new implementation of the destruct tactic,

[helm.git] / matita / library / Fsub / util.ma

diff --git a/matita/library/Fsub/util.ma b/matita/library/Fsub/util.ma
index fa59484f5dd27e94b73bbdd9625badce717984a0..4f5e1542232f753e7fbb209ecc3fbb0a7ec69760 100644 (file)
--- a/matita/library/Fsub/util.ma
+++ b/matita/library/Fsub/util.ma
@@ -19,16 +19,40 @@ include "list/list.ma".
 
 (*** useful definitions and lemmas not really related to Fsub ***)
 
-let rec max m n \def
+definition max \def 
+\lambda m,n.
   match (leb m n) with
      [true \Rightarrow n
      |false \Rightarrow m]. 
+     
+lemma le_n_max_m_n: \forall m,n:nat. n \le max m n.
+intros.
+unfold max.
+apply (leb_elim m n)
+  [simplify.intros.apply le_n
+  |simplify.intros.autobatch
+  ]
+qed.
+  
+lemma le_m_max_m_n: \forall m,n:nat. m \le max m n.
+intros.
+unfold max.
+apply (leb_elim m n)
+  [simplify.intro.assumption
+  |simplify.intros.autobatch
+  ]
+qed.  
+
+inductive in_list (A:Type): A → (list A) → Prop ≝
+| in_Base : ∀ x,l.(in_list A x (x::l))
+| in_Skip : ∀ x,y,l.(in_list A x l) → (in_list A x (y::l)).
 
-inductive in_list (A : Type) : A \to (list A) \to Prop \def
-  | in_Base : \forall x:A.\forall l:(list A).
-              (in_list A x (x :: l))
-  | in_Skip : \forall x,y:A.\forall l:(list A).
-              (in_list A x l) \to (in_list A x (y :: l)).
+notation > "hvbox(x ∈ l)"
+  non associative with precedence 30 for @{ 'inlist $x $l }.
+notation < "hvbox(x \nbsp ∈ \nbsp l)"
+  non associative with precedence 30 for @{ 'inlist $x $l }.
+interpretation "item in list" 'inlist x l =
+  (cic:/matita/Fsub/util/in_list.ind#xpointer(1/1) _ x l).
 
 definition incl : \forall A.(list A) \to (list A) \to Prop \def
   \lambda A,l,m.\forall x.(in_list A x l) \to (in_list A x m).               
@@ -43,14 +67,68 @@ definition map : \forall A,B,f.((list A) \to (list B)) \def
 
 lemma in_list_nil : \forall A,x.\lnot (in_list A x []).
 intros.unfold.intro.inversion H
-  [intros;lapply (sym_eq ? ? ? H2);absurd (a::l = [])
-     [assumption|apply nil_cons]
-  |intros;lapply (sym_eq ? ? ? H4);absurd (a1::l = [])
-     [assumption|apply nil_cons]]
+  [intros;lapply (sym_eq ? ? ? H2);destruct Hletin
+  |intros;destruct H4]
+qed.
+
+lemma in_cons_case : ∀A.∀x,h:A.∀t:list A.x ∈ h::t → x = h ∨ (x ∈ t).
+intros;inversion H;intros
+  [destruct H2;left;symmetry;reflexivity
+  |destruct H4;right;applyS H1]
+qed.
+
+lemma append_nil:\forall A:Type.\forall l:list A. l@[]=l.
+intros.
+elim l;intros;autobatch.
+qed.
+
+lemma append_cons:\forall A.\forall a:A.\forall l,l1. 
+l@(a::l1)=(l@[a])@l1.
+intros.
+rewrite > associative_append.
+reflexivity.
+qed.
+
+lemma in_list_append1: \forall A.\forall x:A.
+\forall l1,l2. x \in l1 \to x \in l1@l2.
+intros.
+elim H
+  [simplify.apply in_Base
+  |simplify.apply in_Skip.assumption
+  ]
+qed.
+
+lemma in_list_append2: \forall A.\forall x:A.
+\forall l1,l2. x \in l2 \to x \in l1@l2.
+intros 3.
+elim l1
+  [simplify.assumption
+  |simplify.apply in_Skip.apply H.assumption
+  ]
+qed.
+
+theorem append_to_or_in_list: \forall A:Type.\forall x:A.
+\forall l,l1. x \in l@l1 \to (x \in l) \lor (x \in l1).
+intros 3.
+elim l
+  [right.apply H
+  |simplify in H1.inversion H1;intros; destruct;
+    [left.apply in_Base
+    | elim (H l2)
+      [left.apply in_Skip. assumption
+      |right.assumption
+      |assumption
+      ]
+    ]
+  ]
 qed.
 
 lemma max_case : \forall m,n.(max m n) = match (leb m n) with
-      [ false \Rightarrow n
-      | true \Rightarrow m ].
+      [ true \Rightarrow n
+      | false \Rightarrow m ].
 intros;elim m;simplify;reflexivity;
+qed.
+
+lemma incl_A_A: ∀T,A.incl T A A.
+intros.unfold incl.intros.assumption.
 qed.
\ No newline at end of file