src/HOL/List.ML
author nipkow
Tue, 12 May 1998 08:36:07 +0200
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(*  Title:      HOL/List
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1994 TU Muenchen
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List lemmas
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*)
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goal thy "!x. xs ~= x#xs";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed_spec_mp "not_Cons_self";
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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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(* Induction over the length of a list: *)
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val [prem] = goal thy
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  "(!!xs. (!ys. length ys < length xs --> P ys) ==> P xs) ==> P(xs)";
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by(rtac measure_induct 1 THEN etac prem 1);
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qed "length_induct";
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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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by (rtac (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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(**  Case analysis **)
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section "Case analysis";
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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) = (length xs) - 1";
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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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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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   156
goal thy "!ys. length xs = length ys | length us = length vs \
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
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   157
\              --> (xs@us = ys@vs) = (xs=ys & us=vs)";
4423
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wenzelm
parents: 4132
diff changeset
   158
by (induct_tac "xs" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   159
 by (rtac allI 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   160
 by (exhaust_tac "ys" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   161
  by (Asm_simp_tac 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   162
 by (fast_tac (claset() addIs [less_add_Suc2] addss simpset()
3860
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nipkow
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diff changeset
   163
                      addEs [less_not_refl2 RSN (2,rev_notE)]) 1);
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   164
by (rtac allI 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   165
by (exhaust_tac "ys" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   166
 by (fast_tac (claset() addIs [less_add_Suc2] addss simpset()
3860
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nipkow
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diff changeset
   167
                      addEs [(less_not_refl2 RS not_sym) RSN (2,rev_notE)]) 1);
4423
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wenzelm
parents: 4132
diff changeset
   168
by (Asm_simp_tac 1);
3860
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nipkow
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   169
qed_spec_mp "append_eq_append_conv";
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   170
Addsimps [append_eq_append_conv];
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diff changeset
   171
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   172
goal thy "(xs @ ys = xs @ zs) = (ys=zs)";
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   173
by (Simp_tac 1);
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diff changeset
   174
qed "same_append_eq";
3860
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diff changeset
   175
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   176
goal thy "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; 
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   177
by (Simp_tac 1);
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   178
qed "append1_eq_conv";
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diff changeset
   179
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   180
goal thy "(ys @ xs = zs @ xs) = (ys=zs)";
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   181
by (Simp_tac 1);
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   182
qed "append_same_eq";
2608
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diff changeset
   183
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   184
AddSIs
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   185
 [same_append_eq RS iffD2, append1_eq_conv RS iffD2, append_same_eq RS iffD2];
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diff changeset
   186
AddSDs
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   187
 [same_append_eq RS iffD1, append1_eq_conv RS iffD1, append_same_eq RS iffD1];
3571
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diff changeset
   188
4647
42af8ae6e2c1 Added some lemmas.
nipkow
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   189
goal thy "(xs @ ys = ys) = (xs=[])";
42af8ae6e2c1 Added some lemmas.
nipkow
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diff changeset
   190
by(cut_inst_tac [("zs","[]")] append_same_eq 1);
42af8ae6e2c1 Added some lemmas.
nipkow
parents: 4643
diff changeset
   191
by(Asm_full_simp_tac 1);
42af8ae6e2c1 Added some lemmas.
nipkow
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diff changeset
   192
qed "append_self_conv2";
42af8ae6e2c1 Added some lemmas.
nipkow
parents: 4643
diff changeset
   193
42af8ae6e2c1 Added some lemmas.
nipkow
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diff changeset
   194
goal thy "(ys = xs @ ys) = (xs=[])";
42af8ae6e2c1 Added some lemmas.
nipkow
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diff changeset
   195
by(simp_tac (simpset() addsimps
42af8ae6e2c1 Added some lemmas.
nipkow
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diff changeset
   196
     [simplify (simpset()) (read_instantiate[("ys","[]")]append_same_eq)]) 1);
42af8ae6e2c1 Added some lemmas.
nipkow
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diff changeset
   197
by(Blast_tac 1);
42af8ae6e2c1 Added some lemmas.
nipkow
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diff changeset
   198
qed "self_append_conv2";
42af8ae6e2c1 Added some lemmas.
nipkow
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diff changeset
   199
AddIffs [append_self_conv2,self_append_conv2];
42af8ae6e2c1 Added some lemmas.
nipkow
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diff changeset
   200
3011
a3b73ba44a11 Tidied up.
nipkow
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   201
goal thy "xs ~= [] --> hd xs # tl xs = xs";
3457
a8ab7c64817c Ran expandshort
paulson
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diff changeset
   202
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   203
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
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   204
qed_spec_mp "hd_Cons_tl";
450c9b682a92 New class "order" and accompanying changes.
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   205
Addsimps [hd_Cons_tl];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   206
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   207
goal thy "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
3040
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nipkow
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diff changeset
   208
by (induct_tac "xs" 1);
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   209
by (ALLGOALS Asm_simp_tac);
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
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diff changeset
   210
qed "hd_append";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   211
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   212
goal thy "!!xs. xs ~= [] ==> hd(xs @ ys) = hd xs";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   213
by (asm_simp_tac (simpset() addsimps [hd_append]
4069
d6d06a03a2e9 expand_list_case -> split_list_case
nipkow
parents: 4032
diff changeset
   214
                           addsplits [split_list_case]) 1);
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
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diff changeset
   215
qed "hd_append2";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   216
Addsimps [hd_append2];
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nipkow
parents: 3468
diff changeset
   217
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diff changeset
   218
goal thy "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
4089
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wenzelm
parents: 4069
diff changeset
   219
by (simp_tac (simpset() addsplits [split_list_case]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
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diff changeset
   220
qed "tl_append";
450c9b682a92 New class "order" and accompanying changes.
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diff changeset
   221
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nipkow
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diff changeset
   222
goal thy "!!xs. xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   223
by (asm_simp_tac (simpset() addsimps [tl_append]
4069
d6d06a03a2e9 expand_list_case -> split_list_case
nipkow
parents: 4032
diff changeset
   224
                           addsplits [split_list_case]) 1);
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
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diff changeset
   225
qed "tl_append2";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
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diff changeset
   226
Addsimps [tl_append2];
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nipkow
parents: 3468
diff changeset
   227
4830
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nipkow
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diff changeset
   228
bd73675adbed Added a few lemmas.
nipkow
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   229
(** Snoc exhaustion and induction **)
bd73675adbed Added a few lemmas.
nipkow
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   230
section "Snoc exhaustion and induction";
bd73675adbed Added a few lemmas.
nipkow
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diff changeset
   231
bd73675adbed Added a few lemmas.
nipkow
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   232
goal thy "xs ~= [] --> (? ys y. xs = ys@[y])";
bd73675adbed Added a few lemmas.
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   233
by(induct_tac "xs" 1);
bd73675adbed Added a few lemmas.
nipkow
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diff changeset
   234
by(Simp_tac 1);
bd73675adbed Added a few lemmas.
nipkow
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diff changeset
   235
by(exhaust_tac "list" 1);
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   236
 by(Asm_simp_tac 1);
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   237
 by(res_inst_tac [("x","[]")] exI 1);
bd73675adbed Added a few lemmas.
nipkow
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diff changeset
   238
 by(Simp_tac 1);
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   239
by(Asm_full_simp_tac 1);
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   240
by(Clarify_tac 1);
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   241
by(res_inst_tac [("x","a#ys")] exI 1);
bd73675adbed Added a few lemmas.
nipkow
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diff changeset
   242
by(Asm_simp_tac 1);
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   243
val lemma = result();
bd73675adbed Added a few lemmas.
nipkow
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diff changeset
   244
bd73675adbed Added a few lemmas.
nipkow
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diff changeset
   245
goal thy  "(xs = [] --> P) -->  (!ys y. xs = ys@[y] --> P) --> P";
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   246
by(cut_facts_tac [lemma] 1);
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   247
by(Blast_tac 1);
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   248
bind_thm ("snoc_exhaust",
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   249
  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp)))));
bd73675adbed Added a few lemmas.
nipkow
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diff changeset
   250
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   251
val prems = goal thy "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs";
4911
6195e4468c54 Removed duplicate list_length_induct
nipkow
parents: 4830
diff changeset
   252
by(res_inst_tac [("xs","xs")] length_induct 1);
4830
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   253
by(res_inst_tac [("xs","xs")] snoc_exhaust 1);
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   254
 by(Clarify_tac 1);
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   255
 brs prems 1;
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   256
by(Clarify_tac 1);
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   257
brs prems 1;
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   258
auto();
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   259
qed "snoc_induct";
bd73675adbed Added a few lemmas.
nipkow
parents: 4686
diff changeset
   260
bd73675adbed Added a few lemmas.
nipkow
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diff changeset
   261
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
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diff changeset
   262
(** map **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   263
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   264
section "map";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   265
3011
a3b73ba44a11 Tidied up.
nipkow
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diff changeset
   266
goal thy
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   267
  "(!x. x : set xs --> f x = g x) --> map f xs = map g xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   268
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   269
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   270
bind_thm("map_ext", impI RS (allI RS (result() RS mp)));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   271
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   272
goal thy "map (%x. x) = (%xs. xs)";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   273
by (rtac ext 1);
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   274
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   275
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
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diff changeset
   276
qed "map_ident";
450c9b682a92 New class "order" and accompanying changes.
nipkow
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diff changeset
   277
Addsimps[map_ident];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   278
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   279
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
   280
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   281
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
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diff changeset
   282
qed "map_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   283
Addsimps[map_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   284
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   285
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
   286
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   287
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   288
qed "map_compose";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   289
Addsimps[map_compose];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   290
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   291
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
   292
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   293
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   294
qed "rev_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   295
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   296
(* a congruence rule for map: *)
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   297
goal thy
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   298
 "(xs=ys) --> (!x. x : set ys --> f x = g x) --> map f xs = map g ys";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   299
by (rtac impI 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   300
by (hyp_subst_tac 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   301
by (induct_tac "ys" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   302
by (ALLGOALS Asm_simp_tac);
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   303
val lemma = result();
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   304
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
   305
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   306
goal List.thy "(map f xs = []) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   307
by (induct_tac "xs" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   308
by (ALLGOALS Asm_simp_tac);
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   309
qed "map_is_Nil_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   310
AddIffs [map_is_Nil_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   311
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   312
goal List.thy "([] = map f xs) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   313
by (induct_tac "xs" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   314
by (ALLGOALS Asm_simp_tac);
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   315
qed "Nil_is_map_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   316
AddIffs [Nil_is_map_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   317
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   318
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   319
(** rev **)
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   320
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   321
section "rev";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   322
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   323
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
   324
by (induct_tac "xs" 1);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   325
by (ALLGOALS Asm_simp_tac);
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   326
qed "rev_append";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   327
Addsimps[rev_append];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   328
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   329
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
   330
by (induct_tac "l" 1);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   331
by (ALLGOALS Asm_simp_tac);
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   332
qed "rev_rev_ident";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   333
Addsimps[rev_rev_ident];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   334
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   335
goal thy "(rev xs = []) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   336
by (induct_tac "xs" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   337
by (ALLGOALS Asm_simp_tac);
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   338
qed "rev_is_Nil_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   339
AddIffs [rev_is_Nil_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   340
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   341
goal thy "([] = rev xs) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   342
by (induct_tac "xs" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   343
by (ALLGOALS Asm_simp_tac);
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   344
qed "Nil_is_rev_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   345
AddIffs [Nil_is_rev_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   346
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   347
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   348
(** mem **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   349
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   350
section "mem";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   351
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   352
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
   353
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   354
by (ALLGOALS Asm_simp_tac);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   355
qed "mem_append";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   356
Addsimps[mem_append];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   357
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   358
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
   359
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   360
by (ALLGOALS Asm_simp_tac);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   361
qed "mem_filter";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   362
Addsimps[mem_filter];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   363
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   364
(** set **)
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   365
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   366
section "set";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   367
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   368
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
   369
by (induct_tac "xs" 1);
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   370
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   371
qed "set_append";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   372
Addsimps[set_append];
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   373
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   374
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
   375
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   376
by (ALLGOALS Asm_simp_tac);
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2739
diff changeset
   377
by (Blast_tac 1);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   378
qed "set_mem_eq";
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   379
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   380
goal thy "set l <= set (x#l)";
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
paulson
parents: 1908
diff changeset
   381
by (Simp_tac 1);
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2739
diff changeset
   382
by (Blast_tac 1);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   383
qed "set_subset_Cons";
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
paulson
parents: 1908
diff changeset
   384
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   385
goal thy "(set xs = {}) = (xs = [])";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   386
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   387
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   388
qed "set_empty";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   389
Addsimps [set_empty];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   390
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   391
goal thy "set(rev xs) = set(xs)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   392
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   393
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   394
qed "set_rev";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   395
Addsimps [set_rev];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   396
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   397
goal thy "set(map f xs) = f``(set xs)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   398
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   399
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   400
qed "set_map";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   401
Addsimps [set_map];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   402
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   403
goal thy "set(map f xs) = f``(set xs)";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   404
by (induct_tac "xs" 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   405
by (ALLGOALS Asm_simp_tac);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   406
qed "set_map";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   407
Addsimps [set_map];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   408
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   409
goal thy "(x : set(filter P xs)) = (x : set xs & P x)";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   410
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   411
by (ALLGOALS Asm_simp_tac);
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   412
by(Blast_tac 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   413
qed "in_set_filter";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   414
Addsimps [in_set_filter];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   415
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   416
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   417
(** list_all **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   418
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   419
section "list_all";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   420
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   421
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
   422
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   423
by (ALLGOALS Asm_simp_tac);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   424
qed "list_all_True";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   425
Addsimps [list_all_True];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   426
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   427
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
   428
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   429
by (ALLGOALS Asm_simp_tac);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   430
qed "list_all_append";
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   431
Addsimps [list_all_append];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   432
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   433
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
   434
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   435
by (ALLGOALS Asm_simp_tac);
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2739
diff changeset
   436
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   437
qed "list_all_mem_conv";
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
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   440
(** filter **)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   441
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   442
section "filter";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   443
3383
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   444
goal thy "filter P (xs@ys) = filter P xs @ filter P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   445
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   446
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   447
qed "filter_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   448
Addsimps [filter_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   449
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   450
goal thy "filter (%x. True) xs = xs";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   451
by (induct_tac "xs" 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   452
by (ALLGOALS Asm_simp_tac);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   453
qed "filter_True";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   454
Addsimps [filter_True];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   455
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   456
goal thy "filter (%x. False) xs = []";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   457
by (induct_tac "xs" 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   458
by (ALLGOALS Asm_simp_tac);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   459
qed "filter_False";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   460
Addsimps [filter_False];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   461
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   462
goal thy "length (filter P xs) <= length xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   463
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   464
by (ALLGOALS Asm_simp_tac);
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   465
qed "length_filter";
3383
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   466
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   467
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   468
(** concat **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   469
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   470
section "concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   471
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   472
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
   473
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   474
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   475
qed"concat_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   476
Addsimps [concat_append];
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   477
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   478
goal thy "(concat xss = []) = (!xs:set xss. xs=[])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   479
by (induct_tac "xss" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   480
by (ALLGOALS Asm_simp_tac);
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   481
qed "concat_eq_Nil_conv";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   482
AddIffs [concat_eq_Nil_conv];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   483
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   484
goal thy "([] = concat xss) = (!xs:set xss. xs=[])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   485
by (induct_tac "xss" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   486
by (ALLGOALS Asm_simp_tac);
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   487
qed "Nil_eq_concat_conv";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   488
AddIffs [Nil_eq_concat_conv];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   489
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   490
goal thy  "set(concat xs) = Union(set `` set xs)";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   491
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   492
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   493
qed"set_concat";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   494
Addsimps [set_concat];
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   495
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   496
goal thy "map f (concat xs) = concat (map (map f) xs)"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   497
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   498
by (ALLGOALS Asm_simp_tac);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   499
qed "map_concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   500
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   501
goal thy "filter p (concat xs) = concat (map (filter p) xs)"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   502
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   503
by (ALLGOALS Asm_simp_tac);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   504
qed"filter_concat"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   505
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   506
goal thy "rev(concat xs) = concat (map rev (rev xs))";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   507
by (induct_tac "xs" 1);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   508
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   509
qed "rev_concat";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   510
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   511
(** nth **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   512
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   513
section "nth";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   514
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   515
goal thy
4502
337c073de95e nth -> !
nipkow
parents: 4423
diff changeset
   516
  "!xs. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))";
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 (Asm_simp_tac 1);
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 "nth_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   523
4502
337c073de95e nth -> !
nipkow
parents: 4423
diff changeset
   524
goal thy "!n. n < length xs --> (map f xs)!n = f(xs!n)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   525
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   526
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   527
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   528
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   529
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   530
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   531
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
   532
qed_spec_mp "nth_map";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   533
Addsimps [nth_map];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   534
4502
337c073de95e nth -> !
nipkow
parents: 4423
diff changeset
   535
goal thy "!n. n < length xs --> list_all P xs --> P(xs!n)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   536
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   537
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   538
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   539
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   540
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   541
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   542
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
   543
qed_spec_mp "list_all_nth";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   544
4502
337c073de95e nth -> !
nipkow
parents: 4423
diff changeset
   545
goal thy "!n. n < length xs --> xs!n mem 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);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   547
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   548
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   549
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   550
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   551
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   552
(* case 0 *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   553
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   554
(* case Suc x *)
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   555
by (Asm_full_simp_tac 1);
1485
240cc98b94a7 Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents: 1465
diff changeset
   556
qed_spec_mp "nth_mem";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   557
Addsimps [nth_mem];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   558
4643
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   559
(**  More case analysis and induction **)
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   560
section "More case analysis and induction";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   561
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   562
goal thy "xs ~= [] --> (? ys y. xs = ys@[y])";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   563
by(res_inst_tac [("xs","xs")] length_induct 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   564
by(Clarify_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   565
bd (neq_Nil_conv RS iffD1) 1;
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   566
by(Clarify_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   567
by(rename_tac "ys" 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   568
by(case_tac "ys = []" 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   569
 by(res_inst_tac [("x","[]")] exI 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   570
 by(Asm_full_simp_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   571
by(eres_inst_tac [("x","ys")] allE 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   572
by(Asm_full_simp_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   573
by(REPEAT(etac exE 1));
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   574
by(rename_tac "zs z" 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   575
by(hyp_subst_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   576
by(res_inst_tac [("x","y#zs")] exI 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   577
by(Simp_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   578
qed_spec_mp "neq_Nil_snocD";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   579
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   580
val prems = goal thy
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   581
  "[| xs=[] ==> P []; !!ys y. xs=ys@[y] ==> P(ys@[y]) |] ==> P xs";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   582
by(case_tac "xs = []" 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   583
 by(Asm_simp_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   584
 bes prems 1;
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   585
bd neq_Nil_snocD 1;
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   586
by(REPEAT(etac exE 1));
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   587
by(Asm_simp_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   588
bes prems 1;
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   589
qed "snoc_eq_cases";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   590
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   591
val prems = goal thy
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   592
  "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P(xs)";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   593
by(res_inst_tac [("xs","xs")] length_induct 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   594
by(res_inst_tac [("xs","xs")] snoc_eq_cases 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   595
 brs prems 1;
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   596
by(fast_tac (claset() addIs prems addss simpset()) 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   597
qed "snoc_induct";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   598
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   599
(** last & butlast **)
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   600
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   601
goal thy "last(xs@[x]) = x";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   602
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   603
by (ALLGOALS Asm_simp_tac);
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   604
qed "last_snoc";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   605
Addsimps [last_snoc];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   606
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   607
goal thy "butlast(xs@[x]) = xs";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   608
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   609
by (ALLGOALS Asm_simp_tac);
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   610
qed "butlast_snoc";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   611
Addsimps [butlast_snoc];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   612
4643
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   613
goal thy "length(butlast xs) = length xs - 1";
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   614
by (res_inst_tac [("xs","xs")] snoc_induct 1);
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   615
by (ALLGOALS Asm_simp_tac);
4643
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   616
qed "length_butlast";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   617
Addsimps [length_butlast];
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   618
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   619
goal thy
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   620
  "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   621
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   622
by (ALLGOALS Asm_simp_tac);
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   623
qed_spec_mp "butlast_append";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   624
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   625
goal thy "x:set(butlast xs) --> x:set xs";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   626
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   627
by (ALLGOALS Asm_simp_tac);
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   628
qed_spec_mp "in_set_butlastD";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   629
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   630
goal thy "!!xs. x:set(butlast xs) ==> x:set(butlast(xs@ys))";
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   631
by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   632
by (blast_tac (claset() addDs [in_set_butlastD]) 1);
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   633
qed "in_set_butlast_appendI1";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   634
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   635
goal thy "!!xs. x:set(butlast ys) ==> x:set(butlast(xs@ys))";
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   636
by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   637
by (Clarify_tac 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   638
by (Full_simp_tac 1);
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   639
qed "in_set_butlast_appendI2";
3902
265a5d8ab88f Removed comment.
nipkow
parents: 3896
diff changeset
   640
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   641
(** take  & drop **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   642
section "take & drop";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   643
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   644
goal thy "take 0 xs = []";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   645
by (induct_tac "xs" 1);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   646
by (ALLGOALS Asm_simp_tac);
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   647
qed "take_0";
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   648
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   649
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
   650
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   651
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   652
qed "drop_0";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   653
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   654
goal thy "take (Suc n) (x#xs) = x # take n xs";
1552
6f71b5d46700 Ran expandshort
paulson
parents: 1485
diff changeset
   655
by (Simp_tac 1);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   656
qed "take_Suc_Cons";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   657
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   658
goal thy "drop (Suc n) (x#xs) = drop n xs";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   659
by (Simp_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   660
qed "drop_Suc_Cons";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   661
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   662
Delsimps [take_Cons,drop_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   663
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   664
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   665
goal thy "!xs. length(take n xs) = min (length xs) n";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   666
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   667
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   668
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   669
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   670
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   671
qed_spec_mp "length_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   672
Addsimps [length_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   673
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   674
goal thy "!xs. length(drop n xs) = (length xs - n)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   675
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   676
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   677
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   678
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   679
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   680
qed_spec_mp "length_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   681
Addsimps [length_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   682
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   683
goal thy "!xs. length xs <= n --> take n xs = xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   684
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   685
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   686
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   687
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   688
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   689
qed_spec_mp "take_all";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   690
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   691
goal thy "!xs. length xs <= n --> drop n xs = []";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   692
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   693
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   694
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   695
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   696
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   697
qed_spec_mp "drop_all";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   698
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   699
goal thy 
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   700
  "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   701
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   702
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   703
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   704
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   705
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   706
qed_spec_mp "take_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   707
Addsimps [take_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   708
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   709
goal thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   710
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   711
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   712
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   713
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   714
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   715
qed_spec_mp "drop_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   716
Addsimps [drop_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   717
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   718
goal thy "!xs n. take n (take m xs) = take (min n m) xs"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   719
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   720
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   721
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   722
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   723
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   724
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   725
by (exhaust_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   726
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   727
qed_spec_mp "take_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   728
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   729
goal thy "!xs. drop n (drop m xs) = drop (n + m) xs"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   730
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   731
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   732
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   733
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   734
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   735
qed_spec_mp "drop_drop";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   736
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   737
goal thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   738
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   739
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   740
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   741
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   742
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   743
qed_spec_mp "take_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   744
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   745
goal thy "!xs. take n (map f xs) = map f (take n xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   746
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   747
by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   748
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   749
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   750
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   751
qed_spec_mp "take_map"; 
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   752
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   753
goal thy "!xs. drop n (map f xs) = map f (drop n xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   754
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   755
by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   756
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   757
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   758
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   759
qed_spec_mp "drop_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   760
4502
337c073de95e nth -> !
nipkow
parents: 4423
diff changeset
   761
goal thy "!n i. i < n --> (take n xs)!i = xs!i";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   762
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   763
 by (ALLGOALS Asm_simp_tac);
3708
56facaebf3e3 Changed some proofs to use Clarify_tac
paulson
parents: 3647
diff changeset
   764
by (Clarify_tac 1);
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   765
by (exhaust_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   766
 by (Blast_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   767
by (exhaust_tac "i" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   768
by (ALLGOALS Asm_full_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   769
qed_spec_mp "nth_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   770
Addsimps [nth_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   771
4502
337c073de95e nth -> !
nipkow
parents: 4423
diff changeset
   772
goal thy  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   773
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   774
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   775
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   776
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   777
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   778
qed_spec_mp "nth_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   779
Addsimps [nth_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   780
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   781
(** takeWhile & dropWhile **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   782
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   783
section "takeWhile & dropWhile";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   784
3586
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   785
goal thy "takeWhile P xs @ dropWhile P xs = xs";
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   786
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   787
by (ALLGOALS Asm_full_simp_tac);
3586
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   788
qed "takeWhile_dropWhile_id";
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   789
Addsimps [takeWhile_dropWhile_id];
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   790
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   791
goal thy  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   792
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   793
by (ALLGOALS Asm_full_simp_tac);
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   794
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   795
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   796
Addsimps [takeWhile_append1];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   797
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   798
goal thy
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   799
  "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   800
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   801
by (ALLGOALS Asm_full_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   802
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   803
Addsimps [takeWhile_append2];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   804
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   805
goal thy
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   806
  "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   807
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   808
by (ALLGOALS Asm_full_simp_tac);
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   809
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   810
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   811
Addsimps [dropWhile_append1];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   812
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   813
goal thy
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   814
  "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   815
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   816
by (ALLGOALS Asm_full_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   817
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   818
Addsimps [dropWhile_append2];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   819
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   820
goal thy "x:set(takeWhile P xs) --> x:set xs & P x";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   821
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   822
by (ALLGOALS Asm_full_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   823
qed_spec_mp"set_take_whileD";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   824
4132
daff3c9987cc added zip and nodup
oheimb
parents: 4089
diff changeset
   825
qed_goal "zip_Nil_Nil"   thy "zip []     []     = []" (K [Simp_tac 1]);
daff3c9987cc added zip and nodup
oheimb
parents: 4089
diff changeset
   826
qed_goal "zip_Cons_Cons" thy "zip (x#xs) (y#ys) = (x,y)#zip xs ys" 
daff3c9987cc added zip and nodup
oheimb
parents: 4089
diff changeset
   827
						      (K [Simp_tac 1]);
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   828
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   829
(** nodups & remdups **)
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   830
section "nodups & remdups";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   831
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   832
goal thy "set(remdups xs) = set xs";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   833
by (induct_tac "xs" 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   834
 by (Simp_tac 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   835
by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1);
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   836
qed "set_remdups";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   837
Addsimps [set_remdups];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   838
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   839
goal thy "nodups(remdups xs)";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   840
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   841
by (ALLGOALS Asm_full_simp_tac);
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   842
qed "nodups_remdups";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   843
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   844
goal thy "nodups xs --> nodups (filter P xs)";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   845
by (induct_tac "xs" 1);
4686
74a12e86b20b Removed `addsplits [expand_if]'
nipkow
parents: 4681
diff changeset
   846
by (ALLGOALS Asm_full_simp_tac);
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   847
qed_spec_mp "nodups_filter";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   848
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   849
(** replicate **)
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   850
section "replicate";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   851
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   852
goal thy "set(replicate (Suc n) x) = {x}";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   853
by (induct_tac "n" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   854
by (ALLGOALS Asm_full_simp_tac);
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   855
val lemma = result();
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   856
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   857
goal thy "!!n. n ~= 0 ==> set(replicate n x) = {x}";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   858
by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1);
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   859
qed "set_replicate";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   860
Addsimps [set_replicate];