converted;

--- a/src/HOL/List.ML	Tue May 07 19:54:04 2002 +0200
+++ b/src/HOL/List.ML	Tue May 07 19:54:29 2002 +0200
@@ -1,1555 +1,269 @@
-(*  Title:      HOL/List
-    ID:         $Id$
-    Author:     Tobias Nipkow
-    Copyright   1994 TU Muenchen
 
-List lemmas
-*)
-
-Goal "!x. xs ~= x#xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed_spec_mp "not_Cons_self";
-bind_thm("not_Cons_self2",not_Cons_self RS not_sym);
-Addsimps [not_Cons_self,not_Cons_self2];
-
-Goal "(xs ~= []) = (? y ys. xs = y#ys)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "neq_Nil_conv";
-
-(* Induction over the length of a list: *)
-val [prem] = Goal
-  "(!!xs. (!ys. length ys < length xs --> P ys) ==> P xs) ==> P(xs)";
-by (rtac measure_induct 1 THEN etac prem 1);
-qed "length_induct";
-
-
-(** "lists": the list-forming operator over sets **)
-
-Goalw lists.defs "A<=B ==> lists A <= lists B";
-by (rtac lfp_mono 1);
-by (REPEAT (ares_tac basic_monos 1));
-qed "lists_mono";
-
-bind_thm ("listsE", lists.mk_cases "x#l : lists A");
-AddSEs [listsE];
-AddSIs lists.intrs;
-
-Goal "l: lists A ==> l: lists B --> l: lists (A Int B)";
-by (etac lists.induct 1);
-by (ALLGOALS Blast_tac);
-qed_spec_mp "lists_IntI";
-
-Goal "lists (A Int B) = lists A Int lists B";
-by (rtac (mono_Int RS equalityI) 1);
-by (simp_tac (simpset() addsimps [mono_def, lists_mono]) 1);
-by (blast_tac (claset() addSIs [lists_IntI]) 1);
-qed "lists_Int_eq";
-Addsimps [lists_Int_eq];
-
-Goal "(xs@ys : lists A) = (xs : lists A & ys : lists A)";
-by (induct_tac "xs" 1);
-by (Auto_tac);
-qed "append_in_lists_conv";
-AddIffs [append_in_lists_conv];
-
-(** length **)
-(* needs to come before "@" because of thm append_eq_append_conv *)
-
-section "length";
-
-Goal "length(xs@ys) = length(xs)+length(ys)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed"length_append";
-Addsimps [length_append];
+(** legacy ML bindings **)
 
-Goal "length (map f xs) = length xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "length_map";
-Addsimps [length_map];
-
-Goal "length(rev xs) = length(xs)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "length_rev";
-Addsimps [length_rev];
-
-Goal "length(tl xs) = (length xs) - 1";
-by (case_tac "xs" 1);
-by Auto_tac;
-qed "length_tl";
-Addsimps [length_tl];
-
-Goal "(length xs = 0) = (xs = [])";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "length_0_conv";
-AddIffs [length_0_conv];
-
-Goal "(0 < length xs) = (xs ~= [])";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "length_greater_0_conv";
-AddIffs [length_greater_0_conv];
+structure List =
+struct
 
-Goal "(length xs = Suc n) = (? y ys. xs = y#ys & length ys = n)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "length_Suc_conv";
-
-(** @ - append **)
-
-section "@ - append";
-
-Goal "(xs@ys)@zs = xs@(ys@zs)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "append_assoc";
-Addsimps [append_assoc];
-
-Goal "xs @ [] = xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "append_Nil2";
-Addsimps [append_Nil2];
-
-Goal "(xs@ys = []) = (xs=[] & ys=[])";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "append_is_Nil_conv";
-AddIffs [append_is_Nil_conv];
-
-Goal "([] = xs@ys) = (xs=[] & ys=[])";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "Nil_is_append_conv";
-AddIffs [Nil_is_append_conv];
-
-Goal "(xs @ ys = xs) = (ys=[])";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "append_self_conv";
+val thy = the_context ();
 
-Goal "(xs = xs @ ys) = (ys=[])";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "self_append_conv";
-AddIffs [append_self_conv,self_append_conv];
-
-Goal "!ys. length xs = length ys | length us = length vs \
-\              --> (xs@us = ys@vs) = (xs=ys & us=vs)";
-by (induct_tac "xs" 1);
- by (rtac allI 1);
- by (case_tac "ys" 1);
-  by (Asm_simp_tac 1);
- by (Force_tac 1);
-by (rtac allI 1);
-by (case_tac "ys" 1);
-by (Force_tac 1);
-by (Asm_simp_tac 1);
-qed_spec_mp "append_eq_append_conv";
-Addsimps [append_eq_append_conv];
-
-Goal "(xs @ ys = xs @ zs) = (ys=zs)";
-by (Simp_tac 1);
-qed "same_append_eq";
-
-Goal "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; 
-by (Simp_tac 1);
-qed "append1_eq_conv";
-
-Goal "(ys @ xs = zs @ xs) = (ys=zs)";
-by (Simp_tac 1);
-qed "append_same_eq";
-
-AddIffs [same_append_eq, append1_eq_conv, append_same_eq];
-
-Goal "(xs @ ys = ys) = (xs=[])";
-by (cut_inst_tac [("zs","[]")] append_same_eq 1);
-by Auto_tac;
-qed "append_self_conv2";
-
-Goal "(ys = xs @ ys) = (xs=[])";
-by (simp_tac (simpset() addsimps
-     [simplify (simpset()) (read_instantiate[("ys","[]")]append_same_eq)]) 1);
-by (Blast_tac 1);
-qed "self_append_conv2";
-AddIffs [append_self_conv2,self_append_conv2];
-
-Goal "xs ~= [] --> hd xs # tl xs = xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed_spec_mp "hd_Cons_tl";
-Addsimps [hd_Cons_tl];
-
-Goal "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "hd_append";
-
-Goal "xs ~= [] ==> hd(xs @ ys) = hd xs";
-by (asm_simp_tac (simpset() addsimps [hd_append]
-                           addsplits [list.split]) 1);
-qed "hd_append2";
-Addsimps [hd_append2];
-
-Goal "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
-by (simp_tac (simpset() addsplits [list.split]) 1);
-qed "tl_append";
-
-Goal "xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys";
-by (asm_simp_tac (simpset() addsimps [tl_append]
-                           addsplits [list.split]) 1);
-qed "tl_append2";
-Addsimps [tl_append2];
-
-(* trivial rules for solving @-equations automatically *)
-
-Goal "xs = ys ==> xs = [] @ ys";
-by (Asm_simp_tac 1);
-qed "eq_Nil_appendI";
-
-Goal "[| x#xs1 = ys; xs = xs1 @ zs |] ==> x#xs = ys@zs";
-by (dtac sym 1);
-by (Asm_simp_tac 1);
-qed "Cons_eq_appendI";
-
-Goal "[| xs@xs1 = zs; ys = xs1 @ us |] ==> xs@ys = zs@us";
-by (dtac sym 1);
-by (Asm_simp_tac 1);
-qed "append_eq_appendI";
-
-
-(***
-Simplification procedure for all list equalities.
-Currently only tries to rearranges @ to see if
-- both lists end in a singleton list,
-- or both lists end in the same list.
-***)
-local
-
-val list_eq_pattern =
-  Thm.read_cterm (Theory.sign_of (the_context ())) ("(xs::'a list) = ys",HOLogic.boolT)
-
-fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
-      (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
-  | last (Const("List.op @",_) $ _ $ ys) = last ys
-  | last t = t
-
-fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
-  | list1 _ = false
-
-fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
-      (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
-  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
-  | butlast xs = Const("List.list.Nil",fastype_of xs)
-
-val rearr_tac =
-  simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons])
-
-fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
-  let
-    val lastl = last lhs and lastr = last rhs
-    fun rearr conv =
-      let val lhs1 = butlast lhs and rhs1 = butlast rhs
-          val Type(_,listT::_) = eqT
-          val appT = [listT,listT] ---> listT
-          val app = Const("List.op @",appT)
-          val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
-          val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2)))
-          val thm = prove_goalw_cterm [] ct (K [rearr_tac 1])
-            handle ERROR =>
-            error("The error(s) above occurred while trying to prove " ^
-                  string_of_cterm ct)
-      in Some((conv RS (thm RS trans)) RS eq_reflection) end
-
-  in if list1 lastl andalso list1 lastr
-     then rearr append1_eq_conv
-     else
-     if lastl aconv lastr
-     then rearr append_same_eq
-     else None
-  end
-in
-val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq
+structure list =
+struct
+  val distinct = thms "list.distinct";
+  val inject = thms "list.inject";
+  val exhaust = thm "list.exhaust";
+  val cases = thms "list.cases";
+  val split = thm "list.split";
+  val split_asm = thm "list.split_asm";
+  val induct = thm "list.induct";
+  val recs = thms "list.recs";
+  val simps = thms "list.simps";
+  val size = thms "list.size";
 end;
 
-Addsimprocs [list_eq_simproc];
-
-
-(** map **)
-
-section "map";
-
-Goal "(!x. x : set xs --> f x = g x) --> map f xs = map g xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-bind_thm("map_ext", impI RS (allI RS (result() RS mp)));
-
-Goal "map (%x. x) = (%xs. xs)";
-by (rtac ext 1);
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "map_ident";
-Addsimps[map_ident];
-
-Goal "map f (xs@ys) = map f xs @ map f ys";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "map_append";
-Addsimps[map_append];
-
-Goalw [o_def] "map (f o g) xs = map f (map g xs)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "map_compose";
-(*Addsimps[map_compose];*)
-
-Goal "rev(map f xs) = map f (rev xs)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "rev_map";
-
-(* a congruence rule for map: *)
-Goal "xs=ys ==> (!x. x : set ys --> f x = g x) --> map f xs = map g ys";
-by (hyp_subst_tac 1);
-by (induct_tac "ys" 1);
-by Auto_tac;
-bind_thm("map_cong", impI RSN (2,allI RSN (2, result() RS mp)));
-
-Goal "(map f xs = []) = (xs = [])";
-by (case_tac "xs" 1);
-by Auto_tac;
-qed "map_is_Nil_conv";
-AddIffs [map_is_Nil_conv];
-
-Goal "([] = map f xs) = (xs = [])";
-by (case_tac "xs" 1);
-by Auto_tac;
-qed "Nil_is_map_conv";
-AddIffs [Nil_is_map_conv];
-
-Goal "(map f xs = y#ys) = (? x xs'. xs = x#xs' & f x = y & map f xs' = ys)";
-by (case_tac "xs" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "map_eq_Cons";
-
-Goal "!xs. map f xs = map f ys --> (!x y. f x = f y --> x=y) --> xs=ys";
-by (induct_tac "ys" 1);
- by (Asm_simp_tac 1);
-by (fast_tac (claset() addss (simpset() addsimps [map_eq_Cons])) 1);
-qed_spec_mp "map_injective";
-
-Goal "inj f ==> inj (map f)";
-by (blast_tac (claset() addDs [map_injective,injD] addIs [injI]) 1);
-qed "inj_mapI";
-
-Goalw [inj_on_def] "inj (map f) ==> inj f";
-by (Clarify_tac 1);
-by (eres_inst_tac [("x","[x]")] ballE 1);
- by (eres_inst_tac [("x","[y]")] ballE 1);
-  by (Asm_full_simp_tac 1);
- by (Blast_tac 1);
-by (Blast_tac 1);
-qed "inj_mapD";
-
-Goal "inj (map f) = inj f";
-by (blast_tac (claset() addDs [inj_mapD] addIs [inj_mapI]) 1);
-qed "inj_map";
-
-(** rev **)
-
-section "rev";
-
-Goal "rev(xs@ys) = rev(ys) @ rev(xs)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "rev_append";
-Addsimps[rev_append];
-
-Goal "rev(rev l) = l";
-by (induct_tac "l" 1);
-by Auto_tac;
-qed "rev_rev_ident";
-Addsimps[rev_rev_ident];
-
-Goal "(rev xs = []) = (xs = [])";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "rev_is_Nil_conv";
-AddIffs [rev_is_Nil_conv];
-
-Goal "([] = rev xs) = (xs = [])";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "Nil_is_rev_conv";
-AddIffs [Nil_is_rev_conv];
-
-Goal "!ys. (rev xs = rev ys) = (xs = ys)";
-by (induct_tac "xs" 1);
- by (Force_tac 1);
-by (rtac allI 1);
-by (case_tac "ys" 1);
- by (Asm_simp_tac 1);
-by (Force_tac 1);
-qed_spec_mp "rev_is_rev_conv";
-AddIffs [rev_is_rev_conv];
-
-val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs";
-by (stac (rev_rev_ident RS sym) 1);
-by (res_inst_tac [("list", "rev xs")] list.induct 1);
-by (ALLGOALS Simp_tac);
-by (resolve_tac prems 1);
-by (eresolve_tac prems 1);
-qed "rev_induct";
-
-val rev_induct_tac = induct_thm_tac rev_induct;
-
-Goal  "(xs = [] --> P) -->  (!ys y. xs = ys@[y] --> P) --> P";
-by (rev_induct_tac "xs" 1);
-by Auto_tac;
-qed "rev_exhaust_aux";
-
-bind_thm ("rev_exhaust", ObjectLogic.rulify rev_exhaust_aux);
-
-
-(** set **)
-
-section "set";
-
-Goal "finite (set xs)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "finite_set";
-AddIffs [finite_set];
-
-Goal "set (xs@ys) = (set xs Un set ys)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "set_append";
-Addsimps[set_append];
-
-Goal "set l <= set (x#l)";
-by Auto_tac;
-qed "set_subset_Cons";
-
-Goal "(set xs = {}) = (xs = [])";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "set_empty";
-Addsimps [set_empty];
-
-Goal "set(rev xs) = set(xs)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "set_rev";
-Addsimps [set_rev];
-
-Goal "set(map f xs) = f`(set xs)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "set_map";
-Addsimps [set_map];
-
-Goal "set(filter P xs) = {x. x : set xs & P x}";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "set_filter";
-Addsimps [set_filter];
-
-Goal "set[i..j(] = {k. i <= k & k < j}";
-by (induct_tac "j" 1);
-by (ALLGOALS Asm_simp_tac);
-by (etac ssubst 1);
-by Auto_tac;
-by (arith_tac 1);
-qed "set_upt";
-Addsimps [set_upt];
+structure lists =
+struct
+  val intrs = thms "lists.intros";
+  val elims = thms "lists.cases";
+  val elim = thm "lists.cases";
+  val induct = thm "lists.induct";
+  val mk_cases = InductivePackage.the_mk_cases (the_context ()) "List.lists";
+  val [Nil, Cons] = intrs;
+end;
 
-Goal "(x : set xs) = (? ys zs. xs = ys@x#zs)";
-by (induct_tac "xs" 1);
- by (Simp_tac 1);
-by (Asm_simp_tac 1);
-by (rtac iffI 1);
-by (blast_tac (claset() addIs [eq_Nil_appendI,Cons_eq_appendI]) 1);
-by (REPEAT(etac exE 1));
-by (case_tac "ys" 1);
-by Auto_tac;
-qed "in_set_conv_decomp";
-
-
-(* eliminate `lists' in favour of `set' *)
-
-Goal "(xs : lists A) = (!x : set xs. x : A)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "in_lists_conv_set";
-
-bind_thm("in_listsD",in_lists_conv_set RS iffD1);
-AddSDs [in_listsD];
-bind_thm("in_listsI",in_lists_conv_set RS iffD2);
-AddSIs [in_listsI];
-
-(** mem **)
- 
-section "mem";
-
-Goal "(x mem xs) = (x: set xs)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "set_mem_eq";
-
-
-(** list_all **)
-
-section "list_all";
-
-Goal "list_all P xs = (!x:set xs. P x)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "list_all_conv";
-
-Goal "list_all P (xs@ys) = (list_all P xs & list_all P ys)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "list_all_append";
-Addsimps [list_all_append];
-
-
-(** filter **)
-
-section "filter";
-
-Goal "filter P (xs@ys) = filter P xs @ filter P ys";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "filter_append";
-Addsimps [filter_append];
-
-Goal "filter P (filter Q xs) = filter (%x. Q x & P x) xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "filter_filter";
-Addsimps [filter_filter];
-
-Goal "(!x : set xs. P x) --> filter P xs = xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed_spec_mp "filter_True";
-Addsimps [filter_True];
-
-Goal "(!x : set xs. ~P x) --> filter P xs = []";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed_spec_mp "filter_False";
-Addsimps [filter_False];
-
-Goal "length (filter P xs) <= length xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-by (asm_simp_tac (simpset() addsimps [le_SucI]) 1);
-qed "length_filter";
-Addsimps[length_filter];
-
-Goal "set (filter P xs) <= set xs";
-by Auto_tac;
-qed "filter_is_subset";
-Addsimps [filter_is_subset];
-
-
-section "concat";
-
-Goal  "concat(xs@ys) = concat(xs)@concat(ys)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed"concat_append";
-Addsimps [concat_append];
+end;
 
-Goal "(concat xss = []) = (!xs:set xss. xs=[])";
-by (induct_tac "xss" 1);
-by Auto_tac;
-qed "concat_eq_Nil_conv";
-AddIffs [concat_eq_Nil_conv];
-
-Goal "([] = concat xss) = (!xs:set xss. xs=[])";
-by (induct_tac "xss" 1);
-by Auto_tac;
-qed "Nil_eq_concat_conv";
-AddIffs [Nil_eq_concat_conv];
-
-Goal  "set(concat xs) = Union(set ` set xs)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed"set_concat";
-Addsimps [set_concat];
-
-Goal "map f (concat xs) = concat (map (map f) xs)"; 
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "map_concat";
-
-Goal "filter p (concat xs) = concat (map (filter p) xs)"; 
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed"filter_concat"; 
-
-Goal "rev(concat xs) = concat (map rev (rev xs))";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "rev_concat";
-
-(** nth **)
-
-section "nth";
-
-Goal "(x#xs)!0 = x";
-by Auto_tac;
-qed "nth_Cons_0";
-Addsimps [nth_Cons_0];
-
-Goal "(x#xs)!(Suc n) = xs!n";
-by Auto_tac;
-qed "nth_Cons_Suc";
-Addsimps [nth_Cons_Suc];
-
-Delsimps (thms "nth.simps");
-
-Goal "!n. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))";
-by (induct_tac "xs" 1);
- by (Asm_simp_tac 1);
- by (rtac allI 1);
- by (case_tac "n" 1);
-  by Auto_tac;
-qed_spec_mp "nth_append";
-
-Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)";
-by (induct_tac "xs" 1);
- by (Asm_full_simp_tac 1);
-by (rtac allI 1);
-by (induct_tac "n" 1);
-by Auto_tac;
-qed_spec_mp "nth_map";
-Addsimps [nth_map];
-
-Goal "set xs = {xs!i |i. i < length xs}";
-by (induct_tac "xs" 1);
- by (Simp_tac 1);
-by (Asm_simp_tac 1);
-by Safe_tac;
-  by (res_inst_tac [("x","0")] exI 1);
-  by (Simp_tac 1);
- by (res_inst_tac [("x","Suc i")] exI 1);
- by (Asm_simp_tac 1);
-by (case_tac "i" 1);
- by (Asm_full_simp_tac 1);
-by (rename_tac "j" 1);
- by (res_inst_tac [("x","j")] exI 1);
-by (Asm_simp_tac 1);
-qed "set_conv_nth";
-
-Goal "n < length xs ==> Ball (set xs) P --> P(xs!n)";
-by (simp_tac (simpset() addsimps [set_conv_nth]) 1);
-by (Blast_tac 1);
-qed_spec_mp "list_ball_nth";
-
-Goal "n < length xs ==> xs!n : set xs";
-by (simp_tac (simpset() addsimps [set_conv_nth]) 1);
-by (Blast_tac 1);
-qed_spec_mp "nth_mem";
-Addsimps [nth_mem];
-
-Goal "(!i. i < length xs --> P(xs!i)) --> (!x : set xs. P x)";
-by (simp_tac (simpset() addsimps [set_conv_nth]) 1);
-by (Blast_tac 1);
-qed_spec_mp "all_nth_imp_all_set";
-
-Goal "(!x : set xs. P x) = (!i. i<length xs --> P (xs ! i))";
-by (simp_tac (simpset() addsimps [set_conv_nth]) 1);
-by (Blast_tac 1);
-qed_spec_mp "all_set_conv_all_nth";
+open List;
 
 
-(** list update **)
-
-section "list update";
-
-Goal "!i. length(xs[i:=x]) = length xs";
-by (induct_tac "xs" 1);
-by (Simp_tac 1);
-by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1);
-qed_spec_mp "length_list_update";
-Addsimps [length_list_update];
-
-Goal "!i j. i < length xs  --> (xs[i:=x])!j = (if i=j then x else xs!j)";
-by (induct_tac "xs" 1);
- by (Simp_tac 1);
-by (auto_tac (claset(), simpset() addsimps [nth_Cons] addsplits [nat.split]));
-qed_spec_mp "nth_list_update";
-
-Goal "i < length xs  ==> (xs[i:=x])!i = x";
-by (asm_simp_tac (simpset() addsimps [nth_list_update]) 1);
-qed "nth_list_update_eq";
-Addsimps [nth_list_update_eq];
-
-Goal "!i j. i ~= j --> xs[i:=x]!j = xs!j";
-by (induct_tac "xs" 1);
- by (Simp_tac 1);
-by (auto_tac (claset(), simpset() addsimps [nth_Cons] addsplits [nat.split]));
-qed_spec_mp "nth_list_update_neq";
-Addsimps [nth_list_update_neq];
-
-Goal "!i. i < size xs --> xs[i:=x, i:=y] = xs[i:=y]";
-by (induct_tac "xs" 1);
- by (Simp_tac 1);
-by (asm_simp_tac (simpset() addsplits [nat.split]) 1);
-qed_spec_mp "list_update_overwrite";
-Addsimps [list_update_overwrite];
-
-Goal "!i < length xs. (xs[i := x] = xs) = (xs!i = x)";
-by (induct_tac "xs" 1);
- by (Simp_tac 1);
-by (simp_tac (simpset() addsplits [nat.split]) 1);
-by (Blast_tac 1);
-qed_spec_mp "list_update_same_conv";
-
-Goal "!i xy xs. length xs = length ys --> \
-\     (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])";
-by (induct_tac "ys" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by (auto_tac (claset(), simpset() addsplits [nat.split]));
-qed_spec_mp "update_zip";
-
-Goal "!i. set(xs[i:=x]) <= insert x (set xs)";
-by (induct_tac "xs" 1);
- by (asm_full_simp_tac (simpset() addsimps []) 1);
-by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1);
-by (Fast_tac  1);
-qed_spec_mp "set_update_subset_insert";
-
-Goal "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A";
-by (fast_tac (claset() addSDs [set_update_subset_insert RS subsetD]) 1);
-qed "set_update_subsetI";
-
-(** last & butlast **)
-
-section "last / butlast";
-
-Goal "last(xs@[x]) = x";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "last_snoc";
-Addsimps [last_snoc];
-
-Goal "butlast(xs@[x]) = xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "butlast_snoc";
-Addsimps [butlast_snoc];
-
-Goal "length(butlast xs) = length xs - 1";
-by (rev_induct_tac "xs" 1);
-by Auto_tac;
-qed "length_butlast";
-Addsimps [length_butlast];
-
-Goal "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed_spec_mp "butlast_append";
-
-Goal "xs ~= [] --> butlast xs @ [last xs] = xs";
-by (induct_tac "xs" 1);
-by (ALLGOALS Asm_simp_tac);
-qed_spec_mp "append_butlast_last_id";
-Addsimps [append_butlast_last_id];
-
-Goal "x:set(butlast xs) --> x:set xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed_spec_mp "in_set_butlastD";
-
-Goal "x:set(butlast xs) | x:set(butlast ys) ==> x:set(butlast(xs@ys))";
-by (auto_tac (claset() addDs [in_set_butlastD],
-	      simpset() addsimps [butlast_append]));
-qed "in_set_butlast_appendI";
-
-(** take  & drop **)
-section "take & drop";
-
-Goal "take 0 xs = []";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "take_0";
-
-Goal "drop 0 xs = xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "drop_0";
-
-Goal "take (Suc n) (x#xs) = x # take n xs";
-by (Simp_tac 1);
-qed "take_Suc_Cons";
-
-Goal "drop (Suc n) (x#xs) = drop n xs";
-by (Simp_tac 1);
-qed "drop_Suc_Cons";
-
-Delsimps [take_Cons,drop_Cons];
-Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
-
-Goal "!xs. length(take n xs) = min (length xs) n";
-by (induct_tac "n" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by Auto_tac;
-qed_spec_mp "length_take";
-Addsimps [length_take];
-
-Goal "!xs. length(drop n xs) = (length xs - n)";
-by (induct_tac "n" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by Auto_tac;
-qed_spec_mp "length_drop";
-Addsimps [length_drop];
-
-Goal "!xs. length xs <= n --> take n xs = xs";
-by (induct_tac "n" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by Auto_tac;
-qed_spec_mp "take_all";
-Addsimps [take_all];
-
-Goal "!xs. length xs <= n --> drop n xs = []";
-by (induct_tac "n" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by Auto_tac;
-qed_spec_mp "drop_all";
-Addsimps [drop_all];
-
-Goal "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
-by (induct_tac "n" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by Auto_tac;
-qed_spec_mp "take_append";
-Addsimps [take_append];
-
-Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
-by (induct_tac "n" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by Auto_tac;
-qed_spec_mp "drop_append";
-Addsimps [drop_append];
-
-Goal "!xs n. take n (take m xs) = take (min n m) xs"; 
-by (induct_tac "m" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by Auto_tac;
-by (case_tac "na" 1);
- by Auto_tac;
-qed_spec_mp "take_take";
-Addsimps [take_take];
-
-Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; 
-by (induct_tac "m" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by Auto_tac;
-qed_spec_mp "drop_drop";
-Addsimps [drop_drop];
-
-Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
-by (induct_tac "m" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by Auto_tac;
-qed_spec_mp "take_drop";
-
-Goal "!xs. take n xs @ drop n xs = xs";
-by (induct_tac "n" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by Auto_tac;
-qed_spec_mp "append_take_drop_id";
-Addsimps [append_take_drop_id];
-
-Goal "!xs. take n (map f xs) = map f (take n xs)"; 
-by (induct_tac "n" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by Auto_tac;
-qed_spec_mp "take_map"; 
-
-Goal "!xs. drop n (map f xs) = map f (drop n xs)"; 
-by (induct_tac "n" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by Auto_tac;
-qed_spec_mp "drop_map";
-
-Goal "!i. rev (take i xs) = drop (length xs - i) (rev xs)";
-by(induct_tac "xs" 1);
- by Auto_tac;
-by (case_tac "i" 1);
- by Auto_tac;
-qed_spec_mp "rev_take";
-
-Goal "!i. rev (drop i xs) = take (length xs - i) (rev xs)";
-by(induct_tac "xs" 1);
- by Auto_tac;
-by (case_tac "i" 1);
- by Auto_tac;
-qed_spec_mp "rev_drop";
-
-Goal "!n i. i < n --> (take n xs)!i = xs!i";
-by (induct_tac "xs" 1);
- by Auto_tac;
-by (case_tac "n" 1);
- by (Blast_tac 1);
-by (case_tac "i" 1);
- by Auto_tac;
-qed_spec_mp "nth_take";
-Addsimps [nth_take];
-
-Goal  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
-by (induct_tac "n" 1);
- by Auto_tac;
-by (case_tac "xs" 1);
- by Auto_tac;
-qed_spec_mp "nth_drop";
-Addsimps [nth_drop];
-
-
-Goal
- "!zs. (xs@ys = zs) = (xs = take (length xs) zs & ys = drop (length xs) zs)";
-by (induct_tac "xs" 1);
- by (Simp_tac 1);
-by (Asm_full_simp_tac 1);
-by (Clarify_tac 1);
-by (case_tac "zs" 1);
-by (Auto_tac);
-qed_spec_mp "append_eq_conv_conj";
-
-(** takeWhile & dropWhile **)
-
-section "takeWhile & dropWhile";
-
-Goal "takeWhile P xs @ dropWhile P xs = xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "takeWhile_dropWhile_id";
-Addsimps [takeWhile_dropWhile_id];
-
-Goal  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-bind_thm("takeWhile_append1", conjI RS (result() RS mp));
-Addsimps [takeWhile_append1];
-
-Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
-by (induct_tac "xs" 1);
-by Auto_tac;
-bind_thm("takeWhile_append2", ballI RS (result() RS mp));
-Addsimps [takeWhile_append2];
-
-Goal "~P(x) ==> takeWhile P (xs @ (x#l)) = takeWhile P xs";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "takeWhile_tail";
-
-Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
-by (induct_tac "xs" 1);
-by Auto_tac;
-bind_thm("dropWhile_append1", conjI RS (result() RS mp));
-Addsimps [dropWhile_append1];
-
-Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
-by (induct_tac "xs" 1);
-by Auto_tac;
-bind_thm("dropWhile_append2", ballI RS (result() RS mp));
-Addsimps [dropWhile_append2];
-
-Goal "x:set(takeWhile P xs) --> x:set xs & P x";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed_spec_mp"set_take_whileD";
-
-(** zip **)
-section "zip";
-
-Goal "zip [] ys = []";
-by (induct_tac "ys" 1);
-by Auto_tac;
-qed "zip_Nil";
-Addsimps [zip_Nil];
-
-Goal "zip (x#xs) (y#ys) = (x,y)#zip xs ys";
-by (Simp_tac 1);
-qed "zip_Cons_Cons";
-Addsimps [zip_Cons_Cons];
-
-Delsimps(tl (thms"zip.simps"));
-
-Goal "!xs. length (zip xs ys) = min (length xs) (length ys)";
-by (induct_tac "ys" 1);
- by (Simp_tac 1);
-by (Clarify_tac 1);
-by (case_tac "xs" 1);
- by (Auto_tac);
-qed_spec_mp "length_zip";
-Addsimps [length_zip];
-
-Goal
- "!xs. zip (xs@ys) zs = \
-\      zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)";
-by (induct_tac "zs" 1);
- by (Simp_tac 1);
-by (Clarify_tac 1);
-by (case_tac "xs" 1);
- by (Asm_simp_tac 1);
-by (Asm_simp_tac 1);
-qed_spec_mp "zip_append1";
-
-Goal
- "!ys. zip xs (ys@zs) = \
-\      zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs";
-by (induct_tac "xs" 1);
- by (Simp_tac 1);
-by (Clarify_tac 1);
-by (case_tac "ys" 1);
- by (Asm_simp_tac 1);
-by (Asm_simp_tac 1);
-qed_spec_mp "zip_append2";
-
-Goal
- "[| length xs = length us; length ys = length vs |] ==> \
-\ zip (xs@ys) (us@vs) = zip xs us @ zip ys vs";
-by (asm_simp_tac (simpset() addsimps [zip_append1]) 1);
-qed_spec_mp "zip_append";
-Addsimps [zip_append];
-
-Goal "!xs. length xs = length ys --> zip (rev xs) (rev ys) = rev (zip xs ys)";
-by (induct_tac "ys" 1);
- by (Asm_full_simp_tac 1);
-by (Asm_full_simp_tac 1);
-by (Clarify_tac 1);
-by (case_tac "xs" 1);
- by (Auto_tac);
-qed_spec_mp "zip_rev";
-
-
-Goal
-"!i xs. i < length xs --> i < length ys --> (zip xs ys)!i = (xs!i, ys!i)";
-by (induct_tac "ys" 1);
- by (Simp_tac 1);
-by (Clarify_tac 1);
-by (case_tac "xs" 1);
- by (Auto_tac);
-by (asm_full_simp_tac (simpset() addsimps (thms"nth.simps") addsplits [nat.split]) 1);
-qed_spec_mp "nth_zip";
-Addsimps [nth_zip];
-
-Goal "set(zip xs ys) = {(xs!i,ys!i) |i. i < min (length xs) (length ys)}";
-by (simp_tac (simpset() addsimps [set_conv_nth]addcongs [rev_conj_cong]) 1);
-qed_spec_mp "set_zip";
-
-Goal
- "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]";
-by (rtac sym 1);
-by (asm_simp_tac (simpset() addsimps [update_zip]) 1);
-qed_spec_mp "zip_update";
-
-Goal "!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)";
-by (induct_tac "i" 1);
- by (Auto_tac);
-by (case_tac "j" 1);
- by (Auto_tac);
-qed "zip_replicate";
-Addsimps [zip_replicate];
-
-(** list_all2 **)
-section "list_all2";
-
-Goalw [list_all2_def] "list_all2 P xs ys ==> length xs = length ys";
-by (Asm_simp_tac 1);
-qed "list_all2_lengthD";
-
-Goalw [list_all2_def] "list_all2 P [] ys = (ys=[])";
-by (Simp_tac 1);
-qed "list_all2_Nil";
-AddIffs [list_all2_Nil];
-
-Goalw [list_all2_def] "list_all2 P xs [] = (xs=[])";
-by (Simp_tac 1);
-qed "list_all2_Nil2";
-AddIffs [list_all2_Nil2];
-
-Goalw [list_all2_def]
- "list_all2 P (x#xs) (y#ys) = (P x y & list_all2 P xs ys)";
-by (Auto_tac);
-qed "list_all2_Cons";
-AddIffs[list_all2_Cons];
-
-Goalw [list_all2_def]
- "list_all2 P (x#xs) ys = (? z zs. ys = z#zs & P x z & list_all2 P xs zs)";
-by (case_tac "ys" 1);
-by (Auto_tac);
-qed "list_all2_Cons1";
-
-Goalw [list_all2_def]
- "list_all2 P xs (y#ys) = (? z zs. xs = z#zs & P z y & list_all2 P zs ys)";
-by (case_tac "xs" 1);
-by (Auto_tac);
-qed "list_all2_Cons2";
-
-Goalw [list_all2_def]
- "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys";
-by (asm_full_simp_tac (simpset() addsimps [zip_rev] addcongs [conj_cong]) 1);
-qed "list_all2_rev";
-AddIffs[list_all2_rev];
-
-Goalw [list_all2_def]
- "list_all2 P (xs@ys) zs = \
-\ (EX us vs. zs = us@vs & length us = length xs & length vs = length ys & \
-\            list_all2 P xs us & list_all2 P ys vs)";
-by (simp_tac (simpset() addsimps [zip_append1]) 1);
-by (rtac iffI 1);
- by (res_inst_tac [("x","take (length xs) zs")] exI 1);
- by (res_inst_tac [("x","drop (length xs) zs")] exI 1);
- by (force_tac (claset(),
-		simpset() addsplits [nat_diff_split] addsimps [min_def]) 1);
-by (Clarify_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [ball_Un]) 1);
-qed "list_all2_append1";
-
-Goalw [list_all2_def]
- "list_all2 P xs (ys@zs) = \
-\ (EX us vs. xs = us@vs & length us = length ys & length vs = length zs & \
-\            list_all2 P us ys & list_all2 P vs zs)";
-by (simp_tac (simpset() addsimps [zip_append2]) 1);
-by (rtac iffI 1);
- by (res_inst_tac [("x","take (length ys) xs")] exI 1);
- by (res_inst_tac [("x","drop (length ys) xs")] exI 1);
- by (force_tac (claset(),
-		simpset() addsplits [nat_diff_split] addsimps [min_def]) 1);
-by (Clarify_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [ball_Un]) 1);
-qed "list_all2_append2";
-
-Goalw [list_all2_def]
-  "list_all2 P xs ys = \
-\  (length xs = length ys & (!i<length xs. P (xs!i) (ys!i)))";
-by (force_tac (claset(), simpset() addsimps [set_zip]) 1);
-qed "list_all2_conv_all_nth";
-
-Goal "ALL a b c. P1 a b --> P2 b c --> P3 a c ==> \
-\ ALL bs cs. list_all2 P1 as bs --> list_all2 P2 bs cs --> list_all2 P3 as cs";
-by (induct_tac "as" 1);
-by  (Simp_tac 1);
-by (rtac allI 1);
-by (induct_tac "bs" 1);
-by  (Simp_tac 1);
-by (rtac allI 1);
-by (induct_tac "cs" 1);
-by Auto_tac;
-qed_spec_mp "list_all2_trans";
-
-
-section "foldl";
-
-Goal "!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed_spec_mp "foldl_append";
-Addsimps [foldl_append];
-
-(* Note: `n <= foldl op+ n ns' looks simpler, but is more difficult to use
-   because it requires an additional transitivity step
-*)
-Goal "!n::nat. m <= n --> m <= foldl op+ n ns";
-by (induct_tac "ns" 1);
-by Auto_tac;
-qed_spec_mp "start_le_sum";
-
-Goal "!!n::nat. n : set ns ==> n <= foldl op+ 0 ns";
-by (force_tac (claset() addIs [start_le_sum],
-              simpset() addsimps [in_set_conv_decomp]) 1);
-qed "elem_le_sum";
-
-Goal "!m::nat. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))";
-by (induct_tac "ns" 1);
-by Auto_tac;
-qed_spec_mp "sum_eq_0_conv";
-AddIffs [sum_eq_0_conv];
-
-(** upto **)
-
-(* Does not terminate! *)
-Goal "[i..j(] = (if i<j then i#[Suc i..j(] else [])";
-by (induct_tac "j" 1);
-by Auto_tac;
-qed "upt_rec";
-
-Goal "j<=i ==> [i..j(] = []";
-by (stac upt_rec 1);
-by (Asm_simp_tac 1);
-qed "upt_conv_Nil";
-Addsimps [upt_conv_Nil];
-
-(*Only needed if upt_Suc is deleted from the simpset*)
-Goal "i<=j ==> [i..(Suc j)(] = [i..j(]@[j]";
-by (Asm_simp_tac 1);
-qed "upt_Suc_append";
-
-Goal "i<j ==> [i..j(] = i#[Suc i..j(]";
-by (rtac trans 1);
-by (stac upt_rec 1);
-by (rtac refl 2);
-by (Asm_simp_tac 1);
-qed "upt_conv_Cons";
-
-(*LOOPS as a simprule, since j<=j*)
-Goal "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]";
-by (induct_tac "k" 1);
-by Auto_tac;
-qed "upt_add_eq_append";
-
-Goal "length [i..j(] = j-i";
-by (induct_tac "j" 1);
- by (Simp_tac 1);
-by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1);
-qed "length_upt";
-Addsimps [length_upt];
-
-Goal "i+k < j --> [i..j(] ! k = i+k";
-by (induct_tac "j" 1);
- by (asm_simp_tac (simpset() addsimps [less_Suc_eq, nth_append] 
-                             addsplits [nat_diff_split]) 2);
-by (Simp_tac 1);
-qed_spec_mp "nth_upt";
-Addsimps [nth_upt];
-
-Goal "!i. i+m <= n --> take m [i..n(] = [i..i+m(]";
-by (induct_tac "m" 1);
- by (Simp_tac 1);
-by (Clarify_tac 1);
-by (stac upt_rec 1);
-by (rtac sym 1);
-by (stac upt_rec 1);
-by (asm_simp_tac (simpset() delsimps (thms"upt.simps")) 1);
-qed_spec_mp "take_upt";
-Addsimps [take_upt];
-
-Goal "map Suc [m..n(] = [Suc m..n]";
-by (induct_tac "n" 1);
-by Auto_tac;
-qed "map_Suc_upt";
-
-Goal "ALL i. i < n-m --> (map f [m..n(]) ! i = f(m+i)";
-by (induct_thm_tac diff_induct "n m" 1);
-by (stac (map_Suc_upt RS sym) 3);
-by (auto_tac (claset(), simpset() addsimps [less_diff_conv, nth_upt]));
-qed_spec_mp "nth_map_upt";
-
-Goal "ALL xs ys. k <= length xs --> k <= length ys -->  \
-\        (ALL i. i < k --> xs!i = ys!i)  \
-\     --> take k xs = take k ys";
-by (induct_tac "k" 1);
-by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq_0_disj, 
-						all_conj_distrib])));
-by (Clarify_tac 1);
-(*Both lists must be non-empty*)
-by (case_tac "xs" 1);
-by (case_tac "ys" 2);
-by (ALLGOALS Clarify_tac);
-(*prenexing's needed, not miniscoping*)
-by (ALLGOALS (full_simp_tac (simpset() addsimps (all_simps RL [sym])  
-                                       delsimps (all_simps))));
-by (Blast_tac 1);
-qed_spec_mp "nth_take_lemma";
-
-Goal "[| length xs = length ys;  \
-\        ALL i. i < length xs --> xs!i = ys!i |]  \
-\     ==> xs = ys";
-by (forward_tac [[le_refl, eq_imp_le] MRS nth_take_lemma] 1);
-by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [take_all])));
-qed_spec_mp "nth_equalityI";
-
-(*The famous take-lemma*)
-Goal "(ALL i. take i xs = take i ys) ==> xs = ys";
-by (dres_inst_tac [("x", "max (length xs) (length ys)")] spec 1);
-by (full_simp_tac (simpset() addsimps [le_max_iff_disj, take_all]) 1);
-qed_spec_mp "take_equalityI";
-
-
-(** distinct & remdups **)
-section "distinct & remdups";
-
-Goal  "distinct(xs@ys) = (distinct xs & distinct ys & set xs Int set ys = {})";
-by(induct_tac "xs" 1);
-by Auto_tac;
-qed "distinct_append";
-Addsimps [distinct_append];
-
-Goal "set(remdups xs) = set xs";
-by (induct_tac "xs" 1);
- by (Simp_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1);
-qed "set_remdups";
-Addsimps [set_remdups];
-
-Goal "distinct(remdups xs)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed "distinct_remdups";
-
-Goal "distinct xs --> distinct (filter P xs)";
-by (induct_tac "xs" 1);
-by Auto_tac;
-qed_spec_mp "distinct_filter";
-
-(** replicate **)
-section "replicate";
-
-Goal "length(replicate n x) = n";
-by (induct_tac "n" 1);
-by Auto_tac;
-qed "length_replicate";
-Addsimps [length_replicate];
-
-Goal "map f (replicate n x) = replicate n (f x)";
-by (induct_tac "n" 1);
-by Auto_tac;
-qed "map_replicate";
-Addsimps [map_replicate];
-
-Goal "(replicate n x) @ (x#xs) = x # replicate n x @ xs";
-by (induct_tac "n" 1);
-by Auto_tac;
-qed "replicate_app_Cons_same";
-
-Goal "rev(replicate n x) = replicate n x";
-by (induct_tac "n" 1);
- by (Simp_tac 1);
-by (asm_simp_tac (simpset() addsimps [replicate_app_Cons_same]) 1);
-qed "rev_replicate";
-Addsimps [rev_replicate];
-
-Goal "replicate (n+m) x = replicate n x @ replicate m x";
-by (induct_tac "n" 1);
-by Auto_tac;
-qed "replicate_add";
-
-Goal"n ~= 0 --> hd(replicate n x) = x";
-by (induct_tac "n" 1);
-by Auto_tac;
-qed_spec_mp "hd_replicate";
-Addsimps [hd_replicate];
-
-Goal "n ~= 0 --> tl(replicate n x) = replicate (n - 1) x";
-by (induct_tac "n" 1);
-by Auto_tac;
-qed_spec_mp "tl_replicate";
-Addsimps [tl_replicate];
-
-Goal "n ~= 0 --> last(replicate n x) = x";
-by (induct_tac "n" 1);
-by Auto_tac;
-qed_spec_mp "last_replicate";
-Addsimps [last_replicate];
-
-Goal "!i. i<n --> (replicate n x)!i = x";
-by (induct_tac "n" 1);
- by (Simp_tac 1);
-by (asm_simp_tac (simpset() addsimps [nth_Cons] addsplits [nat.split]) 1);
-qed_spec_mp "nth_replicate";
-Addsimps [nth_replicate];
-
-Goal "set(replicate (Suc n) x) = {x}";
-by (induct_tac "n" 1);
-by Auto_tac;
-val lemma = result();
-
-Goal "n ~= 0 ==> set(replicate n x) = {x}";
-by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1);
-qed "set_replicate";
-Addsimps [set_replicate];
-
-Goal "set(replicate n x) = (if n=0 then {} else {x})";
-by (Auto_tac);
-qed "set_replicate_conv_if";
-
-Goal "x : set(replicate n y) --> x=y";
-by (asm_simp_tac (simpset() addsimps [set_replicate_conv_if]) 1);
-qed_spec_mp "in_set_replicateD";
-
-
-(*** Lexcicographic orderings on lists ***)
-section"Lexcicographic orderings on lists";
-
-Goal "wf r ==> wf(lexn r n)";
-by (induct_tac "n" 1);
-by (Simp_tac 1);
-by (Simp_tac 1);
-by (rtac wf_subset 1);
-by (rtac Int_lower1 2);
-by (rtac wf_prod_fun_image 1);
-by (rtac injI 2);
-by Auto_tac;
-qed "wf_lexn";
-
-Goal "!xs ys. (xs,ys) : lexn r n --> length xs = n & length ys = n";
-by (induct_tac "n" 1);
-by Auto_tac;
-qed_spec_mp "lexn_length";
-
-Goalw [lex_def] "wf r ==> wf(lex r)";
-by (rtac wf_UN 1);
-by (blast_tac (claset() addIs [wf_lexn]) 1);
-by (Clarify_tac 1);
-by (rename_tac "m n" 1);
-by (subgoal_tac "m ~= n" 1);
- by (Blast_tac 2);
-by (blast_tac (claset() addDs [lexn_length,not_sym]) 1);
-qed "wf_lex";
-AddSIs [wf_lex];
-
-
-Goal
- "lexn r n = \
-\ {(xs,ys). length xs = n & length ys = n & \
-\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
-by (induct_tac "n" 1);
- by (Simp_tac 1);
- by (Blast_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [image_Collect, lex_prod_def]) 1);
-by Auto_tac;
-  by (Blast_tac 1);
- by (rename_tac "a xys x xs' y ys'" 1);
- by (res_inst_tac [("x","a#xys")] exI 1);
- by (Simp_tac 1);
-by (case_tac "xys" 1);
- by (ALLGOALS Asm_full_simp_tac);
-by (Blast_tac 1);
-qed "lexn_conv";
-
-Goalw [lex_def]
- "lex r = \
-\ {(xs,ys). length xs = length ys & \
-\           (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}";
-by (force_tac (claset(), simpset() addsimps [lexn_conv]) 1);
-qed "lex_conv";
-
-Goalw [lexico_def] "wf r ==> wf(lexico r)";
-by (Blast_tac 1);
-qed "wf_lexico";
-AddSIs [wf_lexico];
-
-Goalw [lexico_def,diag_def,lex_prod_def,measure_def,inv_image_def]
-"lexico r = {(xs,ys). length xs < length ys | \
-\                     length xs = length ys & (xs,ys) : lex r}";
-by (Simp_tac 1);
-qed "lexico_conv";
-
-Goal "([],ys) ~: lex r";
-by (simp_tac (simpset() addsimps [lex_conv]) 1);
-qed "Nil_notin_lex";
-
-Goal "(xs,[]) ~: lex r";
-by (simp_tac (simpset() addsimps [lex_conv]) 1);
-qed "Nil2_notin_lex";
-
-AddIffs [Nil_notin_lex,Nil2_notin_lex];
-
-Goal "((x#xs,y#ys) : lex r) = \
-\     ((x,y) : r & length xs = length ys | x=y & (xs,ys) : lex r)";
-by (simp_tac (simpset() addsimps [lex_conv]) 1);
-by (rtac iffI 1);
- by (blast_tac (claset() addIs [Cons_eq_appendI]) 2);
-by (Clarify_tac 1);
-by (case_tac "xys" 1);
-by (Asm_full_simp_tac 1);
-by (Asm_full_simp_tac 1);
-by (Blast_tac 1);
-qed "Cons_in_lex";
-AddIffs [Cons_in_lex];
-
-
-(*** sublist (a generalization of nth to sets) ***)
-
-Goalw [sublist_def] "sublist l {} = []";
-by Auto_tac;
-qed "sublist_empty";
-
-Goalw [sublist_def] "sublist [] A = []";
-by Auto_tac;
-qed "sublist_nil";
-
-Goal "map fst [p:zip xs [i..i + length xs(] . snd p : A] =     \
-\     map fst [p:zip xs [0..length xs(] . snd p + i : A]";
-by (rev_induct_tac "xs" 1);
- by (asm_simp_tac (simpset() addsimps [add_commute]) 2);
-by (Simp_tac 1);
-qed "sublist_shift_lemma";
-
-Goalw [sublist_def]
-     "sublist (l@l') A = sublist l A @ sublist l' {j. j + length l : A}";
-by (rev_induct_tac "l'" 1);
-by (Simp_tac 1);
-by (asm_simp_tac (simpset() addsimps [inst "i" "0" upt_add_eq_append, 
-	                              zip_append, sublist_shift_lemma]) 1);
-by (asm_simp_tac (simpset() addsimps [add_commute]) 1);
-qed "sublist_append";
-
-Addsimps [sublist_empty, sublist_nil];
-
-Goal "sublist (x#l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}";
-by (rev_induct_tac "l" 1);
- by (asm_simp_tac (simpset() delsimps [append_Cons]
-	 		     addsimps [append_Cons RS sym, sublist_append]) 2);
-by (simp_tac (simpset() addsimps [sublist_def]) 1);
-qed "sublist_Cons";
-
-Goal "sublist [x] A = (if 0 : A then [x] else [])";
-by (simp_tac (simpset() addsimps [sublist_Cons]) 1);
-qed "sublist_singleton";
-Addsimps [sublist_singleton];
-
-Goal "sublist l {..n(} = take n l";
-by (rev_induct_tac "l" 1);
- by (asm_simp_tac (simpset() addsplits [nat_diff_split]
-                             addsimps [sublist_append]) 2);
-by (Simp_tac 1);
-qed "sublist_upt_eq_take";
-Addsimps [sublist_upt_eq_take];
-
-
-Goal "take n (x#xs) = (if n=0 then [] else x # take (n - 1) xs)";
-by (case_tac "n" 1);
-by (ALLGOALS 
-    (asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1])));
-qed "take_Cons'";
-
-Goal "drop n (x#xs) = (if n=0 then x#xs else drop (n - 1) xs)";
-by (case_tac "n" 1);
-by (ALLGOALS
-    (asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1])));
-qed "drop_Cons'";
-
-Goal "(x#xs)!n = (if n=0 then x else xs!(n - 1))";
-by (case_tac "n" 1);
-by (ALLGOALS
-    (asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1])));
-qed "nth_Cons'";
-
-Addsimps (map (inst "n" "number_of ?v") [take_Cons', drop_Cons', nth_Cons']);
-
+val Cons_eq_appendI = thm "Cons_eq_appendI";
+val Cons_in_lex = thm "Cons_in_lex";
+val Nil2_notin_lex = thm "Nil2_notin_lex";
+val Nil_eq_concat_conv = thm "Nil_eq_concat_conv";
+val Nil_is_append_conv = thm "Nil_is_append_conv";
+val Nil_is_map_conv = thm "Nil_is_map_conv";
+val Nil_is_rev_conv = thm "Nil_is_rev_conv";
+val Nil_notin_lex = thm "Nil_notin_lex";
+val all_nth_imp_all_set = thm "all_nth_imp_all_set";
+val all_set_conv_all_nth = thm "all_set_conv_all_nth";
+val append1_eq_conv = thm "append1_eq_conv";
+val append_Cons = thm "append_Cons";
+val append_Nil = thm "append_Nil";
+val append_Nil2 = thm "append_Nil2";
+val append_assoc = thm "append_assoc";
+val append_butlast_last_id = thm "append_butlast_last_id";
+val append_eq_appendI = thm "append_eq_appendI";
+val append_eq_append_conv = thm "append_eq_append_conv";
+val append_eq_conv_conj = thm "append_eq_conv_conj";
+val append_in_lists_conv = thm "append_in_lists_conv";
+val append_is_Nil_conv = thm "append_is_Nil_conv";
+val append_same_eq = thm "append_same_eq";
+val append_self_conv = thm "append_self_conv";
+val append_self_conv2 = thm "append_self_conv2";
+val append_take_drop_id = thm "append_take_drop_id";
+val butlast_append = thm "butlast_append";
+val butlast_snoc = thm "butlast_snoc";
+val concat_append = thm "concat_append";
+val concat_eq_Nil_conv = thm "concat_eq_Nil_conv";
+val distinct_append = thm "distinct_append";
+val distinct_filter = thm "distinct_filter";
+val distinct_remdups = thm "distinct_remdups";
+val dropWhile_append1 = thm "dropWhile_append1";
+val dropWhile_append2 = thm "dropWhile_append2";
+val drop_0 = thm "drop_0";
+val drop_Cons = thm "drop_Cons";
+val drop_Cons' = thm "drop_Cons'";
+val drop_Nil = thm "drop_Nil";
+val drop_Suc_Cons = thm "drop_Suc_Cons";
+val drop_all = thm "drop_all";
+val drop_append = thm "drop_append";
+val drop_drop = thm "drop_drop";
+val drop_map = thm "drop_map";
+val elem_le_sum = thm "elem_le_sum";
+val eq_Nil_appendI = thm "eq_Nil_appendI";
+val filter_False = thm "filter_False";
+val filter_True = thm "filter_True";
+val filter_append = thm "filter_append";
+val filter_concat = thm "filter_concat"; 
+val filter_filter = thm "filter_filter";
+val filter_is_subset = thm "filter_is_subset";
+val finite_set = thm "finite_set";
+val foldl_Cons = thm "foldl_Cons";
+val foldl_Nil = thm "foldl_Nil";
+val foldl_append = thm "foldl_append";
+val hd_Cons_tl = thm "hd_Cons_tl";
+val hd_append = thm "hd_append";
+val hd_append2 = thm "hd_append2";
+val hd_replicate = thm "hd_replicate";
+val in_listsD = thm "in_listsD";
+val in_listsI = thm "in_listsI";
+val in_lists_conv_set = thm "in_lists_conv_set";
+val in_set_butlastD = thm "in_set_butlastD";
+val in_set_butlast_appendI = thm "in_set_butlast_appendI";
+val in_set_conv_decomp = thm "in_set_conv_decomp";
+val in_set_replicateD = thm "in_set_replicateD";
+val inj_map = thm "inj_map";
+val inj_mapD = thm "inj_mapD";
+val inj_mapI = thm "inj_mapI";
+val last_replicate = thm "last_replicate";
+val last_snoc = thm "last_snoc";
+val length_0_conv = thm "length_0_conv";
+val length_Suc_conv = thm "length_Suc_conv";
+val length_append = thm "length_append";
+val length_butlast = thm "length_butlast";
+val length_drop = thm "length_drop";
+val length_filter = thm "length_filter";
+val length_greater_0_conv = thm "length_greater_0_conv";
+val length_induct = thm "length_induct";
+val length_list_update = thm "length_list_update";
+val length_map = thm "length_map";
+val length_replicate = thm "length_replicate";
+val length_rev = thm "length_rev";
+val length_take = thm "length_take";
+val length_tl = thm "length_tl";
+val length_upt = thm "length_upt";
+val length_zip = thm "length_zip";
+val lex_conv = thm "lex_conv";
+val lex_def = thm "lex_def";
+val lexico_conv = thm "lexico_conv";
+val lexico_def = thm "lexico_def";
+val lexn_conv = thm "lexn_conv";
+val lexn_length = thm "lexn_length";
+val list_all2_Cons = thm "list_all2_Cons";
+val list_all2_Cons1 = thm "list_all2_Cons1";
+val list_all2_Cons2 = thm "list_all2_Cons2";
+val list_all2_Nil = thm "list_all2_Nil";
+val list_all2_Nil2 = thm "list_all2_Nil2";
+val list_all2_append1 = thm "list_all2_append1";
+val list_all2_append2 = thm "list_all2_append2";
+val list_all2_conv_all_nth = thm "list_all2_conv_all_nth";
+val list_all2_def = thm "list_all2_def";
+val list_all2_lengthD = thm "list_all2_lengthD";
+val list_all2_rev = thm "list_all2_rev";
+val list_all2_trans = thm "list_all2_trans";
+val list_all_Cons = thm "list_all_Cons";
+val list_all_Nil = thm "list_all_Nil";
+val list_all_append = thm "list_all_append";
+val list_all_conv = thm "list_all_conv";
+val list_ball_nth = thm "list_ball_nth";
+val list_update_overwrite = thm "list_update_overwrite";
+val list_update_same_conv = thm "list_update_same_conv";
+val listsE = thm "listsE";
+val lists_IntI = thm "lists_IntI";
+val lists_Int_eq = thm "lists_Int_eq";
+val lists_mono = thm "lists_mono";
+val map_Suc_upt = thm "map_Suc_upt";
+val map_append = thm "map_append";
+val map_compose = thm "map_compose";
+val map_concat = thm "map_concat";
+val map_cong = thm "map_cong";
+val map_eq_Cons = thm "map_eq_Cons";
+val map_ext = thm "map_ext";
+val map_ident = thm "map_ident";
+val map_injective = thm "map_injective";
+val map_is_Nil_conv = thm "map_is_Nil_conv";
+val map_replicate = thm "map_replicate";
+val neq_Nil_conv = thm "neq_Nil_conv";
+val not_Cons_self = thm "not_Cons_self";
+val not_Cons_self2 = thm "not_Cons_self2";
+val nth_Cons = thm "nth_Cons";
+val nth_Cons' = thm "nth_Cons'";
+val nth_Cons_0 = thm "nth_Cons_0";
+val nth_Cons_Suc = thm "nth_Cons_Suc";
+val nth_append = thm "nth_append";
+val nth_drop = thm "nth_drop";
+val nth_equalityI = thm "nth_equalityI";
+val nth_list_update = thm "nth_list_update";
+val nth_list_update_eq = thm "nth_list_update_eq";
+val nth_list_update_neq = thm "nth_list_update_neq";
+val nth_map = thm "nth_map";
+val nth_map_upt = thm "nth_map_upt";
+val nth_mem = thm "nth_mem";
+val nth_replicate = thm "nth_replicate";
+val nth_take = thm "nth_take";
+val nth_take_lemma = thm "nth_take_lemma";
+val nth_upt = thm "nth_upt";
+val nth_zip = thm "nth_zip";
+val replicate_0 = thm "replicate_0";
+val replicate_Suc = thm "replicate_Suc";
+val replicate_add = thm "replicate_add";
+val replicate_app_Cons_same = thm "replicate_app_Cons_same";
+val rev_append = thm "rev_append";
+val rev_concat = thm "rev_concat";
+val rev_drop = thm "rev_drop";
+val rev_exhaust = thm "rev_exhaust";
+val rev_exhaust_aux = thm "rev_exhaust_aux";
+val rev_induct = thm "rev_induct";
+val rev_is_Nil_conv = thm "rev_is_Nil_conv";
+val rev_is_rev_conv = thm "rev_is_rev_conv";
+val rev_map = thm "rev_map";
+val rev_replicate = thm "rev_replicate";
+val rev_rev_ident = thm "rev_rev_ident";
+val rev_take = thm "rev_take";
+val same_append_eq = thm "same_append_eq";
+val self_append_conv = thm "self_append_conv";
+val self_append_conv2 = thm "self_append_conv2";
+val set_append = thm "set_append";
+val set_concat = thm "set_concat";
+val set_conv_nth = thm "set_conv_nth";
+val set_empty = thm "set_empty";
+val set_filter = thm "set_filter";
+val set_map = thm "set_map";
+val set_mem_eq = thm "set_mem_eq";
+val set_remdups = thm "set_remdups";
+val set_replicate = thm "set_replicate";
+val set_replicate_conv_if = thm "set_replicate_conv_if";
+val set_rev = thm "set_rev";
+val set_subset_Cons = thm "set_subset_Cons";
+val set_take_whileD = thm "set_take_whileD";
+val set_update_subsetI = thm "set_update_subsetI";
+val set_update_subset_insert = thm "set_update_subset_insert";
+val set_upt = thm "set_upt";
+val set_zip = thm "set_zip";
+val start_le_sum = thm "start_le_sum";
+val sublist_Cons = thm "sublist_Cons";
+val sublist_append = thm "sublist_append";
+val sublist_def = thm "sublist_def";
+val sublist_empty = thm "sublist_empty";
+val sublist_nil = thm "sublist_nil";
+val sublist_shift_lemma = thm "sublist_shift_lemma";
+val sublist_singleton = thm "sublist_singleton";
+val sublist_upt_eq_take = thm "sublist_upt_eq_take";
+val sum_eq_0_conv = thm "sum_eq_0_conv";
+val takeWhile_append1 = thm "takeWhile_append1";
+val takeWhile_append2 = thm "takeWhile_append2";
+val takeWhile_dropWhile_id = thm "takeWhile_dropWhile_id";
+val takeWhile_tail = thm "takeWhile_tail";
+val take_0 = thm "take_0";
+val take_Cons = thm "take_Cons";
+val take_Cons' = thm "take_Cons'";
+val take_Nil = thm "take_Nil";
+val take_Suc_Cons = thm "take_Suc_Cons";
+val take_all = thm "take_all";
+val take_append = thm "take_append";
+val take_drop = thm "take_drop";
+val take_equalityI = thm "take_equalityI";
+val take_map = thm "take_map"; 
+val take_take = thm "take_take";
+val take_upt = thm "take_upt";
+val tl_append = thm "tl_append";
+val tl_append2 = thm "tl_append2";
+val tl_replicate = thm "tl_replicate";
+val update_zip = thm "update_zip";
+val upt_0 = thm "upt_0";
+val upt_Suc = thm "upt_Suc";
+val upt_Suc_append = thm "upt_Suc_append";
+val upt_add_eq_append = thm "upt_add_eq_append";
+val upt_conv_Cons = thm "upt_conv_Cons";
+val upt_conv_Nil = thm "upt_conv_Nil";
+val upt_rec = thm "upt_rec";
+val wf_lex = thm "wf_lex";
+val wf_lexico = thm "wf_lexico";
+val wf_lexn = thm "wf_lexn";
+val zip_Cons_Cons = thm "zip_Cons_Cons";
+val zip_Nil = thm "zip_Nil";
+val zip_append = thm "zip_append";
+val zip_append1 = thm "zip_append1";
+val zip_append2 = thm "zip_append2";
+val zip_replicate = thm "zip_replicate";
+val zip_rev = thm "zip_rev";
+val zip_update = thm "zip_update";

--- a/src/HOL/List.thy	Tue May 07 19:54:04 2002 +0200
+++ b/src/HOL/List.thy	Tue May 07 19:54:29 2002 +0200
@@ -2,56 +2,59 @@
     ID:         $Id$
     Author:     Tobias Nipkow
     Copyright   1994 TU Muenchen
-
-The datatype of finite lists.
 *)
 
-List = PreList +
+header {* The datatype of finite lists *}
+theory List1 = PreList:
 
-datatype 'a list = Nil ("[]") | Cons 'a ('a list) (infixr "#" 65)
+datatype 'a list = Nil ("[]") | Cons 'a "'a list" (infixr "#" 65)
 
 consts
-  "@"         :: ['a list, 'a list] => 'a list            (infixr 65)
-  filter      :: ['a => bool, 'a list] => 'a list
-  concat      :: 'a list list => 'a list
-  foldl       :: [['b,'a] => 'b, 'b, 'a list] => 'b
-  foldr       :: [['a,'b] => 'b, 'a list, 'b] => 'b
-  hd, last    :: 'a list => 'a
-  set         :: 'a list => 'a set
-  list_all    :: ('a => bool) => ('a list => bool)
-  list_all2   :: ('a => 'b => bool) => 'a list => 'b list => bool
-  map         :: ('a=>'b) => ('a list => 'b list)
-  mem         :: ['a, 'a list] => bool                    (infixl 55)
-  nth         :: ['a list, nat] => 'a			  (infixl "!" 100)
-  list_update :: 'a list => nat => 'a => 'a list
-  take, drop  :: [nat, 'a list] => 'a list
-  takeWhile,
-  dropWhile   :: ('a => bool) => 'a list => 'a list
-  tl, butlast :: 'a list => 'a list
-  rev         :: 'a list => 'a list
-  zip	      :: "'a list => 'b list => ('a * 'b) list"
-  upt         :: nat => nat => nat list                   ("(1[_../_'(])")
-  remdups     :: "'a list => 'a list"
-  null, "distinct" :: "'a list => bool"
-  replicate   :: nat => 'a => 'a list
+  "@"         :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"            (infixr 65)
+  filter      :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"
+  concat      :: "'a list list \<Rightarrow> 'a list"
+  foldl       :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"
+  foldr       :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
+  hd          :: "'a list \<Rightarrow> 'a"
+  tl          :: "'a list \<Rightarrow> 'a list"
+  last        :: "'a list \<Rightarrow> 'a"
+  butlast     :: "'a list \<Rightarrow> 'a list"
+  set         :: "'a list \<Rightarrow> 'a set"
+  list_all    :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)"
+  list_all2   :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"
+  map         :: "('a\<Rightarrow>'b) \<Rightarrow> ('a list \<Rightarrow> 'b list)"
+  mem         :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"                    (infixl 55)
+  nth         :: "'a list \<Rightarrow> nat \<Rightarrow> 'a"			  (infixl "!" 100)
+  list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list"
+  take        :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
+  drop        :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
+  takeWhile   :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"
+  dropWhile   :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"
+  rev         :: "'a list \<Rightarrow> 'a list"
+  zip	      :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a * 'b) list"
+  upt         :: "nat \<Rightarrow> nat \<Rightarrow> nat list"                   ("(1[_../_'(])")
+  remdups     :: "'a list \<Rightarrow> 'a list"
+  null        :: "'a list \<Rightarrow> bool"
+  "distinct"  :: "'a list \<Rightarrow> bool"
+  replicate   :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list"
 
 nonterminals
   lupdbinds  lupdbind
 
 syntax
   (* list Enumeration *)
-  "@list"     :: args => 'a list                          ("[(_)]")
+  "@list"     :: "args \<Rightarrow> 'a list"                          ("[(_)]")
 
   (* Special syntax for filter *)
-  "@filter"   :: [pttrn, 'a list, bool] => 'a list        ("(1[_:_./ _])")
+  "@filter"   :: "[pttrn, 'a list, bool] \<Rightarrow> 'a list"        ("(1[_:_./ _])")
 
   (* list update *)
-  "_lupdbind"      :: ['a, 'a] => lupdbind            ("(2_ :=/ _)")
-  ""               :: lupdbind => lupdbinds           ("_")
-  "_lupdbinds"     :: [lupdbind, lupdbinds] => lupdbinds ("_,/ _")
-  "_LUpdate"       :: ['a, lupdbinds] => 'a           ("_/[(_)]" [900,0] 900)
+  "_lupdbind"      :: "['a, 'a] \<Rightarrow> lupdbind"            ("(2_ :=/ _)")
+  ""               :: "lupdbind \<Rightarrow> lupdbinds"           ("_")
+  "_lupdbinds"     :: "[lupdbind, lupdbinds] \<Rightarrow> lupdbinds" ("_,/ _")
+  "_LUpdate"       :: "['a, lupdbinds] \<Rightarrow> 'a"           ("_/[(_)]" [900,0] 900)
 
-  upto        :: nat => nat => nat list                   ("(1[_../_])")
+  upto        :: "nat \<Rightarrow> nat \<Rightarrow> nat list"                   ("(1[_../_])")
 
 translations
   "[x, xs]"     == "x#[xs]"
@@ -65,22 +68,32 @@
 
 
 syntax (xsymbols)
-  "@filter"   :: [pttrn, 'a list, bool] => 'a list        ("(1[_\\<in>_ ./ _])")
+  "@filter"   :: "[pttrn, 'a list, bool] \<Rightarrow> 'a list"        ("(1[_\<in>_ ./ _])")
 
 
 consts
-  lists        :: 'a set => 'a list set
+  lists        :: "'a set \<Rightarrow> 'a list set"
 
-  inductive "lists A"
-  intrs
-    Nil  "[]: lists A"
-    Cons "[| a: A;  l: lists A |] ==> a#l : lists A"
+inductive "lists A"
+intros
+Nil:  "[]: lists A"
+Cons: "\<lbrakk> a: A;  l: lists A \<rbrakk> \<Longrightarrow> a#l : lists A"
 
 
 (*Function "size" is overloaded for all datatypes.  Users may refer to the
   list version as "length".*)
-syntax   length :: 'a list => nat
-translations  "length"  =>  "size:: _ list => nat"
+syntax   length :: "'a list \<Rightarrow> nat"
+translations  "length"  =>  "size:: _ list \<Rightarrow> nat"
+
+(* translating size::list -> length *)
+typed_print_translation
+{*
+let
+fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] =
+      Syntax.const "length" $ t
+  | size_tr' _ _ _ = raise Match;
+in [("size", size_tr')] end
+*}
 
 primrec
   "hd(x#xs) = x"
@@ -102,14 +115,14 @@
   "set [] = {}"
   "set (x#xs) = insert x (set xs)"
 primrec
-  list_all_Nil  "list_all P [] = True"
-  list_all_Cons "list_all P (x#xs) = (P(x) & list_all P xs)"
+  list_all_Nil:  "list_all P [] = True"
+  list_all_Cons: "list_all P (x#xs) = (P(x) & list_all P xs)"
 primrec
   "map f []     = []"
   "map f (x#xs) = f(x)#map f xs"
 primrec
-  append_Nil  "[]    @ys = ys"
-  append_Cons "(x#xs)@ys = x#(xs@ys)"
+  append_Nil:  "[]    @ys = ys"
+  append_Cons: "(x#xs)@ys = x#(xs@ys)"
 primrec
   "rev([])   = []"
   "rev(x#xs) = rev(xs) @ [x]"
@@ -117,8 +130,8 @@
   "filter P []     = []"
   "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
 primrec
-  foldl_Nil  "foldl f a [] = a"
-  foldl_Cons "foldl f a (x#xs) = foldl f (f a x) xs"
+  foldl_Nil:  "foldl f a [] = a"
+  foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
 primrec
   "foldr f [] a     = a"
   "foldr f (x#xs) a = f x (foldr f xs a)"
@@ -126,23 +139,23 @@
   "concat([])   = []"
   "concat(x#xs) = x @ concat(xs)"
 primrec
-  drop_Nil  "drop n [] = []"
-  drop_Cons "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
+  drop_Nil:  "drop n [] = []"
+  drop_Cons: "drop n (x#xs) = (case n of 0 \<Rightarrow> x#xs | Suc(m) \<Rightarrow> drop m xs)"
   (* Warning: simpset does not contain this definition but separate theorems 
      for n=0 / n=Suc k*)
 primrec
-  take_Nil  "take n [] = []"
-  take_Cons "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
+  take_Nil:  "take n [] = []"
+  take_Cons: "take n (x#xs) = (case n of 0 \<Rightarrow> [] | Suc(m) \<Rightarrow> x # take m xs)"
   (* Warning: simpset does not contain this definition but separate theorems 
      for n=0 / n=Suc k*)
 primrec 
-  nth_Cons  "(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
+  nth_Cons:  "(x#xs)!n = (case n of 0 \<Rightarrow> x | (Suc k) \<Rightarrow> xs!k)"
   (* Warning: simpset does not contain this definition but separate theorems 
      for n=0 / n=Suc k*)
 primrec
  "    [][i:=v] = []"
- "(x#xs)[i:=v] = (case i of 0     => v # xs 
-			  | Suc j => x # xs[j:=v])"
+ "(x#xs)[i:=v] = (case i of 0     \<Rightarrow> v # xs 
+			  | Suc j \<Rightarrow> x # xs[j:=v])"
 primrec
   "takeWhile P []     = []"
   "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
@@ -151,12 +164,13 @@
   "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
 primrec
   "zip xs []     = []"
-  "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
+zip_Cons:
+  "zip xs (y#ys) = (case xs of [] \<Rightarrow> [] | z#zs \<Rightarrow> (z,y)#zip zs ys)"
   (* Warning: simpset does not contain this definition but separate theorems 
      for xs=[] / xs=z#zs *)
 primrec
-  upt_0   "[i..0(] = []"
-  upt_Suc "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])"
+  upt_0:   "[i..0(] = []"
+  upt_Suc: "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])"
 primrec
   "distinct []     = True"
   "distinct (x#xs) = (x ~: set xs & distinct xs)"
@@ -164,46 +178,1170 @@
   "remdups [] = []"
   "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
 primrec
-  replicate_0   "replicate  0      x = []"
-  replicate_Suc "replicate (Suc n) x = x # replicate n x"
+  replicate_0:   "replicate  0      x = []"
+  replicate_Suc: "replicate (Suc n) x = x # replicate n x"
 defs
- list_all2_def
+ list_all2_def:
  "list_all2 P xs ys == length xs = length ys & (!(x,y):set(zip xs ys). P x y)"
 
 
 (** Lexicographic orderings on lists **)
 
 consts
- lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
+ lexn :: "('a * 'a)set \<Rightarrow> nat \<Rightarrow> ('a list * 'a list)set"
 primrec
 "lexn r 0       = {}"
 "lexn r (Suc n) = (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
                   {(xs,ys). length xs = Suc n & length ys = Suc n}"
 
 constdefs
-  lex :: "('a * 'a)set => ('a list * 'a list)set"
+  lex :: "('a * 'a)set \<Rightarrow> ('a list * 'a list)set"
     "lex r == UN n. lexn r n"
 
-  lexico :: "('a * 'a)set => ('a list * 'a list)set"
+  lexico :: "('a * 'a)set \<Rightarrow> ('a list * 'a list)set"
     "lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
 
-  sublist :: "['a list, nat set] => 'a list"
+  sublist :: "['a list, nat set] \<Rightarrow> 'a list"
     "sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))"
 
-end
+
+lemma not_Cons_self[simp]: "\<And>x. xs ~= x#xs"
+by(induct_tac "xs", auto)
+
+lemmas not_Cons_self2[simp] = not_Cons_self[THEN not_sym]
+
+lemma neq_Nil_conv: "(xs ~= []) = (? y ys. xs = y#ys)";
+by(induct_tac "xs", auto)
+
+(* Induction over the length of a list: *)
+(* "(!!xs. (!ys. length ys < length xs --> P ys) ==> P xs) ==> P(xs)" *)
+lemmas length_induct = measure_induct[of length]
+
+
+(** "lists": the list-forming operator over sets **)
+
+lemma lists_mono: "A<=B ==> lists A <= lists B"
+apply(unfold lists.defs)
+apply(blast intro!:lfp_mono)
+done
+
+inductive_cases listsE[elim!]: "x#l : lists A"
+declare lists.intros[intro!]
+
+lemma lists_IntI[rule_format]:
+ "l: lists A ==> l: lists B --> l: lists (A Int B)";
+apply(erule lists.induct)
+apply blast+
+done
+
+lemma lists_Int_eq[simp]: "lists (A Int B) = lists A Int lists B"
+apply(rule mono_Int[THEN equalityI])
+apply(simp add:mono_def lists_mono)
+apply(blast intro!: lists_IntI)
+done
+
+lemma append_in_lists_conv[iff]:
+ "(xs@ys : lists A) = (xs : lists A & ys : lists A)"
+by(induct_tac "xs", auto)
+
+(** length **)
+(* needs to come before "@" because of thm append_eq_append_conv *)
+
+section "length"
+
+lemma length_append[simp]: "length(xs@ys) = length(xs)+length(ys)"
+by(induct_tac "xs", auto)
+
+lemma length_map[simp]: "length (map f xs) = length xs"
+by(induct_tac "xs", auto)
+
+lemma length_rev[simp]: "length(rev xs) = length(xs)"
+by(induct_tac "xs", auto)
+
+lemma length_tl[simp]: "length(tl xs) = (length xs) - 1"
+by(case_tac "xs", auto)
+
+lemma length_0_conv[iff]: "(length xs = 0) = (xs = [])"
+by(induct_tac "xs", auto)
+
+lemma length_greater_0_conv[iff]: "(0 < length xs) = (xs ~= [])"
+by(induct_tac xs, auto)
+
+lemma length_Suc_conv:
+ "(length xs = Suc n) = (? y ys. xs = y#ys & length ys = n)"
+by(induct_tac "xs", auto)
+
+(** @ - append **)
+
+section "@ - append"
+
+lemma append_assoc[simp]: "(xs@ys)@zs = xs@(ys@zs)"
+by(induct_tac "xs", auto)
 
-ML
+lemma append_Nil2[simp]: "xs @ [] = xs"
+by(induct_tac "xs", auto)
+
+lemma append_is_Nil_conv[iff]: "(xs@ys = []) = (xs=[] & ys=[])"
+by(induct_tac "xs", auto)
+
+lemma Nil_is_append_conv[iff]: "([] = xs@ys) = (xs=[] & ys=[])"
+by(induct_tac "xs", auto)
+
+lemma append_self_conv[iff]: "(xs @ ys = xs) = (ys=[])"
+by(induct_tac "xs", auto)
+
+lemma self_append_conv[iff]: "(xs = xs @ ys) = (ys=[])"
+by(induct_tac "xs", auto)
+
+lemma append_eq_append_conv[rule_format,simp]:
+ "!ys. length xs = length ys | length us = length vs
+       --> (xs@us = ys@vs) = (xs=ys & us=vs)"
+apply(induct_tac "xs")
+ apply(rule allI)
+ apply(case_tac "ys")
+  apply simp
+ apply force
+apply(rule allI)
+apply(case_tac "ys")
+ apply force
+apply simp
+done
+
+lemma same_append_eq[iff]: "(xs @ ys = xs @ zs) = (ys=zs)"
+by simp
+
+lemma append1_eq_conv[iff]: "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)" 
+by simp
+
+lemma append_same_eq[iff]: "(ys @ xs = zs @ xs) = (ys=zs)"
+by simp
 
+lemma append_self_conv2[iff]: "(xs @ ys = ys) = (xs=[])"
+by(insert append_same_eq[of _ _ "[]"], auto)
+
+lemma self_append_conv2[iff]: "(ys = xs @ ys) = (xs=[])"
+by(auto simp add: append_same_eq[of "[]", simplified])
+
+lemma hd_Cons_tl[rule_format,simp]: "xs ~= [] --> hd xs # tl xs = xs"
+by(induct_tac "xs", auto)
+
+lemma hd_append: "hd(xs@ys) = (if xs=[] then hd ys else hd xs)"
+by(induct_tac "xs", auto)
+
+lemma hd_append2[simp]: "xs ~= [] ==> hd(xs @ ys) = hd xs"
+by(simp add: hd_append split: list.split)
+
+lemma tl_append: "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)"
+by(simp split: list.split)
+
+lemma tl_append2[simp]: "xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys"
+by(simp add: tl_append split: list.split)
+
+(* trivial rules for solving @-equations automatically *)
+
+lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
+by simp
+
+lemma Cons_eq_appendI: "[| x#xs1 = ys; xs = xs1 @ zs |] ==> x#xs = ys@zs"
+by(drule sym, simp)
+
+lemma append_eq_appendI: "[| xs@xs1 = zs; ys = xs1 @ us |] ==> xs@ys = zs@us"
+by(drule sym, simp)
+
+
+(***
+Simplification procedure for all list equalities.
+Currently only tries to rearrange @ to see if
+- both lists end in a singleton list,
+- or both lists end in the same list.
+***)
+ML_setup{*
 local
 
-(* translating size::list -> length *)
+val list_eq_pattern =
+  Thm.read_cterm (Theory.sign_of (the_context ())) ("(xs::'a list) = ys",HOLogic.boolT)
+
+fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
+      (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
+  | last (Const("List.op @",_) $ _ $ ys) = last ys
+  | last t = t
+
+fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
+  | list1 _ = false
+
+fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
+      (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
+  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
+  | butlast xs = Const("List.list.Nil",fastype_of xs)
+
+val rearr_tac =
+  simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons])
+
+fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
+  let
+    val lastl = last lhs and lastr = last rhs
+    fun rearr conv =
+      let val lhs1 = butlast lhs and rhs1 = butlast rhs
+          val Type(_,listT::_) = eqT
+          val appT = [listT,listT] ---> listT
+          val app = Const("List.op @",appT)
+          val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
+          val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2)))
+          val thm = prove_goalw_cterm [] ct (K [rearr_tac 1])
+            handle ERROR =>
+            error("The error(s) above occurred while trying to prove " ^
+                  string_of_cterm ct)
+      in Some((conv RS (thm RS trans)) RS eq_reflection) end
+
+  in if list1 lastl andalso list1 lastr
+     then rearr append1_eq_conv
+     else
+     if lastl aconv lastr
+     then rearr append_same_eq
+     else None
+  end
+in
+val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq
+end;
+
+Addsimprocs [list_eq_simproc];
+*}
+
+
+(** map **)
+
+section "map"
+
+lemma map_ext: "(\<And>x. x : set xs \<longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g xs"
+by (induct xs, simp_all)
+
+lemma map_ident[simp]: "map (%x. x) = (%xs. xs)"
+by(rule ext, induct_tac "xs", auto)
+
+lemma map_append[simp]: "map f (xs@ys) = map f xs @ map f ys"
+by(induct_tac "xs", auto)
+
+lemma map_compose(*[simp]*): "map (f o g) xs = map f (map g xs)"
+by(unfold o_def, induct_tac "xs", auto)
+
+lemma rev_map: "rev(map f xs) = map f (rev xs)"
+by(induct_tac xs, auto)
+
+(* a congruence rule for map: *)
+lemma map_cong:
+ "xs=ys ==> (!!x. x : set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys"
+by (clarify, induct ys, auto)
+
+lemma map_is_Nil_conv[iff]: "(map f xs = []) = (xs = [])"
+by(case_tac xs, auto)
+
+lemma Nil_is_map_conv[iff]: "([] = map f xs) = (xs = [])"
+by(case_tac xs, auto)
+
+lemma map_eq_Cons:
+ "(map f xs = y#ys) = (? x xs'. xs = x#xs' & f x = y & map f xs' = ys)"
+by(case_tac xs, auto)
+
+lemma map_injective:
+ "\<And>xs. map f xs = map f ys \<Longrightarrow> (!x y. f x = f y --> x=y) \<Longrightarrow> xs=ys"
+by(induct "ys", simp, fastsimp simp add:map_eq_Cons)
+
+lemma inj_mapI: "inj f ==> inj (map f)"
+by(blast dest:map_injective injD intro:injI)
+
+lemma inj_mapD: "inj (map f) ==> inj f"
+apply(unfold inj_on_def)
+apply clarify
+apply(erule_tac x = "[x]" in ballE)
+ apply(erule_tac x = "[y]" in ballE)
+  apply simp
+ apply blast
+apply blast
+done
+
+lemma inj_map: "inj (map f) = inj f"
+by(blast dest:inj_mapD intro:inj_mapI)
+
+(** rev **)
+
+section "rev"
+
+lemma rev_append[simp]: "rev(xs@ys) = rev(ys) @ rev(xs)"
+by(induct_tac xs, auto)
+
+lemma rev_rev_ident[simp]: "rev(rev xs) = xs"
+by(induct_tac xs, auto)
+
+lemma rev_is_Nil_conv[iff]: "(rev xs = []) = (xs = [])"
+by(induct_tac xs, auto)
+
+lemma Nil_is_rev_conv[iff]: "([] = rev xs) = (xs = [])"
+by(induct_tac xs, auto)
+
+lemma rev_is_rev_conv[iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
+apply(induct "xs" )
+ apply force
+apply(case_tac ys)
+ apply simp
+apply force
+done
+
+lemma rev_induct: "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs"
+apply(subst rev_rev_ident[symmetric])
+apply(rule_tac list = "rev xs" in list.induct, simp_all)
+done
+
+(* Instead of (rev_induct_tac xs) use (induct_tac xs rule: rev_induct) *)
+
+lemma rev_exhaust: "(xs = [] \<Longrightarrow> P) \<Longrightarrow>  (!!ys y. xs = ys@[y] \<Longrightarrow> P) \<Longrightarrow> P"
+by(induct xs rule: rev_induct, auto)
+
+
+(** set **)
+
+section "set"
+
+lemma finite_set[iff]: "finite (set xs)"
+by(induct_tac xs, auto)
+
+lemma set_append[simp]: "set (xs@ys) = (set xs Un set ys)"
+by(induct_tac xs, auto)
+
+lemma set_subset_Cons: "set xs \<subseteq> set (x#xs)"
+by auto
+
+lemma set_empty[iff]: "(set xs = {}) = (xs = [])"
+by(induct_tac xs, auto)
+
+lemma set_rev[simp]: "set(rev xs) = set(xs)"
+by(induct_tac xs, auto)
+
+lemma set_map[simp]: "set(map f xs) = f`(set xs)"
+by(induct_tac xs, auto)
+
+lemma set_filter[simp]: "set(filter P xs) = {x. x : set xs & P x}"
+by(induct_tac xs, auto)
+
+lemma set_upt[simp]: "set[i..j(] = {k. i <= k & k < j}"
+apply(induct_tac j)
+ apply simp_all
+apply(erule ssubst)
+apply auto
+apply arith
+done
+
+lemma in_set_conv_decomp: "(x : set xs) = (? ys zs. xs = ys@x#zs)"
+apply(induct_tac "xs")
+ apply simp
+apply simp
+apply(rule iffI)
+ apply(blast intro: eq_Nil_appendI Cons_eq_appendI)
+apply(erule exE)+
+apply(case_tac "ys")
+apply auto
+done
+
+
+(* eliminate `lists' in favour of `set' *)
+
+lemma in_lists_conv_set: "(xs : lists A) = (!x : set xs. x : A)"
+by(induct_tac xs, auto)
+
+lemmas in_listsD[dest!] = in_lists_conv_set[THEN iffD1]
+lemmas in_listsI[intro!] = in_lists_conv_set[THEN iffD2]
+
+
+(** mem **)
+ 
+section "mem"
+
+lemma set_mem_eq: "(x mem xs) = (x : set xs)"
+by(induct_tac xs, auto)
+
+
+(** list_all **)
+
+section "list_all"
+
+lemma list_all_conv: "list_all P xs = (!x:set xs. P x)"
+by(induct_tac xs, auto)
+
+lemma list_all_append[simp]:
+ "list_all P (xs@ys) = (list_all P xs & list_all P ys)"
+by(induct_tac xs, auto)
+
+
+(** filter **)
+
+section "filter"
+
+lemma filter_append[simp]: "filter P (xs@ys) = filter P xs @ filter P ys"
+by(induct_tac xs, auto)
+
+lemma filter_filter[simp]: "filter P (filter Q xs) = filter (%x. Q x & P x) xs"
+by(induct_tac xs, auto)
+
+lemma filter_True[simp]: "!x : set xs. P x \<Longrightarrow> filter P xs = xs"
+by(induct xs, auto)
+
+lemma filter_False[simp]: "!x : set xs. ~P x \<Longrightarrow> filter P xs = []"
+by(induct xs, auto)
+
+lemma length_filter[simp]: "length (filter P xs) <= length xs"
+by(induct xs, auto simp add: le_SucI)
+
+lemma filter_is_subset[simp]: "set (filter P xs) <= set xs"
+by auto
+
+
+section "concat"
+
+lemma concat_append[simp]: "concat(xs@ys) = concat(xs)@concat(ys)"
+by(induct xs, auto)
+
+lemma concat_eq_Nil_conv[iff]: "(concat xss = []) = (!xs:set xss. xs=[])"
+by(induct xss, auto)
+
+lemma Nil_eq_concat_conv[iff]: "([] = concat xss) = (!xs:set xss. xs=[])"
+by(induct xss, auto)
+
+lemma set_concat[simp]: "set(concat xs) = Union(set ` set xs)"
+by(induct xs, auto)
+
+lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" 
+by(induct xs, auto)
+
+lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" 
+by(induct xs, auto)
+
+lemma rev_concat: "rev(concat xs) = concat (map rev (rev xs))"
+by(induct xs, auto)
+
+(** nth **)
+
+section "nth"
+
+lemma nth_Cons_0[simp]: "(x#xs)!0 = x"
+by auto
+
+lemma nth_Cons_Suc[simp]: "(x#xs)!(Suc n) = xs!n"
+by auto
+
+declare nth.simps[simp del]
+
+lemma nth_append:
+ "!!n. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
+apply(induct "xs")
+ apply simp
+apply(case_tac "n" )
+ apply auto
+done
+
+lemma nth_map[simp]: "!!n. n < length xs \<Longrightarrow> (map f xs)!n = f(xs!n)"
+apply(induct "xs" )
+ apply simp
+apply(case_tac "n")
+ apply auto
+done
+
+lemma set_conv_nth: "set xs = {xs!i |i. i < length xs}"
+apply(induct_tac "xs")
+ apply simp
+apply simp
+apply safe
+  apply(rule_tac x = 0 in exI)
+  apply simp
+ apply(rule_tac x = "Suc i" in exI)
+ apply simp
+apply(case_tac "i")
+ apply simp
+apply(rename_tac "j")
+apply(rule_tac x = "j" in exI)
+apply simp
+done
+
+lemma list_ball_nth: "\<lbrakk> n < length xs; !x : set xs. P x \<rbrakk> \<Longrightarrow> P(xs!n)"
+by(simp add:set_conv_nth, blast)
+
+lemma nth_mem[simp]: "n < length xs ==> xs!n : set xs"
+by(simp add:set_conv_nth, blast)
+
+lemma all_nth_imp_all_set:
+ "\<lbrakk> !i < length xs. P(xs!i); x : set xs \<rbrakk> \<Longrightarrow> P x"
+by(simp add:set_conv_nth, blast)
+
+lemma all_set_conv_all_nth:
+ "(!x : set xs. P x) = (!i. i<length xs --> P (xs ! i))"
+by(simp add:set_conv_nth, blast)
+
+
+(** list update **)
+
+section "list update"
+
+lemma length_list_update[simp]: "!!i. length(xs[i:=x]) = length xs"
+by(induct xs, simp, simp split:nat.split)
+
+lemma nth_list_update:
+ "!!i j. i < length xs  \<Longrightarrow> (xs[i:=x])!j = (if i=j then x else xs!j)"
+by(induct xs, simp, auto simp add:nth_Cons split:nat.split)
+
+lemma nth_list_update_eq[simp]: "i < length xs  ==> (xs[i:=x])!i = x"
+by(simp add:nth_list_update)
+
+lemma nth_list_update_neq[simp]: "!!i j. i ~= j \<Longrightarrow> xs[i:=x]!j = xs!j"
+by(induct xs, simp, auto simp add:nth_Cons split:nat.split)
+
+lemma list_update_overwrite[simp]:
+ "!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
+by(induct xs, simp, simp split:nat.split)
+
+lemma list_update_same_conv:
+ "!!i. i < length xs \<Longrightarrow> (xs[i := x] = xs) = (xs!i = x)"
+by(induct xs, simp, simp split:nat.split, blast)
+
+lemma update_zip:
+"!!i xy xs. length xs = length ys \<Longrightarrow>
+    (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
+by(induct ys, auto, case_tac xs, auto split:nat.split)
+
+lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
+by(induct xs, simp, simp split:nat.split, fast)
+
+lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
+by(fast dest!:set_update_subset_insert[THEN subsetD])
+
+
+(** last & butlast **)
+
+section "last / butlast"
+
+lemma last_snoc[simp]: "last(xs@[x]) = x"
+by(induct xs, auto)
+
+lemma butlast_snoc[simp]:"butlast(xs@[x]) = xs"
+by(induct xs, auto)
+
+lemma length_butlast[simp]: "length(butlast xs) = length xs - 1"
+by(induct xs rule:rev_induct, auto)
+
+lemma butlast_append:
+ "!!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)"
+by(induct xs, auto)
+
+lemma append_butlast_last_id[simp]:
+ "xs ~= [] --> butlast xs @ [last xs] = xs"
+by(induct xs, auto)
+
+lemma in_set_butlastD: "x:set(butlast xs) ==> x:set xs"
+by(induct xs, auto split:split_if_asm)
+
+lemma in_set_butlast_appendI:
+ "x:set(butlast xs) | x:set(butlast ys) ==> x:set(butlast(xs@ys))"
+by(auto dest:in_set_butlastD simp add:butlast_append)
+
+(** take  & drop **)
+section "take & drop"
+
+lemma take_0[simp]: "take 0 xs = []"
+by(induct xs, auto)
+
+lemma drop_0[simp]: "drop 0 xs = xs"
+by(induct xs, auto)
+
+lemma take_Suc_Cons[simp]: "take (Suc n) (x#xs) = x # take n xs"
+by simp
+
+lemma drop_Suc_Cons[simp]: "drop (Suc n) (x#xs) = drop n xs"
+by simp
+
+declare take_Cons[simp del] drop_Cons[simp del]
+
+lemma length_take[simp]: "!!xs. length(take n xs) = min (length xs) n"
+by(induct n, auto, case_tac xs, auto)
+
+lemma length_drop[simp]: "!!xs. length(drop n xs) = (length xs - n)"
+by(induct n, auto, case_tac xs, auto)
+
+lemma take_all[simp]: "!!xs. length xs <= n ==> take n xs = xs"
+by(induct n, auto, case_tac xs, auto)
+
+lemma drop_all[simp]: "!!xs. length xs <= n ==> drop n xs = []"
+by(induct n, auto, case_tac xs, auto)
+
+lemma take_append[simp]:
+ "!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
+by(induct n, auto, case_tac xs, auto)
+
+lemma drop_append[simp]:
+ "!!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys" 
+by(induct n, auto, case_tac xs, auto)
+
+lemma take_take[simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
+apply(induct m)
+ apply auto
+apply(case_tac xs)
+ apply auto
+apply(case_tac na)
+ apply auto
+done
+
+lemma drop_drop[simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" 
+apply(induct m)
+ apply auto
+apply(case_tac xs)
+ apply auto
+done
+
+lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
+apply(induct m)
+ apply auto
+apply(case_tac xs)
+ apply auto
+done
+
+lemma append_take_drop_id[simp]: "!!xs. take n xs @ drop n xs = xs"
+apply(induct n)
+ apply auto
+apply(case_tac xs)
+ apply auto
+done
+
+lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
+apply(induct n)
+ apply auto
+apply(case_tac xs)
+ apply auto
+done
+
+lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" 
+apply(induct n)
+ apply auto
+apply(case_tac xs)
+ apply auto
+done
+
+lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
+apply(induct xs)
+ apply auto
+apply(case_tac i)
+ apply auto
+done
+
+lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
+apply(induct xs)
+ apply auto
+apply(case_tac i)
+ apply auto
+done
+
+lemma nth_take[simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
+apply(induct xs)
+ apply auto
+apply(case_tac n)
+ apply(blast )
+apply(case_tac i)
+ apply auto
+done
+
+lemma nth_drop[simp]: "!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n+i)"
+apply(induct n)
+ apply auto
+apply(case_tac xs)
+ apply auto
+done
 
-fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] =
-      Syntax.const "length" $ t
-  | size_tr' _ _ _ = raise Match;
+lemma append_eq_conv_conj:
+ "!!zs. (xs@ys = zs) = (xs = take (length xs) zs & ys = drop (length xs) zs)"
+apply(induct xs)
+ apply simp
+apply clarsimp
+apply(case_tac zs)
+apply auto
+done
+
+(** takeWhile & dropWhile **)
+
+section "takeWhile & dropWhile"
+
+lemma takeWhile_dropWhile_id[simp]: "takeWhile P xs @ dropWhile P xs = xs"
+by(induct xs, auto)
+
+lemma  takeWhile_append1[simp]:
+ "\<lbrakk> x:set xs; ~P(x) \<rbrakk> \<Longrightarrow> takeWhile P (xs @ ys) = takeWhile P xs"
+by(induct xs, auto)
+
+lemma takeWhile_append2[simp]:
+ "(!!x. x : set xs \<Longrightarrow> P(x)) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
+by(induct xs, auto)
+
+lemma takeWhile_tail: "~P(x) ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
+by(induct xs, auto)
+
+lemma dropWhile_append1[simp]:
+ "\<lbrakk> x : set xs; ~P(x) \<rbrakk> \<Longrightarrow> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
+by(induct xs, auto)
+
+lemma dropWhile_append2[simp]:
+ "(!!x. x:set xs \<Longrightarrow> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
+by(induct xs, auto)
+
+lemma set_take_whileD: "x:set(takeWhile P xs) ==> x:set xs & P x"
+by(induct xs, auto split:split_if_asm)
+
+
+(** zip **)
+section "zip"
+
+lemma zip_Nil[simp]: "zip [] ys = []"
+by(induct ys, auto)
+
+lemma zip_Cons_Cons[simp]: "zip (x#xs) (y#ys) = (x,y)#zip xs ys"
+by simp
+
+declare zip_Cons[simp del]
+
+lemma length_zip[simp]:
+ "!!xs. length (zip xs ys) = min (length xs) (length ys)"
+apply(induct ys)
+ apply simp
+apply(case_tac xs)
+ apply auto
+done
+
+lemma zip_append1:
+ "!!xs. zip (xs@ys) zs =
+        zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
+apply(induct zs)
+ apply simp
+apply(case_tac xs)
+ apply simp_all
+done
+
+lemma zip_append2:
+ "!!ys. zip xs (ys@zs) =
+        zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
+apply(induct xs)
+ apply simp
+apply(case_tac ys)
+ apply simp_all
+done
+
+lemma zip_append[simp]:
+ "[| length xs = length us; length ys = length vs |] ==> \
+\ zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
+by(simp add: zip_append1)
+
+lemma zip_rev:
+ "!!xs. length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
+apply(induct ys)
+ apply simp
+apply(case_tac xs)
+ apply simp_all
+done
+
+lemma nth_zip[simp]:
+"!!i xs. \<lbrakk> i < length xs; i < length ys \<rbrakk> \<Longrightarrow> (zip xs ys)!i = (xs!i, ys!i)"
+apply(induct ys)
+ apply simp
+apply(case_tac xs)
+ apply (simp_all add: nth.simps split:nat.split)
+done
+
+lemma set_zip:
+ "set(zip xs ys) = {(xs!i,ys!i) |i. i < min (length xs) (length ys)}"
+by(simp add: set_conv_nth cong: rev_conj_cong)
+
+lemma zip_update:
+ "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
+by(rule sym, simp add: update_zip)
+
+lemma zip_replicate[simp]:
+ "!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
+apply(induct i)
+ apply auto
+apply(case_tac j)
+ apply auto
+done
+
+(** list_all2 **)
+section "list_all2"
+
+lemma list_all2_lengthD: "list_all2 P xs ys ==> length xs = length ys"
+by(simp add:list_all2_def)
+
+lemma list_all2_Nil[iff]: "list_all2 P [] ys = (ys=[])"
+by(simp add:list_all2_def)
+
+lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs=[])"
+by(simp add:list_all2_def)
+
+lemma list_all2_Cons[iff]:
+ "list_all2 P (x#xs) (y#ys) = (P x y & list_all2 P xs ys)"
+by(auto simp add:list_all2_def)
+
+lemma list_all2_Cons1:
+ "list_all2 P (x#xs) ys = (? z zs. ys = z#zs & P x z & list_all2 P xs zs)"
+by(case_tac ys, auto)
+
+lemma list_all2_Cons2:
+ "list_all2 P xs (y#ys) = (? z zs. xs = z#zs & P z y & list_all2 P zs ys)"
+by(case_tac xs, auto)
+
+lemma list_all2_rev[iff]:
+ "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
+by(simp add:list_all2_def zip_rev cong:conj_cong)
+
+lemma list_all2_append1:
+ "list_all2 P (xs@ys) zs =
+  (EX us vs. zs = us@vs & length us = length xs & length vs = length ys &
+             list_all2 P xs us & list_all2 P ys vs)"
+apply(simp add:list_all2_def zip_append1)
+apply(rule iffI)
+ apply(rule_tac x = "take (length xs) zs" in exI)
+ apply(rule_tac x = "drop (length xs) zs" in exI)
+ apply(force split: nat_diff_split simp add:min_def)
+apply clarify
+apply(simp add: ball_Un)
+done
+
+lemma list_all2_append2:
+ "list_all2 P xs (ys@zs) =
+  (EX us vs. xs = us@vs & length us = length ys & length vs = length zs &
+             list_all2 P us ys & list_all2 P vs zs)"
+apply(simp add:list_all2_def zip_append2)
+apply(rule iffI)
+ apply(rule_tac x = "take (length ys) xs" in exI)
+ apply(rule_tac x = "drop (length ys) xs" in exI)
+ apply(force split: nat_diff_split simp add:min_def)
+apply clarify
+apply(simp add: ball_Un)
+done
+
+lemma list_all2_conv_all_nth:
+  "list_all2 P xs ys =
+   (length xs = length ys & (!i<length xs. P (xs!i) (ys!i)))"
+by(force simp add:list_all2_def set_zip)
+
+lemma list_all2_trans[rule_format]:
+ "ALL a b c. P1 a b --> P2 b c --> P3 a c ==>
+  ALL bs cs. list_all2 P1 as bs --> list_all2 P2 bs cs --> list_all2 P3 as cs"
+apply(induct_tac as)
+ apply simp
+apply(rule allI)
+apply(induct_tac bs)
+ apply simp
+apply(rule allI)
+apply(induct_tac cs)
+ apply auto
+done
+
+
+section "foldl"
+
+lemma foldl_append[simp]:
+ "!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
+by(induct xs, auto)
+
+(* Note: `n <= foldl op+ n ns' looks simpler, but is more difficult to use
+   because it requires an additional transitivity step
+*)
+lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl op+ n ns"
+by(induct ns, auto)
+
+lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl op+ 0 ns"
+by(force intro: start_le_sum simp add:in_set_conv_decomp)
+
+lemma sum_eq_0_conv[iff]:
+ "!!m::nat. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))"
+by(induct ns, auto)
+
+(** upto **)
+
+(* Does not terminate! *)
+lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
+by(induct j, auto)
+
+lemma upt_conv_Nil[simp]: "j<=i ==> [i..j(] = []"
+by(subst upt_rec, simp)
+
+(*Only needed if upt_Suc is deleted from the simpset*)
+lemma upt_Suc_append: "i<=j ==> [i..(Suc j)(] = [i..j(]@[j]"
+by simp
+
+lemma upt_conv_Cons: "i<j ==> [i..j(] = i#[Suc i..j(]"
+apply(rule trans)
+apply(subst upt_rec)
+ prefer 2 apply(rule refl)
+apply simp
+done
+
+(*LOOPS as a simprule, since j<=j*)
+lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
+by(induct_tac "k", auto)
+
+lemma length_upt[simp]: "length [i..j(] = j-i"
+by(induct_tac j, simp, simp add: Suc_diff_le)
+
+lemma nth_upt[simp]: "i+k < j ==> [i..j(] ! k = i+k"
+apply(induct j)
+apply(auto simp add: less_Suc_eq nth_append split:nat_diff_split)
+done
+
+lemma take_upt[simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
+apply(induct m)
+ apply simp
+apply(subst upt_rec)
+apply(rule sym)
+apply(subst upt_rec)
+apply(simp del: upt.simps)
+done
 
-in
+lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
+by(induct n, auto)
+
+lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
+thm diff_induct
+apply(induct n m rule: diff_induct)
+prefer 3 apply(subst map_Suc_upt[symmetric])
+apply(auto simp add: less_diff_conv nth_upt)
+done
+
+lemma nth_take_lemma[rule_format]:
+ "ALL xs ys. k <= length xs --> k <= length ys
+             --> (ALL i. i < k --> xs!i = ys!i)
+             --> take k xs = take k ys"
+apply(induct_tac k)
+apply(simp_all add: less_Suc_eq_0_disj all_conj_distrib)
+apply clarify
+(*Both lists must be non-empty*)
+apply(case_tac xs)
+ apply simp
+apply(case_tac ys)
+ apply clarify
+ apply(simp (no_asm_use))
+apply clarify
+(*prenexing's needed, not miniscoping*)
+apply(simp (no_asm_use) add: all_simps[symmetric] del: all_simps)
+apply blast
+(*prenexing's needed, not miniscoping*)
+done
+
+lemma nth_equalityI:
+ "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
+apply(frule nth_take_lemma[OF le_refl eq_imp_le])
+apply(simp_all add: take_all)
+done
+
+(*The famous take-lemma*)
+lemma take_equalityI: "(ALL i. take i xs = take i ys) ==> xs = ys"
+apply(drule_tac x = "max (length xs) (length ys)" in spec)
+apply(simp add: le_max_iff_disj take_all)
+done
+
+
+(** distinct & remdups **)
+section "distinct & remdups"
+
+lemma distinct_append[simp]:
+ "distinct(xs@ys) = (distinct xs & distinct ys & set xs Int set ys = {})"
+by(induct xs, auto)
+
+lemma set_remdups[simp]: "set(remdups xs) = set xs"
+by(induct xs, simp, simp add:insert_absorb)
+
+lemma distinct_remdups[iff]: "distinct(remdups xs)"
+by(induct xs, auto)
+
+lemma distinct_filter[simp]: "distinct xs ==> distinct (filter P xs)"
+by(induct xs, auto)
+
+(** replicate **)
+section "replicate"
+
+lemma length_replicate[simp]: "length(replicate n x) = n"
+by(induct n, auto)
+
+lemma map_replicate[simp]: "map f (replicate n x) = replicate n (f x)"
+by(induct n, auto)
+
+lemma replicate_app_Cons_same:
+ "(replicate n x) @ (x#xs) = x # replicate n x @ xs"
+by(induct n, auto)
+
+lemma rev_replicate[simp]: "rev(replicate n x) = replicate n x"
+apply(induct n)
+ apply simp
+apply(simp add: replicate_app_Cons_same)
+done
+
+lemma replicate_add: "replicate (n+m) x = replicate n x @ replicate m x"
+by(induct n, auto)
+
+lemma hd_replicate[simp]: "n ~= 0 ==> hd(replicate n x) = x"
+by(induct n, auto)
+
+lemma tl_replicate[simp]: "n ~= 0 ==> tl(replicate n x) = replicate (n - 1) x"
+by(induct n, auto)
+
+lemma last_replicate[rule_format,simp]:
+ "n ~= 0 --> last(replicate n x) = x"
+by(induct_tac n, auto)
+
+lemma nth_replicate[simp]: "!!i. i<n ==> (replicate n x)!i = x"
+apply(induct n)
+ apply simp
+apply(simp add: nth_Cons split:nat.split)
+done
+
+lemma set_replicate_Suc: "set(replicate (Suc n) x) = {x}"
+by(induct n, auto)
+
+lemma set_replicate[simp]: "n ~= 0 ==> set(replicate n x) = {x}"
+by(fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
+
+lemma set_replicate_conv_if: "set(replicate n x) = (if n=0 then {} else {x})"
+by auto
+
+lemma in_set_replicateD: "x : set(replicate n y) ==> x=y"
+by(simp add: set_replicate_conv_if split:split_if_asm)
+
+
+(*** Lexcicographic orderings on lists ***)
+section"Lexcicographic orderings on lists"
 
-val typed_print_translation = [("size", size_tr')];
+lemma wf_lexn: "wf r ==> wf(lexn r n)"
+apply(induct_tac n)
+ apply simp
+apply simp
+apply(rule wf_subset)
+ prefer 2 apply(rule Int_lower1)
+apply(rule wf_prod_fun_image)
+ prefer 2 apply(rule injI)
+apply auto
+done
+
+lemma lexn_length:
+ "!!xs ys. (xs,ys) : lexn r n ==> length xs = n & length ys = n"
+by(induct n, auto)
+
+lemma wf_lex[intro!]: "wf r ==> wf(lex r)"
+apply(unfold lex_def)
+apply(rule wf_UN)
+apply(blast intro: wf_lexn)
+apply clarify
+apply(rename_tac m n)
+apply(subgoal_tac "m ~= n")
+ prefer 2 apply blast
+apply(blast dest: lexn_length not_sym)
+done
+
+
+lemma lexn_conv:
+ "lexn r n =
+  {(xs,ys). length xs = n & length ys = n &
+            (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}"
+apply(induct_tac n)
+ apply simp
+ apply blast
+apply(simp add: image_Collect lex_prod_def)
+apply auto
+  apply blast
+ apply(rename_tac a xys x xs' y ys')
+ apply(rule_tac x = "a#xys" in exI)
+ apply simp
+apply(case_tac xys)
+ apply simp_all
+apply blast
+done
+
+lemma lex_conv:
+ "lex r =
+  {(xs,ys). length xs = length ys &
+            (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}"
+by(force simp add: lex_def lexn_conv)
+
+lemma wf_lexico[intro!]: "wf r ==> wf(lexico r)"
+by(unfold lexico_def, blast)
+
+lemma lexico_conv:
+"lexico r = {(xs,ys). length xs < length ys |
+                      length xs = length ys & (xs,ys) : lex r}"
+by(simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)
+
+lemma Nil_notin_lex[iff]: "([],ys) ~: lex r"
+by(simp add:lex_conv)
+
+lemma Nil2_notin_lex[iff]: "(xs,[]) ~: lex r"
+by(simp add:lex_conv)
+
+lemma Cons_in_lex[iff]:
+ "((x#xs,y#ys) : lex r) =
+  ((x,y) : r & length xs = length ys | x=y & (xs,ys) : lex r)"
+apply(simp add:lex_conv)
+apply(rule iffI)
+ prefer 2 apply(blast intro: Cons_eq_appendI)
+apply clarify
+apply(case_tac xys)
+ apply simp
+apply simp
+apply blast
+done
+
+
+(*** sublist (a generalization of nth to sets) ***)
+
+lemma sublist_empty[simp]: "sublist xs {} = []"
+by(auto simp add:sublist_def)
+
+lemma sublist_nil[simp]: "sublist [] A = []"
+by(auto simp add:sublist_def)
+
+lemma sublist_shift_lemma:
+ "map fst [p:zip xs [i..i + length xs(] . snd p : A] =
+  map fst [p:zip xs [0..length xs(] . snd p + i : A]"
+apply(induct_tac xs rule: rev_induct)
+ apply simp
+apply(simp add:add_commute)
+done
+
+lemma sublist_append:
+ "sublist (l@l') A = sublist l A @ sublist l' {j. j + length l : A}"
+apply(unfold sublist_def)
+apply(induct_tac l' rule: rev_induct)
+ apply simp
+apply(simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
+apply(simp add:add_commute)
+done
+
+lemma sublist_Cons:
+ "sublist (x#l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
+apply(induct_tac l rule: rev_induct)
+ apply(simp add:sublist_def)
+apply(simp del: append_Cons add: append_Cons[symmetric] sublist_append)
+done
+
+lemma sublist_singleton[simp]: "sublist [x] A = (if 0 : A then [x] else [])"
+by(simp add:sublist_Cons)
+
+lemma sublist_upt_eq_take[simp]: "sublist l {..n(} = take n l"
+apply(induct_tac l rule: rev_induct)
+ apply simp
+apply(simp split:nat_diff_split add:sublist_append)
+done
+
+
+lemma take_Cons': "take n (x#xs) = (if n=0 then [] else x # take (n - 1) xs)"
+by(case_tac n, simp_all)
+
+lemma drop_Cons': "drop n (x#xs) = (if n=0 then x#xs else drop (n - 1) xs)"
+by(case_tac n, simp_all)
+
+lemma nth_Cons': "(x#xs)!n = (if n=0 then x else xs!(n - 1))"
+by(case_tac n, simp_all)
+
+lemmas [simp] = take_Cons'[of "number_of v",standard]
+                drop_Cons'[of "number_of v",standard]
+                nth_Cons'[of "number_of v",standard]
 
 end;