From cf89508024f0a19023bb1bac343012d54e860d9d Mon Sep 17 00:00:00 2001
From: Andrea Asperti <andrea.asperti@unibo.it>
Date: Tue, 16 Feb 2010 09:21:04 +0000
Subject: [PATCH] More theorems

---
 helm/software/matita/nlibrary/basics/list.ma | 20 +++++++++++++++++++-
 1 file changed, 19 insertions(+), 1 deletion(-)

diff --git a/helm/software/matita/nlibrary/basics/list.ma b/helm/software/matita/nlibrary/basics/list.ma
index 62210608a..b5664d134 100644
--- a/helm/software/matita/nlibrary/basics/list.ma
+++ b/helm/software/matita/nlibrary/basics/list.ma
@@ -54,7 +54,12 @@ nlet rec append A (l1: list A) l2 on l1 :=
   [ nil ⇒  l2
   | cons hd tl ⇒  hd :: append A tl l2 ].
 
-ndefinition tail := λA:Type.λl: list A.
+ndefinition hd ≝ λA:Type.λl: list A.λd:A.
+  match l with 
+  [ nil ⇒ d
+  | cons a _ ⇒ a].
+
+ndefinition tail ≝  λA:Type.λl: list A.
   match l with
   [ nil ⇒  []
   | cons hd tl ⇒  tl].
@@ -76,6 +81,19 @@ ntheorem cons_append_commute:
 ntheorem append_cons:∀A.∀a:A.∀l,l1.l@(a::l1)=(l@[a])@l1.
 /2/; nqed.
 
+ntheorem nil_append_elim: ∀A.∀l1,l2: list A.∀P: list A → list A → Prop. 
+  l1@l2 = [] → P (nil A) (nil A) → P l1 l2.
+#A;#l1; #l2; #P; ncases l1; nnormalize;//;
+#a; #l3; #heq; ndestruct;
+nqed.
+
+ntheorem nil_to_nil:  ∀A.∀l1,l2:list A.
+  l1@l2 = [] → l1 = [] ∧ l2 = [].
+#A; #l1; #l2; #isnil; napply (nil_append_elim A l1 l2);/2/;
+nqed.
+
+(* ierators *)
+
 nlet 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)].
   
-- 
2.39.2