src/HOL/List.ML
author nipkow
Thu, 16 Oct 1997 14:12:15 +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 List.thy "!!xs. xs ~= [] ==> length(tl xs) = pred(length xs)";
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by(exhaust_tac "xs" 1);
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by(ALLGOALS Asm_full_simp_tac);
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qed "length_tl";
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Addsimps [length_tl];
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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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   155
by (ALLGOALS Asm_simp_tac);
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nipkow
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diff changeset
   156
by (Blast_tac 1);
5995ab73d790 Added a few lemmas.
nipkow
parents: 3571
diff changeset
   157
qed "self_append_conv";
5995ab73d790 Added a few lemmas.
nipkow
parents: 3571
diff changeset
   158
AddIffs [append_self_conv,self_append_conv];
5995ab73d790 Added a few lemmas.
nipkow
parents: 3571
diff changeset
   159
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
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   160
goal thy "!ys. length xs = length ys | length us = length vs \
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
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diff changeset
   161
\              --> (xs@us = ys@vs) = (xs=ys & us=vs)";
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nipkow
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diff changeset
   162
by(induct_tac "xs" 1);
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nipkow
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diff changeset
   163
 by(rtac allI 1);
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nipkow
parents: 3842
diff changeset
   164
 by(exhaust_tac "ys" 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   165
  by(Asm_simp_tac 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   166
 by(fast_tac (!claset addIs [less_add_Suc2] addss !simpset
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   167
                      addEs [less_not_refl2 RSN (2,rev_notE)]) 1);
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nipkow
parents: 3842
diff changeset
   168
by(rtac allI 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   169
by(exhaust_tac "ys" 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   170
 by(fast_tac (!claset addIs [less_add_Suc2] addss !simpset
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   171
                      addEs [(less_not_refl2 RS not_sym) RSN (2,rev_notE)]) 1);
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nipkow
parents: 3842
diff changeset
   172
by(Asm_simp_tac 1);
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nipkow
parents: 3842
diff changeset
   173
qed_spec_mp "append_eq_append_conv";
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nipkow
parents: 3842
diff changeset
   174
Addsimps [append_eq_append_conv];
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nipkow
parents: 3842
diff changeset
   175
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
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diff changeset
   176
goal thy "(xs @ ys = xs @ zs) = (ys=zs)";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
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   177
by (Simp_tac 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
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diff changeset
   178
qed "same_append_eq";
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   179
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
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diff changeset
   180
goal thy "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; 
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
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   181
by (Simp_tac 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
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   182
qed "append1_eq_conv";
2608
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diff changeset
   183
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
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diff changeset
   184
goal thy "(ys @ xs = zs @ xs) = (ys=zs)";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
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diff changeset
   185
by (Simp_tac 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   186
qed "append_same_eq";
2608
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parents: 2512
diff changeset
   187
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
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diff changeset
   188
AddSIs
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   189
 [same_append_eq RS iffD2, append1_eq_conv RS iffD2, append_same_eq RS iffD2];
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nipkow
parents: 3860
diff changeset
   190
AddSDs
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   191
 [same_append_eq RS iffD1, append1_eq_conv RS iffD1, append_same_eq RS iffD1];
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
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diff changeset
   192
3011
a3b73ba44a11 Tidied up.
nipkow
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diff changeset
   193
goal thy "xs ~= [] --> hd xs # tl xs = xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   194
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   195
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
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diff changeset
   196
qed_spec_mp "hd_Cons_tl";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   197
Addsimps [hd_Cons_tl];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   198
3011
a3b73ba44a11 Tidied up.
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diff changeset
   199
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
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diff changeset
   200
by (induct_tac "xs" 1);
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   201
by (ALLGOALS Asm_simp_tac);
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   202
qed "hd_append";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   203
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
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diff changeset
   204
goal thy "!!xs. xs ~= [] ==> hd(xs @ ys) = hd xs";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   205
by (asm_simp_tac (!simpset addsimps [hd_append]
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   206
                           setloop (split_tac [expand_list_case])) 1);
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   207
qed "hd_append2";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   208
Addsimps [hd_append2];
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   209
3011
a3b73ba44a11 Tidied up.
nipkow
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diff changeset
   210
goal thy "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   211
by (simp_tac (!simpset setloop(split_tac[expand_list_case])) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
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diff changeset
   212
qed "tl_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   213
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   214
goal thy "!!xs. xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   215
by (asm_simp_tac (!simpset addsimps [tl_append]
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   216
                           setloop (split_tac [expand_list_case])) 1);
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   217
qed "tl_append2";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   218
Addsimps [tl_append2];
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   219
2608
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   220
(** map **)
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parents: 2512
diff changeset
   221
3467
a0797ba03dfe More concat lemmas.
nipkow
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diff changeset
   222
section "map";
a0797ba03dfe More concat lemmas.
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diff changeset
   223
3011
a3b73ba44a11 Tidied up.
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parents: 2891
diff changeset
   224
goal thy
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   225
  "(!x. x : set xs --> f x = g x) --> map f xs = map g xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   226
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   227
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
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diff changeset
   228
bind_thm("map_ext", impI RS (allI RS (result() RS mp)));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   229
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   230
goal thy "map (%x. x) = (%xs. xs)";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   231
by (rtac ext 1);
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   232
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
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diff changeset
   233
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   234
qed "map_ident";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   235
Addsimps[map_ident];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   236
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   237
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
   238
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   239
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   240
qed "map_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   241
Addsimps[map_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   242
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   243
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
   244
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   245
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   246
qed "map_compose";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   247
Addsimps[map_compose];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   248
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   249
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
   250
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   251
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   252
qed "rev_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   253
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   254
(* a congruence rule for map: *)
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   255
goal thy
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   256
 "(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
   257
by(rtac impI 1);
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   258
by(hyp_subst_tac 1);
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   259
by(induct_tac "ys" 1);
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   260
by(ALLGOALS Asm_simp_tac);
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   261
val lemma = result();
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   262
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
   263
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   264
goal List.thy "(map f xs = []) = (xs = [])";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   265
by(induct_tac "xs" 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   266
by(ALLGOALS Asm_simp_tac);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   267
qed "map_is_Nil_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   268
AddIffs [map_is_Nil_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   269
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   270
goal List.thy "([] = map f xs) = (xs = [])";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   271
by(induct_tac "xs" 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   272
by(ALLGOALS Asm_simp_tac);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   273
qed "Nil_is_map_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   274
AddIffs [Nil_is_map_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   275
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   276
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   277
(** rev **)
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   278
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   279
section "rev";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   280
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   281
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
   282
by (induct_tac "xs" 1);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   283
by (ALLGOALS Asm_simp_tac);
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   284
qed "rev_append";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   285
Addsimps[rev_append];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   286
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   287
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
   288
by (induct_tac "l" 1);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   289
by (ALLGOALS Asm_simp_tac);
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   290
qed "rev_rev_ident";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   291
Addsimps[rev_rev_ident];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   292
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   293
goal thy "(rev xs = []) = (xs = [])";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   294
by(induct_tac "xs" 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   295
by(ALLGOALS Asm_simp_tac);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   296
qed "rev_is_Nil_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   297
AddIffs [rev_is_Nil_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   298
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   299
goal thy "([] = rev xs) = (xs = [])";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   300
by(induct_tac "xs" 1);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   301
by(ALLGOALS Asm_simp_tac);
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   302
qed "Nil_is_rev_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   303
AddIffs [Nil_is_rev_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   304
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   305
923
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parents:
diff changeset
   306
(** mem **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   307
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   308
section "mem";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   309
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   310
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
   311
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   312
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
923
ff1574a81019 new version of HOL with curried function application
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parents:
diff changeset
   313
qed "mem_append";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   314
Addsimps[mem_append];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   315
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   316
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
   317
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   318
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
   319
qed "mem_filter";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   320
Addsimps[mem_filter];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   321
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   322
(** set **)
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   323
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   324
section "set";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   325
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   326
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
   327
by (induct_tac "xs" 1);
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   328
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   329
qed "set_append";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   330
Addsimps[set_append];
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   331
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   332
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
   333
by (induct_tac "xs" 1);
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   334
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2739
diff changeset
   335
by (Blast_tac 1);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   336
qed "set_mem_eq";
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   337
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   338
goal thy "set l <= set (x#l)";
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
paulson
parents: 1908
diff changeset
   339
by (Simp_tac 1);
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2739
diff changeset
   340
by (Blast_tac 1);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   341
qed "set_subset_Cons";
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
paulson
parents: 1908
diff changeset
   342
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   343
goal thy "(set xs = {}) = (xs = [])";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   344
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   345
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   346
qed "set_empty";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   347
Addsimps [set_empty];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   348
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   349
goal thy "set(rev xs) = set(xs)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   350
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   351
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   352
qed "set_rev";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   353
Addsimps [set_rev];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   354
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   355
goal thy "set(map f xs) = f``(set xs)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   356
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   357
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   358
qed "set_map";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   359
Addsimps [set_map];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   360
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   361
923
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clasohm
parents:
diff changeset
   362
(** list_all **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   363
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   364
section "list_all";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   365
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   366
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
   367
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   368
by (ALLGOALS Asm_simp_tac);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   369
qed "list_all_True";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   370
Addsimps [list_all_True];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   371
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   372
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
   373
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   374
by (ALLGOALS Asm_simp_tac);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   375
qed "list_all_append";
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   376
Addsimps [list_all_append];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   377
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   378
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
   379
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   380
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2739
diff changeset
   381
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   382
qed "list_all_mem_conv";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   383
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   384
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   385
(** filter **)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   386
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   387
section "filter";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   388
3383
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   389
goal thy "filter P (xs@ys) = filter P xs @ filter P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   390
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   391
 by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   392
qed "filter_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   393
Addsimps [filter_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   394
3383
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   395
goal thy "size (filter P xs) <= size xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   396
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   397
 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
   398
qed "filter_size";
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   399
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   400
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   401
(** concat **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   402
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   403
section "concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   404
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   405
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
   406
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   407
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   408
qed"concat_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   409
Addsimps [concat_append];
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   410
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   411
goal thy "(concat xss = []) = (!xs:set xss. xs=[])";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   412
by(induct_tac "xss" 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   413
by(ALLGOALS Asm_simp_tac);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   414
qed "concat_eq_Nil_conv";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   415
AddIffs [concat_eq_Nil_conv];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   416
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   417
goal thy "([] = concat xss) = (!xs:set xss. xs=[])";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   418
by(induct_tac "xss" 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   419
by(ALLGOALS Asm_simp_tac);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   420
qed "Nil_eq_concat_conv";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   421
AddIffs [Nil_eq_concat_conv];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   422
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   423
goal thy  "set(concat xs) = Union(set `` set xs)";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   424
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   425
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   426
qed"set_concat";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   427
Addsimps [set_concat];
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   428
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   429
goal thy "map f (concat xs) = concat (map (map f) 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 "map_concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   433
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   434
goal thy "filter p (concat xs) = concat (map (filter p) xs)"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   435
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   436
by (ALLGOALS Asm_simp_tac);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   437
qed"filter_concat"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   438
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   439
goal thy "rev(concat xs) = concat (map rev (rev xs))";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   440
by (induct_tac "xs" 1);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   441
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   442
qed "rev_concat";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   443
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   444
(** nth **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   445
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   446
section "nth";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   447
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   448
goal thy
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   449
  "!xs. nth n (xs@ys) = \
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   450
\          (if n < length xs then nth n xs else nth (n - length xs) ys)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   451
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   452
 by (Asm_simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   453
 by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   454
 by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   455
  by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   456
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   457
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   458
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   459
qed_spec_mp "nth_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   460
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   461
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
   462
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   463
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   464
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   465
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   466
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   467
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   468
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
   469
qed_spec_mp "nth_map";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   470
Addsimps [nth_map];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   471
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   472
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
   473
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   474
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   475
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   476
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   477
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   478
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   479
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
   480
qed_spec_mp "list_all_nth";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   481
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   482
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
   483
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   484
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   485
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   486
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   487
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   488
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   489
(* case 0 *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   490
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   491
(* case Suc x *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   492
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
   493
qed_spec_mp "nth_mem";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   494
Addsimps [nth_mem];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   495
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   496
(** last & butlast **)
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   497
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   498
goal thy "last(xs@[x]) = x";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   499
by(induct_tac "xs" 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   500
by(ALLGOALS (asm_simp_tac (!simpset setloop (split_tac[expand_if]))));
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   501
qed "last_snoc";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   502
Addsimps [last_snoc];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   503
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   504
goal thy "butlast(xs@[x]) = xs";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   505
by(induct_tac "xs" 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   506
by(ALLGOALS (asm_simp_tac (!simpset setloop (split_tac[expand_if]))));
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   507
qed "butlast_snoc";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   508
Addsimps [butlast_snoc];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   509
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   510
goal thy
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   511
  "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   512
by(induct_tac "xs" 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   513
by(ALLGOALS(asm_simp_tac (!simpset setloop (split_tac[expand_if]))));
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   514
qed_spec_mp "butlast_append";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   515
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   516
goal thy "x:set(butlast xs) --> x:set xs";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   517
by(induct_tac "xs" 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   518
by(ALLGOALS (asm_simp_tac (!simpset setloop (split_tac[expand_if]))));
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   519
qed_spec_mp "in_set_butlastD";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   520
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   521
goal thy "!!xs. x:set(butlast xs) ==> x:set(butlast(xs@ys))";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   522
by(asm_simp_tac (!simpset addsimps [butlast_append]
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   523
                          setloop (split_tac[expand_if])) 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   524
by(blast_tac (!claset addDs [in_set_butlastD]) 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   525
qed "in_set_butlast_appendI1";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   526
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   527
goal thy "!!xs. x:set(butlast ys) ==> x:set(butlast(xs@ys))";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   528
by(asm_simp_tac (!simpset addsimps [butlast_append]
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   529
                          setloop (split_tac[expand_if])) 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   530
by(Clarify_tac 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   531
by(Full_simp_tac 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   532
qed "in_set_butlast_appendI2";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   533
(* FIXME
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   534
Addsimps [in_set_butlast_appendI1,in_set_butlast_appendI2];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   535
AddIs    [in_set_butlast_appendI1,in_set_butlast_appendI2];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   536
*)
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   537
(** take  & drop **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   538
section "take & drop";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   539
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   540
goal thy "take 0 xs = []";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   541
by (induct_tac "xs" 1);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   542
by (ALLGOALS Asm_simp_tac);
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   543
qed "take_0";
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   544
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   545
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
   546
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   547
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   548
qed "drop_0";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   549
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   550
goal thy "take (Suc n) (x#xs) = x # take n xs";
1552
6f71b5d46700 Ran expandshort
paulson
parents: 1485
diff changeset
   551
by (Simp_tac 1);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   552
qed "take_Suc_Cons";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   553
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   554
goal thy "drop (Suc n) (x#xs) = drop n xs";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   555
by (Simp_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   556
qed "drop_Suc_Cons";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   557
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   558
Delsimps [take_Cons,drop_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   559
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   560
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   561
goal thy "!xs. length(take n xs) = min (length xs) n";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   562
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   563
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   564
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   565
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   566
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   567
qed_spec_mp "length_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   568
Addsimps [length_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   569
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   570
goal thy "!xs. length(drop n xs) = (length xs - n)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   571
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   572
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   573
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   574
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   575
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   576
qed_spec_mp "length_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   577
Addsimps [length_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   578
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   579
goal thy "!xs. length xs <= n --> take n xs = xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   580
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   581
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   582
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   583
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   584
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   585
qed_spec_mp "take_all";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   586
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   587
goal thy "!xs. length xs <= n --> drop n xs = []";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   588
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   589
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   590
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   591
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   592
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   593
qed_spec_mp "drop_all";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   594
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   595
goal thy 
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   596
  "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
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_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   603
Addsimps [take_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   604
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   605
goal thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   606
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   607
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   608
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   609
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   610
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   611
qed_spec_mp "drop_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   612
Addsimps [drop_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   613
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   614
goal thy "!xs n. take n (take m xs) = take (min n m) xs"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   615
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   616
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   617
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   618
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   619
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   620
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   621
by (exhaust_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   622
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   623
qed_spec_mp "take_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   624
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   625
goal thy "!xs. drop n (drop m xs) = drop (n + m) xs"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   626
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   627
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   628
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   629
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   630
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   631
qed_spec_mp "drop_drop";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   632
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   633
goal thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   634
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   635
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   636
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   637
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   638
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   639
qed_spec_mp "take_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   640
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   641
goal thy "!xs. take n (map f xs) = map f (take n xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   642
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   643
by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   644
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   645
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   646
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   647
qed_spec_mp "take_map"; 
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   648
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   649
goal thy "!xs. drop n (map f xs) = map f (drop n xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   650
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   651
by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   652
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   653
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   654
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   655
qed_spec_mp "drop_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   656
3283
0db086394024 Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents: 3196
diff changeset
   657
goal thy "!n i. i < n --> nth i (take n xs) = nth i xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   658
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   659
 by (ALLGOALS Asm_simp_tac);
3708
56facaebf3e3 Changed some proofs to use Clarify_tac
paulson
parents: 3647
diff changeset
   660
by (Clarify_tac 1);
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   661
by (exhaust_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   662
 by (Blast_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   663
by (exhaust_tac "i" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   664
by (ALLGOALS Asm_full_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   665
qed_spec_mp "nth_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   666
Addsimps [nth_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   667
3585
5b2dcdc1829c Generalized nth_drop (Conny).
nipkow
parents: 3574
diff changeset
   668
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
   669
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   670
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   671
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   672
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   673
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   674
qed_spec_mp "nth_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   675
Addsimps [nth_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   676
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   677
(** takeWhile & dropWhile **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   678
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   679
section "takeWhile & dropWhile";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   680
3586
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   681
goal thy "takeWhile P xs @ dropWhile P xs = xs";
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   682
by (induct_tac "xs" 1);
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   683
 by (Simp_tac 1);
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   684
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   685
qed "takeWhile_dropWhile_id";
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   686
Addsimps [takeWhile_dropWhile_id];
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   687
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   688
goal thy  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   689
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   690
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   691
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   692
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   693
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   694
Addsimps [takeWhile_append1];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   695
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   696
goal thy
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   697
  "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   698
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   699
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   700
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
   701
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   702
Addsimps [takeWhile_append2];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   703
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   704
goal thy
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   705
  "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   706
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   707
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   708
by (asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   709
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   710
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   711
Addsimps [dropWhile_append1];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   712
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   713
goal thy
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   714
  "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   715
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   716
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   717
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
   718
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   719
Addsimps [dropWhile_append2];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   720
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   721
goal thy "x:set(takeWhile P xs) --> x:set xs & P x";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   722
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   723
 by (Simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   724
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
   725
qed_spec_mp"set_take_whileD";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   726
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   727
(** replicate **)
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   728
section "replicate";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   729
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   730
goal thy "set(replicate (Suc n) x) = {x}";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   731
by(induct_tac "n" 1);
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   732
by(ALLGOALS Asm_full_simp_tac);
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   733
val lemma = result();
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   734
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   735
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
   736
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
   737
qed "set_replicate";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   738
Addsimps [set_replicate];