(*  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";

val listsE = lists.mk_cases list.simps  "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];

(**  Case analysis **)

section "Case analysis";

val prems = Goal "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)";

by (induct_tac "xs" 1);

by (REPEAT(resolve_tac prems 1));

qed "list_cases";

Goal "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)";

by (induct_tac "xs" 1);

by (Blast_tac 1);

by (Blast_tac 1);

bind_thm("list_eq_cases",

  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (conjI RS (result() RS mp))))));

(** 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];

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 "xs ~= [] ==> length(tl xs) = (length xs) - 1";

by (exhaust_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 "zero_length_conv";

AddIffs [zero_length_conv];

Goal "(0 < length xs) = (xs ~= [])";

by (induct_tac "xs" 1);

by (Auto_tac);

qed "length_greater_0_conv";

AddIffs [length_greater_0_conv];

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";

AddIffs [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";

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 (exhaust_tac "ys" 1);

  by (Asm_simp_tac 1);

 by (fast_tac (claset() addIs [less_add_Suc2] addss simpset()

                      addEs [less_not_refl2 RSN (2,rev_notE)]) 1);

by (rtac allI 1);

by (exhaust_tac "ys" 1);

by (fast_tac (claset() addIs [less_add_Suc2] 

		       addss (simpset() delsimps [length_Suc_conv])

                       addEs [(less_not_refl2 RS not_sym) RSN (2,rev_notE)]) 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";

AddSIs

 [same_append_eq RS iffD2, append1_eq_conv RS iffD2, append_same_eq RS iffD2];

AddSDs

 [same_append_eq RS iffD1, append1_eq_conv RS iffD1, append_same_eq RS iffD1];

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";

bd sym 1;

by(Asm_simp_tac 1);

qed "Cons_eq_appendI";

Goal "[| xs@xs1 = zs; ys = xs1 @ us |] ==> xs@ys = zs@us";

bd sym 1;

by(Asm_simp_tac 1);

qed "append_eq_appendI";

(** 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 (rtac impI 1);

by (hyp_subst_tac 1);

by (induct_tac "ys" 1);

by (Auto_tac);

val lemma = result();

bind_thm("map_cong",impI RSN (2,allI RSN (2,lemma RS mp RS mp)));

Goal "(map f xs = []) = (xs = [])";

by (induct_tac "xs" 1);

by (Auto_tac);

qed "map_is_Nil_conv";

AddIffs [map_is_Nil_conv];

Goal "([] = map f xs) = (xs = [])";

by (induct_tac "xs" 1);

by (Auto_tac);

qed "Nil_is_map_conv";

AddIffs [Nil_is_map_conv];

(** 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];

val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs";

by (stac (rev_rev_ident RS sym) 1);

br(read_instantiate [("P","%xs. ?P(rev xs)")]list.induct)1;

by (ALLGOALS Simp_tac);

by (resolve_tac prems 1);

by (eresolve_tac prems 1);

qed "rev_induct";

fun rev_induct_tac xs = res_inst_tac [("xs",xs)] rev_induct;

Goal  "(xs = [] --> P) -->  (!ys y. xs = ys@[y] --> P) --> P";

by (res_inst_tac [("xs","xs")] rev_induct 1);

by (Auto_tac);

bind_thm ("rev_exhaust",

  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp)))));

(** mem **)

section "mem";

Goal "x mem (xs@ys) = (x mem xs | x mem ys)";

by (induct_tac "xs" 1);

by (Auto_tac);

qed "mem_append";

Addsimps[mem_append];

Goal "x mem [x:xs. P(x)] = (x mem xs & P(x))";

by (induct_tac "xs" 1);

by (Auto_tac);

qed "mem_filter";

Addsimps[mem_filter];

(** set **)

section "set";

qed_goal "finite_set" thy "finite (set xs)" 

	(K [induct_tac "xs" 1, Auto_tac]);

Addsimps[finite_set];

AddSIs[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 "(x mem xs) = (x: set xs)";

by (induct_tac "xs" 1);

by (Auto_tac);

qed "set_mem_eq";

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 "(x : set(filter P xs)) = (x : set xs & P x)";

by (induct_tac "xs" 1);

by (Auto_tac);

qed "in_set_filter";

Addsimps [in_set_filter];

Goal "(x : set xs) = (? ys zs. xs = ys@x#zs)";

by(induct_tac "xs" 1);

 by(Simp_tac 1);

by(Asm_simp_tac 1);

br iffI 1;

by(blast_tac (claset() addIs [eq_Nil_appendI,Cons_eq_appendI]) 1);

by(REPEAT(etac exE 1));

by(exhaust_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];

(** list_all **)

section "list_all";

Goal "list_all (%x. True) xs = True";

by (induct_tac "xs" 1);

by (Auto_tac);

qed "list_all_True";

Addsimps [list_all_True];

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];

Goal "list_all P xs = (!x. x mem xs --> P(x))";

by (induct_tac "xs" 1);

by (Auto_tac);

qed "list_all_mem_conv";

(** 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 (%x. True) xs = xs";

by (induct_tac "xs" 1);

by (Auto_tac);

qed "filter_True";

Addsimps [filter_True];

Goal "filter (%x. False) xs = []";

by (induct_tac "xs" 1);

by (Auto_tac);

qed "filter_False";

Addsimps [filter_False];

Goal "length (filter P xs) <= length xs";

by (induct_tac "xs" 1);

by (Auto_tac);

qed "length_filter";

(** concat **)

section "concat";

Goal  "concat(xs@ys) = concat(xs)@concat(ys)";

by (induct_tac "xs" 1);

by (Auto_tac);

qed"concat_append";

Addsimps [concat_append];

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 "!xs. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))";

by (induct_tac "n" 1);

 by (Asm_simp_tac 1);

 by (rtac allI 1);

 by (exhaust_tac "xs" 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);

(* case [] *)

by (Asm_full_simp_tac 1);

(* case x#xl *)

by (rtac allI 1);

by (induct_tac "n" 1);

by (Auto_tac);

qed_spec_mp "nth_map";

Addsimps [nth_map];

Goal "!n. n < length xs --> list_all P xs --> P(xs!n)";

by (induct_tac "xs" 1);

(* case [] *)

by (Simp_tac 1);

(* case x#xl *)

by (rtac allI 1);

by (induct_tac "n" 1);

by (Auto_tac);

qed_spec_mp "list_all_nth";

Goal "!n. n < length xs --> xs!n mem xs";

by (induct_tac "xs" 1);

(* case [] *)

by (Simp_tac 1);

(* case x#xl *)

by (rtac allI 1);

by (induct_tac "n" 1);

(* case 0 *)

by (Asm_full_simp_tac 1);

(* case Suc x *)

by (Asm_full_simp_tac 1);

qed_spec_mp "nth_mem";

Addsimps [nth_mem];

(** 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];

(** 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 (res_inst_tac [("xs","xs")] rev_induct 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 "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(xs@ys))";

by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);

by (blast_tac (claset() addDs [in_set_butlastD]) 1);

qed "in_set_butlast_appendI1";

Goal "x:set(butlast ys) ==> x:set(butlast(xs@ys))";

by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);

by (Clarify_tac 1);

by (Full_simp_tac 1);

qed "in_set_butlast_appendI2";

(** 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 (exhaust_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 (exhaust_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 (exhaust_tac "xs" 1);

 by (Auto_tac);

qed_spec_mp "take_all";

Goal "!xs. length xs <= n --> drop n xs = []";

by (induct_tac "n" 1);

 by (Auto_tac);

by (exhaust_tac "xs" 1);

 by (Auto_tac);

qed_spec_mp "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 (exhaust_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 (exhaust_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 (exhaust_tac "xs" 1);

 by (Auto_tac);

by (exhaust_tac "na" 1);

 by (Auto_tac);

qed_spec_mp "take_take";

Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; 

by (induct_tac "m" 1);

 by (Auto_tac);

by (exhaust_tac "xs" 1);

 by (Auto_tac);

qed_spec_mp "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 (exhaust_tac "xs" 1);

 by (Auto_tac);

qed_spec_mp "take_drop";

Goal "!xs. take n (map f xs) = map f (take n xs)"; 

by (induct_tac "n" 1);

 by (Auto_tac);

by (exhaust_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 (exhaust_tac "xs" 1);

 by (Auto_tac);

qed_spec_mp "drop_map";

Goal "!n i. i < n --> (take n xs)!i = xs!i";

by (induct_tac "xs" 1);

 by (Auto_tac);

by (exhaust_tac "n" 1);

 by (Blast_tac 1);

by (exhaust_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 (exhaust_tac "xs" 1);

 by (Auto_tac);

qed_spec_mp "nth_drop";

Addsimps [nth_drop];

(** 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 "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";

qed_goal "zip_Nil_Nil"   thy "zip []     []     = []" (K [Simp_tac 1]);

qed_goal "zip_Cons_Cons" thy "zip (x#xs) (y#ys) = (x,y)#zip xs ys" 

						      (K [Simp_tac 1]);

(** foldl **)

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(Simp_tac 1);

by(Asm_full_simp_tac 1);

by(blast_tac (claset() addIs [trans_le_add1]) 1);

qed_spec_mp "start_le_sum";

Goal "n : set ns ==> n <= foldl op+ 0 ns";

by(auto_tac (claset() addIs [start_le_sum],

             simpset() addsimps [in_set_conv_decomp]));

qed "elem_le_sum";

Goal "!m. (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];

(** nodups & remdups **)

section "nodups & remdups";

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 "nodups(remdups xs)";

by (induct_tac "xs" 1);

by (Auto_tac);

qed "nodups_remdups";

Goal "nodups xs --> nodups (filter P xs)";

by (induct_tac "xs" 1);

by (Auto_tac);

qed_spec_mp "nodups_filter";

(** replicate **)

section "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];

(*** 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);

br wf_subset 1;

br Int_lower1 2;

br wf_prod_fun_image 1;

br 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)";

br 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() delsimps [length_Suc_conv] 

				addsimps [lex_prod_def]) 1);

by(auto_tac (claset(), simpset() delsimps [length_Suc_conv]));

  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(exhaust_tac "xys" 1);

 by(ALLGOALS (asm_full_simp_tac (simpset() delsimps [length_Suc_conv])));

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() delsimps [length_Suc_conv] 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);

br iffI 1;

 by(blast_tac (claset() addIs [Cons_eq_appendI]) 2);

by(REPEAT(eresolve_tac [conjE, exE] 1));

by(exhaust_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];

(***

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 =

  read_cterm (sign_of List.thy) ("(xs::'a list) = ys",HOLogic.boolT);

fun last (cons as Const("List.list.op #",_) $ _ $ xs) =

      (case xs of Const("List.list.[]",_) => cons | _ => last xs)

  | last (Const("List.op @",_) $ _ $ ys) = last ys

  | last t = t;

fun list1 (Const("List.list.op #",_) $ _ $ Const("List.list.[]",_)) = true

  | list1 _ = false;

fun butlast ((cons as Const("List.list.op #",_) $ x) $ xs) =

      (case xs of Const("List.list.[]",_) => xs | _ => cons $ butlast xs)

  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys

  | butlast xs = Const("List.list.[]",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 " ^