nat model ported to the dualized version, but not itself dualized dedekind-sig-compl...

[helm.git] / helm / software / matita / contribs / dama / dama / models / list_support.ma

diff --git a/helm/software/matita/contribs/dama/dama/models/list_support.ma b/helm/software/matita/contribs/dama/dama/models/list_support.ma
index ea701bb9db69cba6b1efa532efc6b01123fef0d8..eb70322a9a30a0307aceb63500bf1d3085060392 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/models/list_support.ma
+++ b/helm/software/matita/contribs/dama/dama/models/list_support.ma
@@ -38,6 +38,11 @@ interpretation "len" 'len = (length _).
 notation < "\len \nbsp term 90 l" with precedence 69 for @{'len_appl $l}.
 interpretation "len appl" 'len_appl l = (length _ l).
 
+lemma mk_list_ext: ∀T:Type.∀f1,f2:nat→T.∀n. (∀x.x<n → f1 x = f2 x) → \mk_list f1 n = \mk_list f2 n.
+intros 4;elim n; [reflexivity] simplify; rewrite > H1; [2: apply le_n]
+apply eq_f; apply H; intros; apply H1; apply (trans_le ??? H2); apply le_S; apply le_n;
+qed.
+
 lemma len_mk_list : ∀T:Type.∀f:nat→T.∀n.\len (\mk_list f n) = n.
 intros; elim n; [reflexivity] simplify; rewrite > H; reflexivity;
 qed.
@@ -242,3 +247,36 @@ intros; cases (find_lemma ? f l t); apply w; qed.
 
 lemma cases_find: ∀T,P,l,d. find_spec T P l d (find T P l d) (find T P l d).
 intros; unfold find; cases (find_lemma T P l d); simplify; assumption; qed.
+
+lemma list_elim_with_len:
+  ∀T:Type.∀P: nat → list T → CProp.
+    P O [] → (∀l,a,n.P (\len l) l → P (S n) (a::l)) →
+     ∀l.P (\len l) l.
+intros;elim l; [assumption] simplify; apply H1; apply H2;
+qed.
+ 
+lemma sorted_near:
+ ∀r,l. sorted r l → ∀i,d. S i < \len l → r (\nth l d i) (\nth l d (S i)).
+ intros 3; elim H; 
+ [1: cases (not_le_Sn_O ? H1);
+ |2: simplify in H1; cases (not_le_Sn_O ? (le_S_S_to_le ?? H1));
+ |3: simplify; cases i in H4; intros; [apply H1]
+     apply H3; apply le_S_S_to_le; apply H4]
+qed.  
+ 
+definition last ≝
+ λT:Type.λd.λl:list T. \nth l d (pred (\len l)).
+
+notation > "\last" non associative with precedence 90 for @{'last}.
+notation < "\last d l" non associative with precedence 70 for @{'last2 $d $l}.
+interpretation "list last" 'last = (last _). 
+interpretation "list last applied" 'last2 d l = (last _ d l).
+
+definition head ≝
+ λT:Type.λd.λl:list T.\nth l d O.
+
+notation > "\hd" non associative with precedence 90 for @{'hd}.
+notation < "\hd d l" non associative with precedence 70 for @{'hd2 $d $l}.
+interpretation "list head" 'hd = (head _).
+interpretation "list head applied" 'hd2 d l = (head _ d l).
+