src/HOL/List.ML
author paulson
Mon May 26 12:38:29 1997 +0200 (1997-05-26)
changeset 3342 ec3b55fcb165
parent 3292 8b143c196d42
child 3383 7707cb7a5054
permissions -rw-r--r--
New operator "lists" for formalizing sets of lists
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(*  Title:      HOL/List
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1994 TU Muenchen
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List lemmas
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*)
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goal thy "!x. xs ~= x#xs";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed_spec_mp "not_Cons_self";
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Addsimps [not_Cons_self];
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goal thy "(xs ~= []) = (? y ys. xs = y#ys)";
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by (induct_tac "xs" 1);
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by (Simp_tac 1);
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by (Asm_simp_tac 1);
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qed "neq_Nil_conv";
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(** List operator over sets **)
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goalw thy lists.defs "!!A B. A<=B ==> lists A <= lists B";
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by (rtac lfp_mono 1);
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by (REPEAT (ares_tac basic_monos 1));
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qed "lists_mono";
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(** list_case **)
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goal thy
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 "P(list_case a f xs) = ((xs=[] --> P(a)) & \
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\                        (!y ys. xs=y#ys --> P(f y ys)))";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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by (Blast_tac 1);
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qed "expand_list_case";
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val prems = goal thy "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)";
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by(induct_tac "xs" 1);
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by(REPEAT(resolve_tac prems 1));
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qed "list_cases";
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goal thy  "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)";
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by (induct_tac "xs" 1);
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by (Blast_tac 1);
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by (Blast_tac 1);
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bind_thm("list_eq_cases",
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  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (conjI RS (result() RS mp))))));
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(** @ - append **)
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goal thy "(xs@ys)@zs = xs@(ys@zs)";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "append_assoc";
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Addsimps [append_assoc];
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goal thy "xs @ [] = xs";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "append_Nil2";
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Addsimps [append_Nil2];
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goal thy "(xs@ys = []) = (xs=[] & ys=[])";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "append_is_Nil_conv";
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AddIffs [append_is_Nil_conv];
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goal thy "([] = xs@ys) = (xs=[] & ys=[])";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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by(Blast_tac 1);
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qed "Nil_is_append_conv";
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AddIffs [Nil_is_append_conv];
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goal thy "(xs @ ys = xs @ zs) = (ys=zs)";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "same_append_eq";
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AddIffs [same_append_eq];
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goal thy "!ys. (xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; 
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by(induct_tac "xs" 1);
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 br allI 1;
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 by(induct_tac "ys" 1);
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  by(ALLGOALS Asm_simp_tac);
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br allI 1;
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by(induct_tac "ys" 1);
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 by(ALLGOALS Asm_simp_tac);
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qed_spec_mp "append1_eq_conv";
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AddIffs [append1_eq_conv];
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goal thy "xs ~= [] --> hd xs # tl xs = xs";
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by(induct_tac "xs" 1);
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by(ALLGOALS Asm_simp_tac);
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qed_spec_mp "hd_Cons_tl";
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Addsimps [hd_Cons_tl];
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goal thy "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "hd_append";
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goal thy "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
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by(simp_tac (!simpset setloop(split_tac[expand_list_case])) 1);
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qed "tl_append";
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(** map **)
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goal thy
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  "(!x. x : set_of_list xs --> f x = g x) --> map f xs = map g xs";
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by(induct_tac "xs" 1);
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by(ALLGOALS Asm_simp_tac);
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bind_thm("map_ext", impI RS (allI RS (result() RS mp)));
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goal thy "map (%x.x) = (%xs.xs)";
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by (rtac ext 1);
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "map_ident";
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Addsimps[map_ident];
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goal thy "map f (xs@ys) = map f xs @ map f ys";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "map_append";
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Addsimps[map_append];
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goalw thy [o_def] "map (f o g) xs = map f (map g xs)";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "map_compose";
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Addsimps[map_compose];
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goal thy "rev(map f xs) = map f (rev xs)";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "rev_map";
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(** rev **)
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goal thy "rev(xs@ys) = rev(ys) @ rev(xs)";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "rev_append";
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Addsimps[rev_append];
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goal thy "rev(rev l) = l";
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by (induct_tac "l" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "rev_rev_ident";
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Addsimps[rev_rev_ident];
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(** mem **)
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goal thy "x mem (xs@ys) = (x mem xs | x mem ys)";
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by (induct_tac "xs" 1);
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by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
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qed "mem_append";
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Addsimps[mem_append];
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goal thy "x mem [x:xs.P(x)] = (x mem xs & P(x))";
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by (induct_tac "xs" 1);
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by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
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qed "mem_filter";
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Addsimps[mem_filter];
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(** set_of_list **)
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goal thy "set_of_list (xs@ys) = (set_of_list xs Un set_of_list ys)";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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by (Blast_tac 1);
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qed "set_of_list_append";
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Addsimps[set_of_list_append];
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goal thy "(x mem xs) = (x: set_of_list xs)";
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by (induct_tac "xs" 1);
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by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
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by (Blast_tac 1);
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qed "set_of_list_mem_eq";
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goal thy "set_of_list l <= set_of_list (x#l)";
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by (Simp_tac 1);
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by (Blast_tac 1);
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qed "set_of_list_subset_Cons";
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goal thy "(set_of_list xs = {}) = (xs = [])";
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by(induct_tac "xs" 1);
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by(ALLGOALS Asm_simp_tac);
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qed "set_of_list_empty";
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Addsimps [set_of_list_empty];
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goal thy "set_of_list(rev xs) = set_of_list(xs)";
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by(induct_tac "xs" 1);
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by(ALLGOALS Asm_simp_tac);
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by(Blast_tac 1);
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qed "set_of_list_rev";
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Addsimps [set_of_list_rev];
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goal thy "set_of_list(map f xs) = f``(set_of_list xs)";
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by(induct_tac "xs" 1);
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by(ALLGOALS Asm_simp_tac);
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qed "set_of_list_map";
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Addsimps [set_of_list_map];
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(** list_all **)
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goal thy "list_all (%x.True) xs = True";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "list_all_True";
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Addsimps [list_all_True];
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goal thy "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "list_all_append";
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Addsimps [list_all_append];
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goal thy "list_all P xs = (!x. x mem xs --> P(x))";
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by (induct_tac "xs" 1);
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by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
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by (Blast_tac 1);
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qed "list_all_mem_conv";
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(** filter **)
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goal thy "[x:xs@ys . P] = [x:xs . P] @ [y:ys . P]";
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by(induct_tac "xs" 1);
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 by(ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
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qed "filter_append";
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Addsimps [filter_append];
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(** concat **)
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goal thy  "concat(xs@ys) = concat(xs)@concat(ys)";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed"concat_append";
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Addsimps [concat_append];
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goal thy "rev(concat ls) = concat (map rev (rev ls))";
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by (induct_tac "ls" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "rev_concat";
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(** length **)
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goal thy "length(xs@ys) = length(xs)+length(ys)";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed"length_append";
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Addsimps [length_append];
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goal thy "length (map f l) = length l";
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by (induct_tac "l" 1);
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by (ALLGOALS Simp_tac);
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qed "length_map";
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Addsimps [length_map];
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goal thy "length(rev xs) = length(xs)";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "length_rev";
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Addsimps [length_rev];
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goal thy "(length xs = 0) = (xs = [])";
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by(induct_tac "xs" 1);
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by(ALLGOALS Asm_simp_tac);
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qed "length_0_conv";
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AddIffs [length_0_conv];
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goal thy "(0 < length xs) = (xs ~= [])";
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by(induct_tac "xs" 1);
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by(ALLGOALS Asm_simp_tac);
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qed "length_greater_0_conv";
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AddIffs [length_greater_0_conv];
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(** nth **)
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goal thy
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  "!xs. nth n (xs@ys) = \
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\          (if n < length xs then nth n xs else nth (n - length xs) ys)";
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by(nat_ind_tac "n" 1);
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 by(Asm_simp_tac 1);
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 br allI 1;
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 by(exhaust_tac "xs" 1);
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  by(ALLGOALS Asm_simp_tac);
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br allI 1;
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by(exhaust_tac "xs" 1);
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 by(ALLGOALS Asm_simp_tac);
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qed_spec_mp "nth_append";
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goal thy "!n. n < length xs --> nth n (map f xs) = f (nth n xs)";
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by (induct_tac "xs" 1);
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(* case [] *)
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by (Asm_full_simp_tac 1);
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(* case x#xl *)
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by (rtac allI 1);
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by (nat_ind_tac "n" 1);
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by (ALLGOALS Asm_full_simp_tac);
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qed_spec_mp "nth_map";
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Addsimps [nth_map];
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goal thy "!n. n < length xs --> list_all P xs --> P(nth n xs)";
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by (induct_tac "xs" 1);
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(* case [] *)
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by (Simp_tac 1);
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(* case x#xl *)
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by (rtac allI 1);
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by (nat_ind_tac "n" 1);
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by (ALLGOALS Asm_full_simp_tac);
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qed_spec_mp "list_all_nth";
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goal thy "!n. n < length xs --> (nth n xs) mem xs";
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by (induct_tac "xs" 1);
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(* case [] *)
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by (Simp_tac 1);
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(* case x#xl *)
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by (rtac allI 1);
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by (nat_ind_tac "n" 1);
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(* case 0 *)
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by (Asm_full_simp_tac 1);
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(* case Suc x *)
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by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 1);
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qed_spec_mp "nth_mem";
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Addsimps [nth_mem];
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(** take  & drop **)
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section "take & drop";
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goal thy "take 0 xs = []";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "take_0";
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goal thy "drop 0 xs = xs";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
nipkow@2608
   351
qed "drop_0";
nipkow@2608
   352
nipkow@1419
   353
goal thy "take (Suc n) (x#xs) = x # take n xs";
paulson@1552
   354
by (Simp_tac 1);
nipkow@1419
   355
qed "take_Suc_Cons";
nipkow@1327
   356
nipkow@2608
   357
goal thy "drop (Suc n) (x#xs) = drop n xs";
nipkow@2608
   358
by (Simp_tac 1);
nipkow@2608
   359
qed "drop_Suc_Cons";
nipkow@2608
   360
nipkow@2608
   361
Delsimps [take_Cons,drop_Cons];
nipkow@2608
   362
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
nipkow@2608
   363
nipkow@3011
   364
goal thy "!xs. length(take n xs) = min (length xs) n";
nipkow@2608
   365
by(nat_ind_tac "n" 1);
nipkow@2608
   366
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   367
br allI 1;
nipkow@3283
   368
by(exhaust_tac "xs" 1);
nipkow@2608
   369
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   370
qed_spec_mp "length_take";
nipkow@2608
   371
Addsimps [length_take];
clasohm@923
   372
nipkow@3011
   373
goal thy "!xs. length(drop n xs) = (length xs - n)";
nipkow@2608
   374
by(nat_ind_tac "n" 1);
nipkow@2608
   375
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   376
br allI 1;
nipkow@3283
   377
by(exhaust_tac "xs" 1);
nipkow@2608
   378
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   379
qed_spec_mp "length_drop";
nipkow@2608
   380
Addsimps [length_drop];
nipkow@2608
   381
nipkow@3011
   382
goal thy "!xs. length xs <= n --> take n xs = xs";
nipkow@2608
   383
by(nat_ind_tac "n" 1);
nipkow@2608
   384
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   385
br allI 1;
nipkow@3283
   386
by(exhaust_tac "xs" 1);
nipkow@2608
   387
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   388
qed_spec_mp "take_all";
clasohm@923
   389
nipkow@3011
   390
goal thy "!xs. length xs <= n --> drop n xs = []";
nipkow@2608
   391
by(nat_ind_tac "n" 1);
nipkow@2608
   392
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   393
br allI 1;
nipkow@3283
   394
by(exhaust_tac "xs" 1);
nipkow@2608
   395
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   396
qed_spec_mp "drop_all";
nipkow@2608
   397
nipkow@3011
   398
goal thy 
nipkow@2608
   399
  "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
nipkow@2608
   400
by(nat_ind_tac "n" 1);
nipkow@2608
   401
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   402
br allI 1;
nipkow@3283
   403
by(exhaust_tac "xs" 1);
nipkow@2608
   404
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   405
qed_spec_mp "take_append";
nipkow@2608
   406
Addsimps [take_append];
nipkow@2608
   407
nipkow@3011
   408
goal thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
nipkow@2608
   409
by(nat_ind_tac "n" 1);
nipkow@2608
   410
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   411
br allI 1;
nipkow@3283
   412
by(exhaust_tac "xs" 1);
nipkow@2608
   413
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   414
qed_spec_mp "drop_append";
nipkow@2608
   415
Addsimps [drop_append];
nipkow@2608
   416
nipkow@3011
   417
goal thy "!xs n. take n (take m xs) = take (min n m) xs"; 
nipkow@2608
   418
by(nat_ind_tac "m" 1);
nipkow@2608
   419
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   420
br allI 1;
nipkow@3283
   421
by(exhaust_tac "xs" 1);
nipkow@2608
   422
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   423
br allI 1;
nipkow@3283
   424
by(exhaust_tac "n" 1);
nipkow@2608
   425
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   426
qed_spec_mp "take_take";
nipkow@2608
   427
nipkow@3011
   428
goal thy "!xs. drop n (drop m xs) = drop (n + m) xs"; 
nipkow@2608
   429
by(nat_ind_tac "m" 1);
nipkow@2608
   430
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   431
br allI 1;
nipkow@3283
   432
by(exhaust_tac "xs" 1);
nipkow@2608
   433
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   434
qed_spec_mp "drop_drop";
clasohm@923
   435
nipkow@3011
   436
goal thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
nipkow@2608
   437
by(nat_ind_tac "m" 1);
nipkow@2608
   438
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   439
br allI 1;
nipkow@3283
   440
by(exhaust_tac "xs" 1);
nipkow@2608
   441
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   442
qed_spec_mp "take_drop";
nipkow@2608
   443
nipkow@3011
   444
goal thy "!xs. take n (map f xs) = map f (take n xs)"; 
nipkow@2608
   445
by(nat_ind_tac "n" 1);
nipkow@2608
   446
by(ALLGOALS Asm_simp_tac);
nipkow@2608
   447
br allI 1;
nipkow@3283
   448
by(exhaust_tac "xs" 1);
nipkow@2608
   449
by(ALLGOALS Asm_simp_tac);
nipkow@2608
   450
qed_spec_mp "take_map"; 
nipkow@2608
   451
nipkow@3011
   452
goal thy "!xs. drop n (map f xs) = map f (drop n xs)"; 
nipkow@2608
   453
by(nat_ind_tac "n" 1);
nipkow@2608
   454
by(ALLGOALS Asm_simp_tac);
nipkow@2608
   455
br allI 1;
nipkow@3283
   456
by(exhaust_tac "xs" 1);
nipkow@2608
   457
by(ALLGOALS Asm_simp_tac);
nipkow@2608
   458
qed_spec_mp "drop_map";
nipkow@2608
   459
nipkow@3283
   460
goal thy "!n i. i < n --> nth i (take n xs) = nth i xs";
nipkow@3040
   461
by(induct_tac "xs" 1);
nipkow@2608
   462
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   463
by(strip_tac 1);
nipkow@3283
   464
by(exhaust_tac "n" 1);
paulson@2891
   465
 by(Blast_tac 1);
nipkow@3283
   466
by(exhaust_tac "i" 1);
nipkow@3292
   467
by(ALLGOALS Asm_full_simp_tac);
nipkow@2608
   468
qed_spec_mp "nth_take";
nipkow@2608
   469
Addsimps [nth_take];
clasohm@923
   470
nipkow@3283
   471
goal thy  "!xs i. n + i < length xs --> nth i (drop n xs) = nth (n + i) xs";
nipkow@2608
   472
by(nat_ind_tac "n" 1);
nipkow@2608
   473
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   474
br allI 1;
nipkow@3283
   475
by(exhaust_tac "xs" 1);
nipkow@2608
   476
 by(ALLGOALS Asm_simp_tac);
nipkow@2608
   477
qed_spec_mp "nth_drop";
nipkow@2608
   478
Addsimps [nth_drop];
nipkow@2608
   479
nipkow@2608
   480
(** takeWhile & dropWhile **)
nipkow@2608
   481
nipkow@3011
   482
goal thy
nipkow@2608
   483
  "x:set_of_list xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
nipkow@3040
   484
by(induct_tac "xs" 1);
nipkow@2608
   485
 by(Simp_tac 1);
nipkow@2608
   486
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
paulson@2891
   487
by(Blast_tac 1);
nipkow@2608
   488
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   489
Addsimps [takeWhile_append1];
clasohm@923
   490
nipkow@3011
   491
goal thy
nipkow@2608
   492
  "(!x:set_of_list xs.P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
nipkow@3040
   493
by(induct_tac "xs" 1);
nipkow@2608
   494
 by(Simp_tac 1);
nipkow@2608
   495
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
nipkow@2608
   496
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   497
Addsimps [takeWhile_append2];
lcp@1169
   498
nipkow@3011
   499
goal thy
nipkow@2608
   500
  "x:set_of_list xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
nipkow@3040
   501
by(induct_tac "xs" 1);
nipkow@2608
   502
 by(Simp_tac 1);
nipkow@2608
   503
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
paulson@2891
   504
by(Blast_tac 1);
nipkow@2608
   505
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
nipkow@2608
   506
Addsimps [dropWhile_append1];
nipkow@2608
   507
nipkow@3011
   508
goal thy
nipkow@2608
   509
  "(!x:set_of_list xs.P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
nipkow@3040
   510
by(induct_tac "xs" 1);
nipkow@2608
   511
 by(Simp_tac 1);
nipkow@2608
   512
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
nipkow@2608
   513
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
nipkow@2608
   514
Addsimps [dropWhile_append2];
nipkow@2608
   515
nipkow@3011
   516
goal thy "x:set_of_list(takeWhile P xs) --> x:set_of_list xs & P x";
nipkow@3040
   517
by(induct_tac "xs" 1);
nipkow@2608
   518
 by(Simp_tac 1);
nipkow@2608
   519
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
nipkow@2608
   520
qed_spec_mp"set_of_list_take_whileD";
nipkow@2608
   521