src/HOL/List.ML
author nipkow
Mon, 04 Aug 1997 11:50:35 +0200
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Added a take/dropWhile lemma.
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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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bind_thm("not_Cons_self2",not_Cons_self RS not_sym);
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Addsimps [not_Cons_self,not_Cons_self2];
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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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(** "lists": the list-forming 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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val listsE = lists.mk_cases list.simps  "x#l : lists A";
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AddSEs [listsE];
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AddSIs lists.intrs;
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goal thy "!!l. l: lists A ==> l: lists B --> l: lists (A Int B)";
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by (etac lists.induct 1);
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by (ALLGOALS Blast_tac);
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qed_spec_mp "lists_IntI";
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goal thy "lists (A Int B) = lists A Int lists B";
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br (mono_Int RS equalityI) 1;
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by (simp_tac (!simpset addsimps [mono_def, lists_mono]) 1);
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by (blast_tac (!claset addSIs [lists_IntI]) 1);
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qed "lists_Int_eq";
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Addsimps [lists_Int_eq];
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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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section "@ - 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) = (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_self_conv";
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goal thy "(xs = xs @ ys) = (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 "self_append_conv";
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AddIffs [append_self_conv,self_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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 by (rtac allI 1);
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 by (induct_tac "ys" 1);
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  by (ALLGOALS Asm_simp_tac);
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by (rtac 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 "!ys zs. (ys @ xs = zs @ xs) = (ys=zs)";
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by (induct_tac "xs" 1);
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by (Simp_tac 1);
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by (strip_tac 1);
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by (subgoal_tac "((ys @ [a]) @ list = (zs @ [a]) @ list) = (ys=zs)" 1);
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by (Asm_full_simp_tac 1);
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by (Blast_tac 1);
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qed_spec_mp "append_same_eq";
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AddIffs [append_same_eq];
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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 "!!xs. xs ~= [] ==> hd(xs @ ys) = hd xs";
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by (asm_simp_tac (!simpset addsimps [hd_append]
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                           setloop (split_tac [expand_list_case])) 1);
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qed "hd_append2";
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Addsimps [hd_append2];
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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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goal thy "!!xs. xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys";
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by (asm_simp_tac (!simpset addsimps [tl_append]
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                           setloop (split_tac [expand_list_case])) 1);
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qed "tl_append2";
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Addsimps [tl_append2];
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(** map **)
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section "map";
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goal thy
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  "(!x. x : set 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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   193
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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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section "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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section "mem";
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   219
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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 **)
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section "set";
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   235
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   236
goal thy "set (xs@ys) = (set xs Un set ys)";
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   237
by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
1908
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qed "set_of_list_append";
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Addsimps[set_of_list_append];
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   241
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   242
goal thy "(x mem xs) = (x: set xs)";
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   243
by (induct_tac "xs" 1);
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by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
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   245
by (Blast_tac 1);
1908
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   246
qed "set_of_list_mem_eq";
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   247
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   248
goal thy "set l <= set (x#l)";
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
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parents: 1908
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   249
by (Simp_tac 1);
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paulson
parents: 2739
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   250
by (Blast_tac 1);
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
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parents: 1908
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   251
qed "set_of_list_subset_Cons";
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   252
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   253
goal thy "(set xs = {}) = (xs = [])";
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by (induct_tac "xs" 1);
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   255
by (ALLGOALS Asm_simp_tac);
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qed "set_of_list_empty";
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   257
Addsimps [set_of_list_empty];
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   258
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   259
goal thy "set(rev xs) = set(xs)";
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   260
by (induct_tac "xs" 1);
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   261
by (ALLGOALS Asm_simp_tac);
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qed "set_of_list_rev";
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   263
Addsimps [set_of_list_rev];
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   264
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   265
goal thy "set(map f xs) = f``(set xs)";
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   266
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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   270
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   271
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(** list_all **)
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section "list_all";
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   275
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goal thy "list_all (%x.True) xs = True";
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   277
by (induct_tac "xs" 1);
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   278
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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   282
goal thy "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
3040
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   283
by (induct_tac "xs" 1);
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   284
by (ALLGOALS Asm_simp_tac);
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   285
qed "list_all_append";
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   286
Addsimps [list_all_append];
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   287
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   288
goal thy "list_all P xs = (!x. x mem xs --> P(x))";
3040
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   289
by (induct_tac "xs" 1);
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clasohm
parents: 1202
diff changeset
   290
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
2891
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parents: 2739
diff changeset
   291
by (Blast_tac 1);
923
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   292
qed "list_all_mem_conv";
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   293
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   294
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   295
(** filter **)
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   296
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   297
section "filter";
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   298
3383
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paulson
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   299
goal thy "filter P (xs@ys) = filter P xs @ filter P ys";
3457
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   300
by (induct_tac "xs" 1);
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   301
 by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
2608
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   302
qed "filter_append";
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   303
Addsimps [filter_append];
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   304
3383
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paulson
parents: 3342
diff changeset
   305
goal thy "size (filter P xs) <= size xs";
3457
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diff changeset
   306
by (induct_tac "xs" 1);
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diff changeset
   307
 by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
3383
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paulson
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   308
qed "filter_size";
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parents: 3342
diff changeset
   309
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   311
(** concat **)
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   312
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   313
section "concat";
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nipkow
parents: 3465
diff changeset
   314
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   315
goal thy  "concat(xs@ys) = concat(xs)@concat(ys)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   316
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   317
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   318
qed"concat_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   319
Addsimps [concat_append];
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   320
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   321
goal thy  "set(concat xs) = Union(set `` set xs)";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   322
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   323
by (ALLGOALS Asm_simp_tac);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   324
qed"set_of_list_concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   325
Addsimps [set_of_list_concat];
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   326
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   327
goal thy "map f (concat xs) = concat (map (map f) xs)"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   328
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   329
by (ALLGOALS Asm_simp_tac);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   330
qed "map_concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   331
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   332
goal thy "filter p (concat xs) = concat (map (filter p) xs)"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   333
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   334
by (ALLGOALS Asm_simp_tac);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   335
qed"filter_concat"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   336
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   337
goal thy "rev(concat xs) = concat (map rev (rev xs))";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   338
by (induct_tac "xs" 1);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   339
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   340
qed "rev_concat";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   341
962
136308504cd9 Added a few thms to nat_ss and list_ss
nipkow
parents: 923
diff changeset
   342
(** length **)
136308504cd9 Added a few thms to nat_ss and list_ss
nipkow
parents: 923
diff changeset
   343
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   344
section "length";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   345
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   346
goal thy "length(xs@ys) = length(xs)+length(ys)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   347
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   348
by (ALLGOALS Asm_simp_tac);
962
136308504cd9 Added a few thms to nat_ss and list_ss
nipkow
parents: 923
diff changeset
   349
qed"length_append";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   350
Addsimps [length_append];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   351
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   352
goal thy "length (map f l) = length l";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   353
by (induct_tac "l" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   354
by (ALLGOALS Simp_tac);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   355
qed "length_map";
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   356
Addsimps [length_map];
962
136308504cd9 Added a few thms to nat_ss and list_ss
nipkow
parents: 923
diff changeset
   357
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   358
goal thy "length(rev xs) = length(xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   359
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   360
by (ALLGOALS Asm_simp_tac);
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   361
qed "length_rev";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   362
Addsimps [length_rev];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   363
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   364
goal thy "(length xs = 0) = (xs = [])";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   365
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   366
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   367
qed "length_0_conv";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   368
AddIffs [length_0_conv];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   369
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   370
goal thy "(0 < length xs) = (xs ~= [])";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   371
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   372
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   373
qed "length_greater_0_conv";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   374
AddIffs [length_greater_0_conv];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   375
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   376
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   377
(** nth **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   378
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   379
section "nth";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   380
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   381
goal thy
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   382
  "!xs. nth n (xs@ys) = \
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   383
\          (if n < length xs then nth n xs else nth (n - length xs) ys)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   384
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   385
 by (Asm_simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   386
 by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   387
 by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   388
  by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   389
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   390
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   391
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   392
qed_spec_mp "nth_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   393
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   394
goal thy "!n. n < length xs --> nth n (map f xs) = f (nth n xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   395
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   396
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   397
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   398
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   399
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   400
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   401
by (ALLGOALS Asm_full_simp_tac);
1485
240cc98b94a7 Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents: 1465
diff changeset
   402
qed_spec_mp "nth_map";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   403
Addsimps [nth_map];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   404
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   405
goal thy "!n. n < length xs --> list_all P xs --> P(nth n xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   406
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   407
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   408
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   409
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   410
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   411
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   412
by (ALLGOALS Asm_full_simp_tac);
1485
240cc98b94a7 Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents: 1465
diff changeset
   413
qed_spec_mp "list_all_nth";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   414
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   415
goal thy "!n. n < length xs --> (nth n xs) mem xs";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   416
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   417
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   418
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   419
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   420
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   421
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   422
(* case 0 *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   423
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   424
(* case Suc x *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   425
by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 1);
1485
240cc98b94a7 Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents: 1465
diff changeset
   426
qed_spec_mp "nth_mem";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   427
Addsimps [nth_mem];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   428
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   429
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   430
(** take  & drop **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   431
section "take & drop";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   432
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   433
goal thy "take 0 xs = []";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   434
by (induct_tac "xs" 1);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   435
by (ALLGOALS Asm_simp_tac);
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   436
qed "take_0";
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   437
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   438
goal thy "drop 0 xs = xs";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   439
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   440
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   441
qed "drop_0";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   442
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   443
goal thy "take (Suc n) (x#xs) = x # take n xs";
1552
6f71b5d46700 Ran expandshort
paulson
parents: 1485
diff changeset
   444
by (Simp_tac 1);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   445
qed "take_Suc_Cons";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   446
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   447
goal thy "drop (Suc n) (x#xs) = drop n xs";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   448
by (Simp_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   449
qed "drop_Suc_Cons";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   450
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   451
Delsimps [take_Cons,drop_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   452
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   453
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   454
goal thy "!xs. length(take n xs) = min (length xs) n";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   455
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   456
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   457
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   458
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   459
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   460
qed_spec_mp "length_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   461
Addsimps [length_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   462
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   463
goal thy "!xs. length(drop n xs) = (length xs - n)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   464
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   465
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   466
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   467
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   468
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   469
qed_spec_mp "length_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   470
Addsimps [length_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   471
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   472
goal thy "!xs. length xs <= n --> take n xs = xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   473
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   474
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   475
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   476
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   477
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   478
qed_spec_mp "take_all";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   479
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   480
goal thy "!xs. length xs <= n --> drop n xs = []";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   481
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   482
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   483
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   484
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   485
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   486
qed_spec_mp "drop_all";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   487
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   488
goal thy 
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   489
  "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   490
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   491
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   492
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   493
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   494
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   495
qed_spec_mp "take_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   496
Addsimps [take_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   497
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   498
goal thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   499
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   500
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   501
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   502
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   503
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   504
qed_spec_mp "drop_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   505
Addsimps [drop_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   506
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   507
goal thy "!xs n. take n (take m xs) = take (min n m) xs"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   508
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   509
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   510
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   511
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   512
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   513
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   514
by (exhaust_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   515
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   516
qed_spec_mp "take_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   517
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   518
goal thy "!xs. drop n (drop m xs) = drop (n + m) xs"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   519
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   520
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   521
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   522
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   523
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   524
qed_spec_mp "drop_drop";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   525
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   526
goal thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   527
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   528
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   529
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   530
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   531
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   532
qed_spec_mp "take_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   533
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   534
goal thy "!xs. take n (map f xs) = map f (take n xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   535
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   536
by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   537
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   538
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   539
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   540
qed_spec_mp "take_map"; 
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   541
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   542
goal thy "!xs. drop n (map f xs) = map f (drop n xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   543
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   544
by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   545
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   546
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   547
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   548
qed_spec_mp "drop_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   549
3283
0db086394024 Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents: 3196
diff changeset
   550
goal thy "!n i. i < n --> nth i (take n xs) = nth i xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   551
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   552
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   553
by (strip_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   554
by (exhaust_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   555
 by (Blast_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   556
by (exhaust_tac "i" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   557
by (ALLGOALS Asm_full_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   558
qed_spec_mp "nth_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   559
Addsimps [nth_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   560
3585
5b2dcdc1829c Generalized nth_drop (Conny).
nipkow
parents: 3574
diff changeset
   561
goal thy  "!xs i. n + i <= length xs --> nth i (drop n xs) = nth (n + i) xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   562
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   563
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   564
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   565
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   566
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   567
qed_spec_mp "nth_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   568
Addsimps [nth_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   569
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   570
(** takeWhile & dropWhile **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   571
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   572
section "takeWhile & dropWhile";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   573
3586
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   574
goal thy "takeWhile P xs @ dropWhile P xs = xs";
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   575
by (induct_tac "xs" 1);
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   576
 by (Simp_tac 1);
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   577
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   578
qed "takeWhile_dropWhile_id";
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   579
Addsimps [takeWhile_dropWhile_id];
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   580
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   581
goal thy  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   582
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   583
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   584
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   585
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   586
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   587
Addsimps [takeWhile_append1];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   588
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   589
goal thy
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   590
  "(!x:set xs.P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   591
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   592
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   593
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   594
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   595
Addsimps [takeWhile_append2];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   596
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   597
goal thy
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   598
  "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   599
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   600
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   601
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   602
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   603
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   604
Addsimps [dropWhile_append1];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   605
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   606
goal thy
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   607
  "(!x:set xs.P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   608
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   609
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   610
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   611
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   612
Addsimps [dropWhile_append2];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   613
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   614
goal thy "x:set(takeWhile P xs) --> x:set xs & P x";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   615
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   616
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   617
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   618
qed_spec_mp"set_of_list_take_whileD";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   619