src/HOL/List.ML
author paulson
Mon, 23 Jun 1997 10:42:03 +0200
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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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 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 "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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   156
Addsimps[rev_rev_ident];
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lcp
parents: 995
diff changeset
   157
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   158
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(** mem **)
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   161
goal thy "x mem (xs@ys) = (x mem xs | x mem ys)";
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by (induct_tac "xs" 1);
1264
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   163
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
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qed "mem_append";
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   165
Addsimps[mem_append];
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   166
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   167
goal thy "x mem [x:xs.P(x)] = (x mem xs & P(x))";
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   168
by (induct_tac "xs" 1);
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clasohm
parents: 1202
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   169
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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1908
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   173
(** set_of_list **)
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   174
1908
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goal thy "set_of_list (xs@ys) = (set_of_list xs Un set_of_list ys)";
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   176
by (induct_tac "xs" 1);
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   177
by (ALLGOALS Asm_simp_tac);
1908
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   178
qed "set_of_list_append";
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   179
Addsimps[set_of_list_append];
1812
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parents: 1760
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   180
1908
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   181
goal thy "(x mem xs) = (x: set_of_list xs)";
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   182
by (induct_tac "xs" 1);
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parents: 1760
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   183
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
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   184
by (Blast_tac 1);
1908
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   185
qed "set_of_list_mem_eq";
1812
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parents: 1760
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   186
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   187
goal thy "set_of_list l <= set_of_list (x#l)";
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
paulson
parents: 1908
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   188
by (Simp_tac 1);
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paulson
parents: 2739
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   189
by (Blast_tac 1);
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
paulson
parents: 1908
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   190
qed "set_of_list_subset_Cons";
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paulson
parents: 1908
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   191
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   192
goal thy "(set_of_list xs = {}) = (xs = [])";
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parents: 3383
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   193
by (induct_tac "xs" 1);
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parents: 3383
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   194
by (ALLGOALS Asm_simp_tac);
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   195
qed "set_of_list_empty";
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   196
Addsimps [set_of_list_empty];
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   197
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   198
goal thy "set_of_list(rev xs) = set_of_list(xs)";
3457
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parents: 3383
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   199
by (induct_tac "xs" 1);
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paulson
parents: 3383
diff changeset
   200
by (ALLGOALS Asm_simp_tac);
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   201
qed "set_of_list_rev";
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   202
Addsimps [set_of_list_rev];
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   203
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   204
goal thy "set_of_list(map f xs) = f``(set_of_list xs)";
3457
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   205
by (induct_tac "xs" 1);
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parents: 3383
diff changeset
   206
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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   209
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paulson
parents: 1760
diff changeset
   210
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(** list_all **)
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   213
goal thy "list_all (%x.True) xs = True";
3040
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   214
by (induct_tac "xs" 1);
1264
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clasohm
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   215
by (ALLGOALS Asm_simp_tac);
923
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qed "list_all_True";
2512
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   217
Addsimps [list_all_True];
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   218
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   219
goal thy "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
3040
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nipkow
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   220
by (induct_tac "xs" 1);
1264
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clasohm
parents: 1202
diff changeset
   221
by (ALLGOALS Asm_simp_tac);
2512
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diff changeset
   222
qed "list_all_append";
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diff changeset
   223
Addsimps [list_all_append];
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   224
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   225
goal thy "list_all P xs = (!x. x mem xs --> P(x))";
3040
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nipkow
parents: 3011
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   226
by (induct_tac "xs" 1);
1264
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clasohm
parents: 1202
diff changeset
   227
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2739
diff changeset
   228
by (Blast_tac 1);
923
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   229
qed "list_all_mem_conv";
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   230
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   231
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   232
(** filter **)
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   233
3383
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   234
goal thy "filter P (xs@ys) = filter P xs @ filter P ys";
3457
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parents: 3383
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   235
by (induct_tac "xs" 1);
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paulson
parents: 3383
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   236
 by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
2608
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   237
qed "filter_append";
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   238
Addsimps [filter_append];
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parents: 2512
diff changeset
   239
3383
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   240
goal thy "size (filter P xs) <= size xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   241
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   242
 by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
3383
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   243
qed "filter_size";
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   244
2608
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   245
450c9b682a92 New class "order" and accompanying changes.
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   246
(** concat **)
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   247
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   248
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
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   249
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   250
by (ALLGOALS Asm_simp_tac);
2608
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   251
qed"concat_append";
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   252
Addsimps [concat_append];
2512
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parents: 1985
diff changeset
   253
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   254
goal thy "rev(concat ls) = concat (map rev (rev ls))";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   255
by (induct_tac "ls" 1);
2512
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nipkow
parents: 1985
diff changeset
   256
by (ALLGOALS Asm_simp_tac);
2608
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parents: 2512
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   257
qed "rev_concat";
923
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parents:
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   258
962
136308504cd9 Added a few thms to nat_ss and list_ss
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parents: 923
diff changeset
   259
(** length **)
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nipkow
parents: 923
diff changeset
   260
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parents: 2891
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   261
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
   262
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   263
by (ALLGOALS Asm_simp_tac);
962
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nipkow
parents: 923
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   264
qed"length_append";
1301
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parents: 1264
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   265
Addsimps [length_append];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   266
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nipkow
parents: 2891
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   267
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
   268
by (induct_tac "l" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   269
by (ALLGOALS Simp_tac);
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nipkow
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diff changeset
   270
qed "length_map";
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   271
Addsimps [length_map];
962
136308504cd9 Added a few thms to nat_ss and list_ss
nipkow
parents: 923
diff changeset
   272
3011
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parents: 2891
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   273
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
   274
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   275
by (ALLGOALS Asm_simp_tac);
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   276
qed "length_rev";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   277
Addsimps [length_rev];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   278
3011
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nipkow
parents: 2891
diff changeset
   279
goal thy "(length xs = 0) = (xs = [])";
3457
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paulson
parents: 3383
diff changeset
   280
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   281
by (ALLGOALS Asm_simp_tac);
2608
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   282
qed "length_0_conv";
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nipkow
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diff changeset
   283
AddIffs [length_0_conv];
450c9b682a92 New class "order" and accompanying changes.
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parents: 2512
diff changeset
   284
3011
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nipkow
parents: 2891
diff changeset
   285
goal thy "(0 < length xs) = (xs ~= [])";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   286
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   287
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
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   288
qed "length_greater_0_conv";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   289
AddIffs [length_greater_0_conv];
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nipkow
parents: 2512
diff changeset
   290
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   291
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parents:
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   292
(** nth **)
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parents:
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   293
3011
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parents: 2891
diff changeset
   294
goal thy
2608
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diff changeset
   295
  "!xs. nth n (xs@ys) = \
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   296
\          (if n < length xs then nth n xs else nth (n - length xs) ys)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   297
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   298
 by (Asm_simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   299
 by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   300
 by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   301
  by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   302
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   303
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   304
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
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   305
qed_spec_mp "nth_append";
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nipkow
parents: 2512
diff changeset
   306
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   307
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
   308
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   309
(* case [] *)
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nipkow
parents: 1264
diff changeset
   310
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   311
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   312
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   313
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   314
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
   315
qed_spec_mp "nth_map";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   316
Addsimps [nth_map];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   317
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   318
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
   319
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   320
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   321
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   322
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   323
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   324
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   325
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
   326
qed_spec_mp "list_all_nth";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   327
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   328
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
   329
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   330
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   331
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   332
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   333
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   334
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   335
(* case 0 *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   336
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   337
(* case Suc x *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   338
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
   339
qed_spec_mp "nth_mem";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   340
Addsimps [nth_mem];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   341
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   342
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   343
(** take  & drop **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   344
section "take & drop";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   345
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   346
goal thy "take 0 xs = []";
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);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   348
by (ALLGOALS Asm_simp_tac);
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   349
qed "take_0";
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   350
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   351
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
   352
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   353
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   354
qed "drop_0";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   355
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   356
goal thy "take (Suc n) (x#xs) = x # take n xs";
1552
6f71b5d46700 Ran expandshort
paulson
parents: 1485
diff changeset
   357
by (Simp_tac 1);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   358
qed "take_Suc_Cons";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   359
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   360
goal thy "drop (Suc n) (x#xs) = drop n xs";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   361
by (Simp_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   362
qed "drop_Suc_Cons";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   363
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   364
Delsimps [take_Cons,drop_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   365
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   366
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   367
goal thy "!xs. length(take n xs) = min (length xs) n";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   368
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   369
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   370
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   371
by (exhaust_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_spec_mp "length_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   374
Addsimps [length_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   375
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   376
goal thy "!xs. length(drop n xs) = (length xs - n)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   377
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   378
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   379
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   380
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   381
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   382
qed_spec_mp "length_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   383
Addsimps [length_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   384
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   385
goal thy "!xs. length xs <= n --> take n xs = xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   386
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   387
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   388
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   389
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   390
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   391
qed_spec_mp "take_all";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   392
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   393
goal thy "!xs. length xs <= n --> drop n xs = []";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   394
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   395
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   396
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   397
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   398
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   399
qed_spec_mp "drop_all";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   400
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   401
goal thy 
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   402
  "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   403
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   404
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   405
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   406
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   407
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   408
qed_spec_mp "take_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   409
Addsimps [take_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   410
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   411
goal thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   412
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   413
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   414
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   415
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   416
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   417
qed_spec_mp "drop_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   418
Addsimps [drop_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   419
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   420
goal thy "!xs n. take n (take m xs) = take (min n m) xs"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   421
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   422
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   423
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   424
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   425
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   426
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   427
by (exhaust_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   428
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   429
qed_spec_mp "take_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   430
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   431
goal thy "!xs. drop n (drop m xs) = drop (n + m) xs"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   432
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   433
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   434
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   435
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   436
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   437
qed_spec_mp "drop_drop";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   438
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   439
goal thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   440
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   441
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   442
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   443
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   444
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   445
qed_spec_mp "take_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   446
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   447
goal thy "!xs. take n (map f xs) = map f (take n xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   448
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   449
by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   450
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   451
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   452
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   453
qed_spec_mp "take_map"; 
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   454
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   455
goal thy "!xs. drop n (map f xs) = map f (drop n xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   456
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   457
by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   458
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   459
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   460
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   461
qed_spec_mp "drop_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   462
3283
0db086394024 Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents: 3196
diff changeset
   463
goal thy "!n i. i < n --> nth i (take n xs) = nth i xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   464
by (induct_tac "xs" 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 (strip_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   467
by (exhaust_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   468
 by (Blast_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   469
by (exhaust_tac "i" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   470
by (ALLGOALS Asm_full_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   471
qed_spec_mp "nth_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   472
Addsimps [nth_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   473
3283
0db086394024 Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents: 3196
diff changeset
   474
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
   475
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   476
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   477
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   478
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   479
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   480
qed_spec_mp "nth_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   481
Addsimps [nth_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   482
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   483
(** takeWhile & dropWhile **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   484
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   485
goal thy
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   486
  "x:set_of_list xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   487
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   488
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   489
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   490
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   491
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   492
Addsimps [takeWhile_append1];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   493
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   494
goal thy
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   495
  "(!x:set_of_list xs.P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   496
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   497
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   498
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
   499
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   500
Addsimps [takeWhile_append2];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   501
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   502
goal thy
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   503
  "x:set_of_list xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   504
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   505
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   506
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   507
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   508
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   509
Addsimps [dropWhile_append1];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   510
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   511
goal thy
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   512
  "(!x:set_of_list xs.P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   513
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   514
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   515
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
   516
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   517
Addsimps [dropWhile_append2];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   518
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   519
goal thy "x:set_of_list(takeWhile P xs) --> x:set_of_list xs & P x";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   520
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   521
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   522
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
   523
qed_spec_mp"set_of_list_take_whileD";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   524