\forall l1,l2:list A.
+    l1 @ l2 = [] \to l1 = [] \land l2 = [].
+*)
+
+(*
+include "nat/nat.ma".
+
+theorem test_notation: [O; S O; S (S O)] = O :: S O :: S (S O) :: []. 
+reflexivity.
+qed.
+
+theorem test_append: [O;O;O;O;O;O] = [O;O;O] @ [O;O] @ [O].
+simplify.
+reflexivity.
+qed.
+*)