src/HOL/List.ML
author nipkow
Tue, 14 Oct 1997 13:58:47 +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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open List;
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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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(** length **)
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(* needs to come before "@" because of thm append_eq_append_conv *)
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section "length";
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goal thy "length(xs@ys) = length(xs)+length(ys)";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed"length_append";
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Addsimps [length_append];
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goal thy "length (map f l) = length l";
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by (induct_tac "l" 1);
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by (ALLGOALS Simp_tac);
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qed "length_map";
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Addsimps [length_map];
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goal thy "length(rev xs) = length(xs)";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "length_rev";
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Addsimps [length_rev];
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goal thy "(length xs = 0) = (xs = [])";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "length_0_conv";
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AddIffs [length_0_conv];
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goal thy "(0 = length xs) = (xs = [])";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "zero_length_conv";
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AddIffs [zero_length_conv];
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goal thy "(0 < length xs) = (xs ~= [])";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "length_greater_0_conv";
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AddIffs [length_greater_0_conv];
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(** @ - 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 "!ys. length xs = length ys | length us = length vs \
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   155
\              --> (xs@us = ys@vs) = (xs=ys & us=vs)";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   156
by(induct_tac "xs" 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   157
 by(rtac allI 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   158
 by(exhaust_tac "ys" 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   159
  by(Asm_simp_tac 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   160
 by(fast_tac (!claset addIs [less_add_Suc2] addss !simpset
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   161
                      addEs [less_not_refl2 RSN (2,rev_notE)]) 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   162
by(rtac allI 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   163
by(exhaust_tac "ys" 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   164
 by(fast_tac (!claset addIs [less_add_Suc2] addss !simpset
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   165
                      addEs [(less_not_refl2 RS not_sym) RSN (2,rev_notE)]) 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   166
by(Asm_simp_tac 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   167
qed_spec_mp "append_eq_append_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   168
Addsimps [append_eq_append_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   169
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   170
(* Still needed? Unconditional and hence AddIffs.
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   171
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   172
goal thy "(xs @ ys = xs @ zs) = (ys=zs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   173
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   174
by (ALLGOALS Asm_simp_tac);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   175
qed "same_append_eq";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   176
AddIffs [same_append_eq];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   177
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   178
goal thy "!ys. (xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   179
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   180
 by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   181
 by (induct_tac "ys" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   182
  by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   183
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   184
by (induct_tac "ys" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   185
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   186
qed_spec_mp "append1_eq_conv";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   187
AddIffs [append1_eq_conv];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   188
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   189
goal thy "!ys zs. (ys @ xs = zs @ xs) = (ys=zs)";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   190
by (induct_tac "xs" 1);
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   191
by (Simp_tac 1);
3708
56facaebf3e3 Changed some proofs to use Clarify_tac
paulson
parents: 3647
diff changeset
   192
by (Clarify_tac 1);
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   193
by (subgoal_tac "((ys @ [a]) @ list = (zs @ [a]) @ list) = (ys=zs)" 1);
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   194
by (Asm_full_simp_tac 1);
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   195
by (Blast_tac 1);
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   196
qed_spec_mp "append_same_eq";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   197
AddIffs [append_same_eq];
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   198
*)
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   199
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   200
goal thy "xs ~= [] --> hd xs # tl xs = xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   201
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   202
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   203
qed_spec_mp "hd_Cons_tl";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   204
Addsimps [hd_Cons_tl];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   205
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   206
goal thy "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   207
by (induct_tac "xs" 1);
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   208
by (ALLGOALS Asm_simp_tac);
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   209
qed "hd_append";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   210
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   211
goal thy "!!xs. xs ~= [] ==> hd(xs @ ys) = hd xs";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   212
by (asm_simp_tac (!simpset addsimps [hd_append]
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   213
                           setloop (split_tac [expand_list_case])) 1);
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   214
qed "hd_append2";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   215
Addsimps [hd_append2];
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   216
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   217
goal thy "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   218
by (simp_tac (!simpset setloop(split_tac[expand_list_case])) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   219
qed "tl_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   220
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   221
goal thy "!!xs. xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   222
by (asm_simp_tac (!simpset addsimps [tl_append]
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   223
                           setloop (split_tac [expand_list_case])) 1);
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   224
qed "tl_append2";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   225
Addsimps [tl_append2];
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   226
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   227
(** map **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   228
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   229
section "map";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   230
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   231
goal thy
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   232
  "(!x. x : set xs --> f x = g x) --> map f xs = map g xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   233
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   234
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   235
bind_thm("map_ext", impI RS (allI RS (result() RS mp)));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   236
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   237
goal thy "map (%x. x) = (%xs. xs)";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   238
by (rtac ext 1);
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   239
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   240
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   241
qed "map_ident";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   242
Addsimps[map_ident];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   243
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   244
goal thy "map f (xs@ys) = map f xs @ map f ys";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   245
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   246
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   247
qed "map_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   248
Addsimps[map_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   249
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   250
goalw thy [o_def] "map (f o g) xs = map f (map g xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   251
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   252
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   253
qed "map_compose";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   254
Addsimps[map_compose];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   255
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   256
goal thy "rev(map f xs) = map f (rev xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   257
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   258
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   259
qed "rev_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   260
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   261
(* a congruence rule for map: *)
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   262
goal thy
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   263
 "(xs=ys) --> (!x. x : set ys --> f x = g x) --> map f xs = map g ys";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   264
by(rtac impI 1);
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   265
by(hyp_subst_tac 1);
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   266
by(induct_tac "ys" 1);
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   267
by(ALLGOALS Asm_simp_tac);
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   268
val lemma = result();
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   269
bind_thm("map_cong",impI RSN (2,allI RSN (2,lemma RS mp RS mp)));
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   270
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   271
goal List.thy "(map f xs = []) = (xs = [])";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   272
by(induct_tac "xs" 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   273
by(ALLGOALS Asm_simp_tac);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   274
qed "map_is_Nil_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   275
AddIffs [map_is_Nil_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   276
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   277
goal List.thy "([] = map f xs) = (xs = [])";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   278
by(induct_tac "xs" 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   279
by(ALLGOALS Asm_simp_tac);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   280
qed "Nil_is_map_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   281
AddIffs [Nil_is_map_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   282
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   283
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   284
(** rev **)
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   285
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   286
section "rev";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   287
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   288
goal thy "rev(xs@ys) = rev(ys) @ rev(xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   289
by (induct_tac "xs" 1);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   290
by (ALLGOALS Asm_simp_tac);
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   291
qed "rev_append";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   292
Addsimps[rev_append];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   293
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   294
goal thy "rev(rev l) = l";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   295
by (induct_tac "l" 1);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   296
by (ALLGOALS Asm_simp_tac);
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   297
qed "rev_rev_ident";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   298
Addsimps[rev_rev_ident];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   299
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   300
goal thy "(rev xs = []) = (xs = [])";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   301
by(induct_tac "xs" 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   302
by(ALLGOALS Asm_simp_tac);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   303
qed "rev_is_Nil_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   304
AddIffs [rev_is_Nil_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   305
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   306
goal thy "([] = rev xs) = (xs = [])";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   307
by(induct_tac "xs" 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   308
by(ALLGOALS Asm_simp_tac);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   309
qed "Nil_is_rev_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   310
AddIffs [Nil_is_rev_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   311
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   312
923
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parents:
diff changeset
   313
(** mem **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   314
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   315
section "mem";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   316
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   317
goal thy "x mem (xs@ys) = (x mem xs | x mem ys)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   318
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   319
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   320
qed "mem_append";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   321
Addsimps[mem_append];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   322
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   323
goal thy "x mem [x:xs. P(x)] = (x mem xs & P(x))";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   324
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   325
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   326
qed "mem_filter";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   327
Addsimps[mem_filter];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   328
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   329
(** set **)
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   330
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   331
section "set";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   332
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   333
goal thy "set (xs@ys) = (set xs Un set ys)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   334
by (induct_tac "xs" 1);
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   335
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   336
qed "set_append";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   337
Addsimps[set_append];
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   338
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   339
goal thy "(x mem xs) = (x: set xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   340
by (induct_tac "xs" 1);
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   341
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2739
diff changeset
   342
by (Blast_tac 1);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   343
qed "set_mem_eq";
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   344
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   345
goal thy "set l <= set (x#l)";
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
paulson
parents: 1908
diff changeset
   346
by (Simp_tac 1);
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2739
diff changeset
   347
by (Blast_tac 1);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   348
qed "set_subset_Cons";
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
paulson
parents: 1908
diff changeset
   349
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   350
goal thy "(set xs = {}) = (xs = [])";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   351
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   352
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   353
qed "set_empty";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   354
Addsimps [set_empty];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   355
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   356
goal thy "set(rev xs) = set(xs)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   357
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   358
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   359
qed "set_rev";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   360
Addsimps [set_rev];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   361
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   362
goal thy "set(map f xs) = f``(set xs)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   363
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   364
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   365
qed "set_map";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   366
Addsimps [set_map];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   367
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   368
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   369
(** list_all **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   370
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   371
section "list_all";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   372
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   373
goal thy "list_all (%x. True) xs = True";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   374
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   375
by (ALLGOALS Asm_simp_tac);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   376
qed "list_all_True";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   377
Addsimps [list_all_True];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   378
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   379
goal thy "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   380
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   381
by (ALLGOALS Asm_simp_tac);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   382
qed "list_all_append";
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   383
Addsimps [list_all_append];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   384
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   385
goal thy "list_all P xs = (!x. x mem xs --> P(x))";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   386
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   387
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2739
diff changeset
   388
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   389
qed "list_all_mem_conv";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   390
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   391
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   392
(** filter **)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   393
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   394
section "filter";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   395
3383
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   396
goal thy "filter P (xs@ys) = filter P xs @ filter P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   397
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   398
 by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   399
qed "filter_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   400
Addsimps [filter_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   401
3383
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   402
goal thy "size (filter P xs) <= size xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   403
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   404
 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
   405
qed "filter_size";
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   406
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   407
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   408
(** concat **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   409
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   410
section "concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   411
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   412
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
   413
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   414
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   415
qed"concat_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   416
Addsimps [concat_append];
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   417
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   418
goal thy  "set(concat xs) = Union(set `` set xs)";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   419
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   420
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   421
qed"set_concat";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   422
Addsimps [set_concat];
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   423
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   424
goal thy "map f (concat xs) = concat (map (map f) xs)"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   425
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   426
by (ALLGOALS Asm_simp_tac);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   427
qed "map_concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   428
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   429
goal thy "filter p (concat xs) = concat (map (filter p) xs)"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   430
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   431
by (ALLGOALS Asm_simp_tac);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   432
qed"filter_concat"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   433
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   434
goal thy "rev(concat xs) = concat (map rev (rev xs))";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   435
by (induct_tac "xs" 1);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   436
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   437
qed "rev_concat";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   438
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   439
(** nth **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   440
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   441
section "nth";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   442
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   443
goal thy
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   444
  "!xs. nth n (xs@ys) = \
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   445
\          (if n < length xs then nth n xs else nth (n - length xs) ys)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   446
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   447
 by (Asm_simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   448
 by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   449
 by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   450
  by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   451
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   452
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   453
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   454
qed_spec_mp "nth_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   455
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   456
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
   457
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   458
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   459
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   460
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   461
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   462
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   463
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
   464
qed_spec_mp "nth_map";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   465
Addsimps [nth_map];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   466
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   467
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
   468
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   469
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   470
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   471
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   472
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   473
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   474
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
   475
qed_spec_mp "list_all_nth";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   476
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   477
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
   478
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   479
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   480
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   481
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   482
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   483
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   484
(* case 0 *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   485
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   486
(* case Suc x *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   487
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
   488
qed_spec_mp "nth_mem";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   489
Addsimps [nth_mem];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   490
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   491
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   492
(** take  & drop **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   493
section "take & drop";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   494
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   495
goal thy "take 0 xs = []";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   496
by (induct_tac "xs" 1);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   497
by (ALLGOALS Asm_simp_tac);
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   498
qed "take_0";
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   499
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   500
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
   501
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   502
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   503
qed "drop_0";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   504
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   505
goal thy "take (Suc n) (x#xs) = x # take n xs";
1552
6f71b5d46700 Ran expandshort
paulson
parents: 1485
diff changeset
   506
by (Simp_tac 1);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   507
qed "take_Suc_Cons";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   508
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   509
goal thy "drop (Suc n) (x#xs) = drop n xs";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   510
by (Simp_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   511
qed "drop_Suc_Cons";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   512
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   513
Delsimps [take_Cons,drop_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   514
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   515
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   516
goal thy "!xs. length(take n xs) = min (length xs) n";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   517
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   518
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   519
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   520
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   521
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   522
qed_spec_mp "length_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   523
Addsimps [length_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   524
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   525
goal thy "!xs. length(drop n xs) = (length xs - n)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   526
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   527
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   528
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   529
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   530
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   531
qed_spec_mp "length_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   532
Addsimps [length_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. length xs <= n --> take n xs = 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_all";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   541
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   542
goal thy "!xs. length xs <= n --> 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_all";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   549
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   550
goal thy 
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   551
  "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   552
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   553
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   554
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   555
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   556
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   557
qed_spec_mp "take_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   558
Addsimps [take_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   559
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   560
goal thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   561
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   562
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   563
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   564
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   565
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   566
qed_spec_mp "drop_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   567
Addsimps [drop_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   568
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   569
goal thy "!xs n. take n (take m xs) = take (min n m) xs"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   570
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   571
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   572
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   573
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   574
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   575
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   576
by (exhaust_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   577
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   578
qed_spec_mp "take_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   579
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   580
goal thy "!xs. drop n (drop m xs) = drop (n + m) xs"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   581
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   582
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   583
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   584
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   585
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   586
qed_spec_mp "drop_drop";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   587
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   588
goal thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   589
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   590
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   591
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   592
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   593
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   594
qed_spec_mp "take_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   595
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   596
goal thy "!xs. take n (map f xs) = map f (take n xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   597
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   598
by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   599
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   600
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   601
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   602
qed_spec_mp "take_map"; 
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   603
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   604
goal thy "!xs. drop n (map f xs) = map f (drop n xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   605
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   606
by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   607
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   608
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   609
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   610
qed_spec_mp "drop_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   611
3283
0db086394024 Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents: 3196
diff changeset
   612
goal thy "!n i. i < n --> nth i (take n xs) = nth i xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   613
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   614
 by (ALLGOALS Asm_simp_tac);
3708
56facaebf3e3 Changed some proofs to use Clarify_tac
paulson
parents: 3647
diff changeset
   615
by (Clarify_tac 1);
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   616
by (exhaust_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   617
 by (Blast_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   618
by (exhaust_tac "i" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   619
by (ALLGOALS Asm_full_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   620
qed_spec_mp "nth_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   621
Addsimps [nth_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   622
3585
5b2dcdc1829c Generalized nth_drop (Conny).
nipkow
parents: 3574
diff changeset
   623
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
   624
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   625
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   626
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   627
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   628
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   629
qed_spec_mp "nth_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   630
Addsimps [nth_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   631
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   632
(** takeWhile & dropWhile **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   633
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   634
section "takeWhile & dropWhile";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   635
3586
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   636
goal thy "takeWhile P xs @ dropWhile P xs = xs";
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   637
by (induct_tac "xs" 1);
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   638
 by (Simp_tac 1);
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   639
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   640
qed "takeWhile_dropWhile_id";
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   641
Addsimps [takeWhile_dropWhile_id];
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   642
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   643
goal thy  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   644
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   645
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   646
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   647
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   648
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   649
Addsimps [takeWhile_append1];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   650
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   651
goal thy
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   652
  "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   653
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   654
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   655
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
   656
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   657
Addsimps [takeWhile_append2];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   658
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   659
goal thy
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   660
  "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   661
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   662
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   663
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   664
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   665
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   666
Addsimps [dropWhile_append1];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   667
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   668
goal thy
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   669
  "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   670
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   671
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   672
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
   673
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   674
Addsimps [dropWhile_append2];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   675
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   676
goal thy "x:set(takeWhile P xs) --> x:set xs & P x";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   677
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   678
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   679
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   680
qed_spec_mp"set_take_whileD";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   681
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   682
(** replicate **)
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   683
section "replicate";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   684
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   685
goal thy "set(replicate (Suc n) x) = {x}";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   686
by(induct_tac "n" 1);
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   687
by(ALLGOALS Asm_full_simp_tac);
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   688
val lemma = result();
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   689
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   690
goal thy "!!n. n ~= 0 ==> set(replicate n x) = {x}";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   691
by(fast_tac (!claset addSDs [not0_implies_Suc] addSIs [lemma]) 1);
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   692
qed "set_replicate";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   693
Addsimps [set_replicate];