(* Title: HOL/List
ID: $Id$
Author: Tobias Nipkow
Copyright 1994 TU Muenchen
List lemmas
*)
open List;
AddIffs list.distinct;
AddIffs list.inject;
bind_thm("Cons_inject", (hd list.inject) RS iffD1 RS conjE);
goal List.thy "!x. xs ~= x#xs";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "not_Cons_self";
Addsimps [not_Cons_self];
goal List.thy "(xs ~= []) = (? y ys. xs = y#ys)";
by (list.induct_tac "xs" 1);
by (Simp_tac 1);
by (Asm_simp_tac 1);
by (REPEAT(resolve_tac [exI,refl,conjI] 1));
qed "neq_Nil_conv";
(** @ - append **)
goal List.thy "(xs@ys)@zs = xs@(ys@zs)";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "append_assoc";
Addsimps [append_assoc];
goal List.thy "xs @ [] = xs";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "append_Nil2";
Addsimps [append_Nil2];
goal List.thy "(xs@ys = []) = (xs=[] & ys=[])";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "append_is_Nil";
Addsimps [append_is_Nil];
goal List.thy "(xs @ ys = xs @ zs) = (ys=zs)";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "same_append_eq";
Addsimps [same_append_eq];
goal List.thy "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "hd_append";
(** rev **)
goal List.thy "rev(xs@ys) = rev(ys) @ rev(xs)";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "rev_append";
Addsimps[rev_append];
goal List.thy "rev(rev l) = l";
by (list.induct_tac "l" 1);
by (ALLGOALS Asm_simp_tac);
qed "rev_rev_ident";
Addsimps[rev_rev_ident];
(** mem **)
goal List.thy "x mem (xs@ys) = (x mem xs | x mem ys)";
by (list.induct_tac "xs" 1);
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
qed "mem_append";
Addsimps[mem_append];
goal List.thy "x mem [x:xs.P(x)] = (x mem xs & P(x))";
by (list.induct_tac "xs" 1);
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
qed "mem_filter";
Addsimps[mem_filter];
(** set_of_list **)
goal thy "set_of_list (xs@ys) = (set_of_list xs Un set_of_list ys)";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
by (Fast_tac 1);
qed "set_of_list_append";
Addsimps[set_of_list_append];
goal thy "(x mem xs) = (x: set_of_list xs)";
by (list.induct_tac "xs" 1);
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
by (Fast_tac 1);
qed "set_of_list_mem_eq";
goal List.thy "set_of_list l <= set_of_list (x#l)";
by (Simp_tac 1);
by (Fast_tac 1);
qed "set_of_list_subset_Cons";
(** list_all **)
goal List.thy "list_all (%x.True) xs = True";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "list_all_True";
Addsimps [list_all_True];
goal List.thy "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "list_all_append";
Addsimps [list_all_append];
goal List.thy "list_all P xs = (!x. x mem xs --> P(x))";
by (list.induct_tac "xs" 1);
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
by (Fast_tac 1);
qed "list_all_mem_conv";
(** list_case **)
goal List.thy
"P(list_case a f xs) = ((xs=[] --> P(a)) & \
\ (!y ys. xs=y#ys --> P(f y ys)))";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
by (Fast_tac 1);
qed "expand_list_case";
val prems = goal List.thy "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)";
by(list.induct_tac "xs" 1);
by(REPEAT(resolve_tac prems 1));
qed "list_cases";
goal List.thy "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)";
by (list.induct_tac "xs" 1);
by (Fast_tac 1);
by (Fast_tac 1);
bind_thm("list_eq_cases",
impI RSN (2,allI RSN (2,allI RSN (2,impI RS (conjI RS (result() RS mp))))));
(** flat **)
goal List.thy "flat(xs@ys) = flat(xs)@flat(ys)";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed"flat_append";
Addsimps [flat_append];
(** length **)
goal List.thy "length(xs@ys) = length(xs)+length(ys)";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed"length_append";
Addsimps [length_append];
goal List.thy "length (map f l) = length l";
by (list.induct_tac "l" 1);
by (ALLGOALS Simp_tac);
qed "length_map";
Addsimps [length_map];
goal List.thy "length(rev xs) = length(xs)";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "length_rev";
Addsimps [length_rev];
(** nth **)
val [nth_0,nth_Suc] = nat_recs nth_def;
store_thm("nth_0",nth_0);
store_thm("nth_Suc",nth_Suc);
Addsimps [nth_0,nth_Suc];
goal List.thy "!n. n < length xs --> nth n (map f xs) = f (nth n xs)";
by (list.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 List.thy "!n. n < length xs --> list_all P xs --> P(nth n xs)";
by (list.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 List.thy "!n. n < length xs --> (nth n xs) mem xs";
by (list.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];
(** drop **)
goal thy "drop 0 xs = xs";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "drop_0";
goal thy "drop (Suc n) (x#xs) = drop n xs";
by (Simp_tac 1);
qed "drop_Suc_Cons";
Delsimps [drop_Cons];
Addsimps [drop_0,drop_Suc_Cons];
(** take **)
goal thy "take 0 xs = []";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "take_0";
goal thy "take (Suc n) (x#xs) = x # take n xs";
by (Simp_tac 1);
qed "take_Suc_Cons";
Delsimps [take_Cons];
Addsimps [take_0,take_Suc_Cons];
(** Additional mapping lemmas **)
goal List.thy "map (%x.x) = (%xs.xs)";
by (rtac ext 1);
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "map_ident";
Addsimps[map_ident];
goal List.thy "map f (xs@ys) = map f xs @ map f ys";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "map_append";
Addsimps[map_append];
goalw List.thy [o_def] "map (f o g) xs = map f (map g xs)";
by (list.induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "map_compose";
Addsimps[map_compose];
goal List.thy "rev(map f l) = map f (rev l)";
by (list.induct_tac "l" 1);
by (ALLGOALS Asm_simp_tac);
qed "rev_map_distrib";
goal List.thy "rev(flat ls) = flat (map rev (rev ls))";
by (list.induct_tac "ls" 1);
by (ALLGOALS Asm_simp_tac);
qed "rev_flat";