(* Title: HOL/List
ID: $Id$
Author: Tobias Nipkow
Copyright 1994 TU Muenchen
List lemmas
*)
goal thy "!x. xs ~= x#xs";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "not_Cons_self";
Addsimps [not_Cons_self];
goal thy "(xs ~= []) = (? y ys. xs = y#ys)";
by (induct_tac "xs" 1);
by (Simp_tac 1);
by (Asm_simp_tac 1);
qed "neq_Nil_conv";
(** List operator over sets **)
goalw thy lists.defs "!!A B. A<=B ==> lists A <= lists B";
by (rtac lfp_mono 1);
by (REPEAT (ares_tac basic_monos 1));
qed "lists_mono";
(** list_case **)
goal thy
"P(list_case a f xs) = ((xs=[] --> P(a)) & \
\ (!y ys. xs=y#ys --> P(f y ys)))";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
by (Blast_tac 1);
qed "expand_list_case";
val prems = goal thy "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)";
by (induct_tac "xs" 1);
by (REPEAT(resolve_tac prems 1));
qed "list_cases";
goal thy "(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))))));
(** @ - append **)
section "@ - append";
goal thy "(xs@ys)@zs = xs@(ys@zs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "append_assoc";
Addsimps [append_assoc];
goal thy "xs @ [] = xs";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "append_Nil2";
Addsimps [append_Nil2];
goal thy "(xs@ys = []) = (xs=[] & ys=[])";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "append_is_Nil_conv";
AddIffs [append_is_Nil_conv];
goal thy "([] = xs@ys) = (xs=[] & ys=[])";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
by (Blast_tac 1);
qed "Nil_is_append_conv";
AddIffs [Nil_is_append_conv];
goal thy "(xs @ ys = xs @ zs) = (ys=zs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "same_append_eq";
AddIffs [same_append_eq];
goal thy "!ys. (xs @ [x] = ys @ [y]) = (xs = ys & x = y)";
by (induct_tac "xs" 1);
by (rtac allI 1);
by (induct_tac "ys" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (induct_tac "ys" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "append1_eq_conv";
AddIffs [append1_eq_conv];
goal thy "xs ~= [] --> hd xs # tl xs = xs";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "hd_Cons_tl";
Addsimps [hd_Cons_tl];
goal thy "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "hd_append";
goal thy "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
by (simp_tac (!simpset setloop(split_tac[expand_list_case])) 1);
qed "tl_append";
(** map **)
section "map";
goal thy
"(!x. x : set xs --> f x = g x) --> map f xs = map g xs";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
bind_thm("map_ext", impI RS (allI RS (result() RS mp)));
goal thy "map (%x.x) = (%xs.xs)";
by (rtac ext 1);
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "map_ident";
Addsimps[map_ident];
goal thy "map f (xs@ys) = map f xs @ map f ys";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "map_append";
Addsimps[map_append];
goalw thy [o_def] "map (f o g) xs = map f (map g xs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "map_compose";
Addsimps[map_compose];
goal thy "rev(map f xs) = map f (rev xs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "rev_map";
(** rev **)
section "rev";
goal thy "rev(xs@ys) = rev(ys) @ rev(xs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "rev_append";
Addsimps[rev_append];
goal thy "rev(rev l) = l";
by (induct_tac "l" 1);
by (ALLGOALS Asm_simp_tac);
qed "rev_rev_ident";
Addsimps[rev_rev_ident];
(** mem **)
section "mem";
goal thy "x mem (xs@ys) = (x mem xs | x mem ys)";
by (induct_tac "xs" 1);
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
qed "mem_append";
Addsimps[mem_append];
goal thy "x mem [x:xs.P(x)] = (x mem xs & P(x))";
by (induct_tac "xs" 1);
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
qed "mem_filter";
Addsimps[mem_filter];
(** set **)
section "set";
goal thy "set (xs@ys) = (set xs Un set ys)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "set_of_list_append";
Addsimps[set_of_list_append];
goal thy "(x mem xs) = (x: set xs)";
by (induct_tac "xs" 1);
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
by (Blast_tac 1);
qed "set_of_list_mem_eq";
goal thy "set l <= set (x#l)";
by (Simp_tac 1);
by (Blast_tac 1);
qed "set_of_list_subset_Cons";
goal thy "(set xs = {}) = (xs = [])";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "set_of_list_empty";
Addsimps [set_of_list_empty];
goal thy "set(rev xs) = set(xs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "set_of_list_rev";
Addsimps [set_of_list_rev];
goal thy "set(map f xs) = f``(set xs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "set_of_list_map";
Addsimps [set_of_list_map];
(** list_all **)
section "list_all";
goal thy "list_all (%x.True) xs = True";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "list_all_True";
Addsimps [list_all_True];
goal thy "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "list_all_append";
Addsimps [list_all_append];
goal thy "list_all P xs = (!x. x mem xs --> P(x))";
by (induct_tac "xs" 1);
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
by (Blast_tac 1);
qed "list_all_mem_conv";
(** filter **)
section "filter";
goal thy "filter P (xs@ys) = filter P xs @ filter P ys";
by (induct_tac "xs" 1);
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
qed "filter_append";
Addsimps [filter_append];
goal thy "size (filter P xs) <= size xs";
by (induct_tac "xs" 1);
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
qed "filter_size";
(** concat **)
section "concat";
goal thy "concat(xs@ys) = concat(xs)@concat(ys)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed"concat_append";
Addsimps [concat_append];
goal thy "set(concat xs) = Union(set `` set xs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed"set_of_list_concat";
Addsimps [set_of_list_concat];
goal thy "map f (concat xs) = concat (map (map f) xs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "map_concat";
goal thy "filter p (concat xs) = concat (map (filter p) xs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed"filter_concat";
goal thy "rev(concat xs) = concat (map rev (rev xs))";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "rev_concat";
(** length **)
section "length";
goal thy "length(xs@ys) = length(xs)+length(ys)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed"length_append";
Addsimps [length_append];
goal thy "length (map f l) = length l";
by (induct_tac "l" 1);
by (ALLGOALS Simp_tac);
qed "length_map";
Addsimps [length_map];
goal thy "length(rev xs) = length(xs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "length_rev";
Addsimps [length_rev];
goal thy "(length xs = 0) = (xs = [])";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "length_0_conv";
AddIffs [length_0_conv];
goal thy "(0 < length xs) = (xs ~= [])";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "length_greater_0_conv";
AddIffs [length_greater_0_conv];
(** nth **)
section "nth";
goal thy
"!xs. nth n (xs@ys) = \
\ (if n < length xs then nth n xs else nth (n - length xs) ys)";
by (nat_ind_tac "n" 1);
by (Asm_simp_tac 1);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "nth_append";
goal thy "!n. n < length xs --> nth n (map f xs) = f (nth n xs)";
by (induct_tac "xs" 1);
(* case [] *)
by (Asm_full_simp_tac 1);
(* case x#xl *)
by (rtac allI 1);
by (nat_ind_tac "n" 1);
by (ALLGOALS Asm_full_simp_tac);
qed_spec_mp "nth_map";
Addsimps [nth_map];
goal thy "!n. n < length xs --> list_all P xs --> P(nth n xs)";
by (induct_tac "xs" 1);
(* case [] *)
by (Simp_tac 1);
(* case x#xl *)
by (rtac allI 1);
by (nat_ind_tac "n" 1);
by (ALLGOALS Asm_full_simp_tac);
qed_spec_mp "list_all_nth";
goal thy "!n. n < length xs --> (nth n xs) mem xs";
by (induct_tac "xs" 1);
(* case [] *)
by (Simp_tac 1);
(* case x#xl *)
by (rtac allI 1);
by (nat_ind_tac "n" 1);
(* case 0 *)
by (Asm_full_simp_tac 1);
(* case Suc x *)
by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 1);
qed_spec_mp "nth_mem";
Addsimps [nth_mem];
(** take & drop **)
section "take & drop";
goal thy "take 0 xs = []";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "take_0";
goal thy "drop 0 xs = xs";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "drop_0";
goal thy "take (Suc n) (x#xs) = x # take n xs";
by (Simp_tac 1);
qed "take_Suc_Cons";
goal thy "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 thy "!xs. length(take n xs) = min (length xs) n";
by (nat_ind_tac "n" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "length_take";
Addsimps [length_take];
goal thy "!xs. length(drop n xs) = (length xs - n)";
by (nat_ind_tac "n" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "length_drop";
Addsimps [length_drop];
goal thy "!xs. length xs <= n --> take n xs = xs";
by (nat_ind_tac "n" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "take_all";
goal thy "!xs. length xs <= n --> drop n xs = []";
by (nat_ind_tac "n" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "drop_all";
goal thy
"!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
by (nat_ind_tac "n" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "take_append";
Addsimps [take_append];
goal thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys";
by (nat_ind_tac "n" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "drop_append";
Addsimps [drop_append];
goal thy "!xs n. take n (take m xs) = take (min n m) xs";
by (nat_ind_tac "m" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "n" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "take_take";
goal thy "!xs. drop n (drop m xs) = drop (n + m) xs";
by (nat_ind_tac "m" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "drop_drop";
goal thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)";
by (nat_ind_tac "m" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "take_drop";
goal thy "!xs. take n (map f xs) = map f (take n xs)";
by (nat_ind_tac "n" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "take_map";
goal thy "!xs. drop n (map f xs) = map f (drop n xs)";
by (nat_ind_tac "n" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "drop_map";
goal thy "!n i. i < n --> nth i (take n xs) = nth i xs";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
by (strip_tac 1);
by (exhaust_tac "n" 1);
by (Blast_tac 1);
by (exhaust_tac "i" 1);
by (ALLGOALS Asm_full_simp_tac);
qed_spec_mp "nth_take";
Addsimps [nth_take];
goal thy "!xs i. n + i < length xs --> nth i (drop n xs) = nth (n + i) xs";
by (nat_ind_tac "n" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed_spec_mp "nth_drop";
Addsimps [nth_drop];
(** takeWhile & dropWhile **)
section "takeWhile & dropWhile";
goal thy
"x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
by (induct_tac "xs" 1);
by (Simp_tac 1);
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
by (Blast_tac 1);
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
Addsimps [takeWhile_append1];
goal thy
"(!x:set xs.P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
by (induct_tac "xs" 1);
by (Simp_tac 1);
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
Addsimps [takeWhile_append2];
goal thy
"x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
by (induct_tac "xs" 1);
by (Simp_tac 1);
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
by (Blast_tac 1);
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
Addsimps [dropWhile_append1];
goal thy
"(!x:set xs.P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
by (induct_tac "xs" 1);
by (Simp_tac 1);
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
Addsimps [dropWhile_append2];
goal thy "x:set(takeWhile P xs) --> x:set xs & P x";
by (induct_tac "xs" 1);
by (Simp_tac 1);
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
qed_spec_mp"set_of_list_take_whileD";