--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "datatypes/bool.ma".
+include "higher_order_defs/functions.ma".
+include "nat/plus.ma".
+include "nat/orders.ma".
+
+inductive list (A:Type) : Type :=
+ | 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/list.ind#xpointer(1/1/1) _).
+interpretation "cons" 'cons hd tl =
+ (cic:/matita/list/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:Type.\forall l:list A.\forall a:A.
+ a::l <> [].
+ intros;
+ unfold Not;
+ intros;
+ destruct 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:Type. \lambda l: list A.
+ match l with
+ [ nil => []
+ | (cons hd tl) => tl].
+
+interpretation "append" 'append l1 l2 = (cic:/matita/list/list/append.con _ l1 l2).
+
+theorem append_nil: \forall A:Type.\forall l:list A.l @ [] = l.
+ intros;
+ elim l;
+ [ reflexivity;
+ | simplify;
+ rewrite > H;
+ reflexivity;
+ ]
+qed.
+
+theorem associative_append: \forall A:Type.associative (list A) (append A).
+ intros; unfold; intros;
+ elim x;
+ [ simplify;
+ reflexivity;
+ | simplify;
+ rewrite > H;
+ reflexivity;
+ ]
+qed.
+
+theorem cons_append_commute:
+ \forall A:Type.\forall l1,l2:list A.\forall a:A.
+ a :: (l1 @ l2) = (a :: l1) @ l2.
+ intros;
+ reflexivity;
+qed.
+
+lemma append_cons:\forall A.\forall a:A.\forall l,l1.
+l@(a::l1)=(l@[a])@l1.
+intros.
+rewrite > associative_append.
+reflexivity.
+qed.
+
+inductive permutation (A:Type) : list A -> list A -> Prop \def
+ | refl : \forall l:list A. permutation ? l l
+ | swap : \forall l:list A. \forall x,y:A.
+ permutation ? (x :: y :: l) (y :: x :: l)
+ | trans : \forall l1,l2,l3:list A.
+ permutation ? l1 l2 -> permut1 ? l2 l3 -> permutation ? l1 l3
+with permut1 : list A -> list A -> Prop \def
+ | step : \forall l1,l2:list A. \forall x,y:A.
+ permut1 ? (l1 @ (x :: y :: l2)) (l1 @ (y :: x :: l2)).
+
+include "nat/nat.ma".
+
+definition x1 \def S O.
+definition x2 \def S x1.
+definition x3 \def S x2.
+
+theorem tmp : permutation nat (x1 :: x2 :: x3 :: []) (x1 :: x3 :: x2 :: []).
+ apply (trans ? (x1 :: x2 :: x3 :: []) (x1 :: x2 :: x3 :: []) ?).
+ apply refl.
+ apply (step ? (x1::[]) [] x2 x3).
+ qed.
+
+
+(*
+theorem nil_append_nil_both:
+ \forall A:Type.\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.
+*)
+
+let rec nth (A:Type) l d n on n ≝
+ match n with
+ [ O ⇒
+ match l with
+ [ nil ⇒ d
+ | cons (x : A) _ ⇒ x
+ ]
+ | S n' ⇒ nth A (tail ? l) d n'
+ ].
+
+let rec map (A,B:Type) (f: A → B) (l : list A) on l : list B ≝
+ match l with [ nil ⇒ nil ? | cons x tl ⇒ f x :: (map A B f tl)].
+
+let rec foldr (A,B:Type) (f : A → B → B) (b : B) (l : list A) on l : B :=
+ match l with [ nil ⇒ b | (cons a l) ⇒ f a (foldr ? ? f b l)].
+
+definition length ≝ λT:Type.λl:list T.foldr T nat (λx,c.S c) O l.
+
+definition filter \def
+ \lambda T:Type.\lambda l:list T.\lambda p:T \to bool.
+ foldr T (list T)
+ (\lambda x,l0.match (p x) with [ true => x::l0 | false => l0]) [] l.
+
+definition iota : nat → nat → list nat ≝
+ λn,m. nat_rect (λ_.list ?) (nil ?) (λx,acc.cons ? (n+x) acc) m.
+
+(* ### induction principle for functions visiting 2 lists in parallel *)
+lemma list_ind2 :
+ ∀T1,T2:Type.∀l1:list T1.∀l2:list T2.∀P:list T1 → list T2 → Prop.
+ length ? l1 = length ? l2 →
+ (P (nil ?) (nil ?)) →
+ (∀tl1,tl2,hd1,hd2. P tl1 tl2 → P (hd1::tl1) (hd2::tl2)) →
+ P l1 l2.
+intros (T1 T2 l1 l2 P Hl Pnil Pcons);
+generalize in match Hl; clear Hl; generalize in match l2; clear l2;
+elim l1 1 (l2 x1); [ cases l2; intros (Hl); [assumption| simplify in Hl; destruct Hl]]
+intros 3 (tl1 IH l2); cases l2; [1: simplify; intros 1 (Hl); destruct Hl]
+intros 1 (Hl); apply Pcons; apply IH; simplify in Hl; destruct Hl; assumption;
+qed.
+
+lemma eq_map : ∀A,B,f,g,l. (∀x.f x = g x) → map A B f l = map A B g l.
+intros (A B f g l Efg); elim l; simplify; [1: reflexivity ];
+rewrite > (Efg t); rewrite > H; reflexivity;
+qed.
+
+lemma le_length_filter : \forall A,l,p.length A (filter A l p) \leq length A l.
+intros;elim l
+ [simplify;apply le_n
+ |simplify;apply (bool_elim ? (p t));intro
+ [simplify;apply le_S_S;assumption
+ |simplify;apply le_S;assumption]]
+qed.
\ No newline at end of file
projects / helm.git / blobdiff