Dummy dependent types are no longer cleaned in inductive type arities.

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

diff --git a/helm/software/matita/library/list/list.ma b/helm/software/matita/library/list/list.ma
index f832d502002f3c9a59855a13acea0ac966fa69bc..ad44bd63a287846d917a08d278b57f01378fd205 100644 (file)
--- a/helm/software/matita/library/list/list.ma
+++ b/helm/software/matita/library/list/list.ma
@@ -14,7 +14,10 @@
 
 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
@@ -32,9 +35,9 @@ 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 "nil" 'nil = (cic:/matita/list/list/list.ind#xpointer(1/1/1) _).
 interpretation "cons" 'cons hd tl =
-  (cic:/matita/list/list.ind#xpointer(1/1/2) _ 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) :: []. *)
 
@@ -62,7 +65,7 @@ definition tail := \lambda A:Type. \lambda l: list A.
   [ nil => []
   | (cons hd tl) => tl].
 
-interpretation "append" 'append l1 l2 = (cic:/matita/list/append.con _ l1 l2).
+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;
@@ -92,6 +95,13 @@ theorem cons_append_commute:
   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. 
@@ -143,3 +153,46 @@ let rec nth (A:Type) l d n on n ≝
       ]
   | 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 a); 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 a));intro
+     [simplify;apply le_S_S;assumption
+     |simplify;apply le_S;assumption]]
+qed.
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