include "nat/minus.ma".
include "list/list.ma".
-interpretation "list nth" 'nth = (nth _).
-interpretation "list nth" 'nth_appl l d i = (nth _ l d i).
+interpretation "list nth" 'nth = (nth ?).
+interpretation "list nth" 'nth_appl l d i = (nth ? l d i).
notation "\nth" with precedence 90 for @{'nth}.
notation < "\nth \nbsp term 90 l \nbsp term 90 d \nbsp term 90 i"
with precedence 69 for @{'nth_appl $l $d $i}.
match n with [ O ⇒ nil ? | S m ⇒ def m :: make_list m]
in make_list.
-interpretation "\mk_list appl" 'mk_list_appl f n = (make_list _ f n).
-interpretation "\mk_list" 'mk_list = (make_list _).
+interpretation "\mk_list appl" 'mk_list_appl f n = (make_list ? f n).
+interpretation "\mk_list" 'mk_list = (make_list ?).
notation "\mk_list" with precedence 90 for @{'mk_list}.
notation < "\mk_list \nbsp term 90 f \nbsp term 90 n"
with precedence 69 for @{'mk_list_appl $f $n}.
notation "\len" with precedence 90 for @{'len}.
-interpretation "len" 'len = (length _).
+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).
+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;
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).
+