src/HOL/List.ML
author nipkow
Sun, 22 Feb 1998 14:12:23 +0100
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permissions -rw-r--r--
New induction schemas for lists (length and snoc).
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(*  Title:      HOL/List
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1994 TU Muenchen
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List lemmas
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*)
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goal thy "!x. xs ~= x#xs";
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by (induct_tac "xs" 1);
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by (ALLGOALS Asm_simp_tac);
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qed_spec_mp "not_Cons_self";
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bind_thm("not_Cons_self2",not_Cons_self RS not_sym);
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Addsimps [not_Cons_self,not_Cons_self2];
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goal thy "(xs ~= []) = (? y ys. xs = y#ys)";
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by (induct_tac "xs" 1);
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by (Simp_tac 1);
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by (Asm_simp_tac 1);
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qed "neq_Nil_conv";
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(** "lists": the list-forming operator over sets **)
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goalw thy lists.defs "!!A B. A<=B ==> lists A <= lists B";
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by (rtac lfp_mono 1);
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by (REPEAT (ares_tac basic_monos 1));
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qed "lists_mono";
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val listsE = lists.mk_cases list.simps  "x#l : lists A";
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AddSEs [listsE];
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AddSIs lists.intrs;
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goal thy "!!l. l: lists A ==> l: lists B --> l: lists (A Int B)";
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by (etac lists.induct 1);
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by (ALLGOALS Blast_tac);
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qed_spec_mp "lists_IntI";
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goal thy "lists (A Int B) = lists A Int lists B";
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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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goal thy "!ys. length xs = length ys | length us = length vs \
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\              --> (xs@us = ys@vs) = (xs=ys & us=vs)";
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by (induct_tac "xs" 1);
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 by (rtac allI 1);
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 by (exhaust_tac "ys" 1);
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  by (Asm_simp_tac 1);
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wenzelm
parents: 4132
diff changeset
   156
 by (fast_tac (claset() addIs [less_add_Suc2] addss simpset()
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   157
                      addEs [less_not_refl2 RSN (2,rev_notE)]) 1);
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   158
by (rtac allI 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   159
by (exhaust_tac "ys" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   160
 by (fast_tac (claset() addIs [less_add_Suc2] addss simpset()
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   161
                      addEs [(less_not_refl2 RS not_sym) RSN (2,rev_notE)]) 1);
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   162
by (Asm_simp_tac 1);
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   163
qed_spec_mp "append_eq_append_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   164
Addsimps [append_eq_append_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   165
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   166
goal thy "(xs @ ys = xs @ zs) = (ys=zs)";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   167
by (Simp_tac 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   168
qed "same_append_eq";
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   169
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   170
goal thy "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; 
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   171
by (Simp_tac 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   172
qed "append1_eq_conv";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   173
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   174
goal thy "(ys @ xs = zs @ xs) = (ys=zs)";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   175
by (Simp_tac 1);
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   176
qed "append_same_eq";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   177
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   178
AddSIs
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   179
 [same_append_eq RS iffD2, append1_eq_conv RS iffD2, append_same_eq RS iffD2];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   180
AddSDs
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   181
 [same_append_eq RS iffD1, append1_eq_conv RS iffD1, append_same_eq RS iffD1];
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   182
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   183
goal thy "xs ~= [] --> hd xs # tl xs = xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   184
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   185
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   186
qed_spec_mp "hd_Cons_tl";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   187
Addsimps [hd_Cons_tl];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   188
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   189
goal thy "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   190
by (induct_tac "xs" 1);
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   191
by (ALLGOALS Asm_simp_tac);
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   192
qed "hd_append";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   193
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   194
goal thy "!!xs. xs ~= [] ==> hd(xs @ ys) = hd xs";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   195
by (asm_simp_tac (simpset() addsimps [hd_append]
4069
d6d06a03a2e9 expand_list_case -> split_list_case
nipkow
parents: 4032
diff changeset
   196
                           addsplits [split_list_case]) 1);
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   197
qed "hd_append2";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   198
Addsimps [hd_append2];
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   199
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   200
goal thy "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   201
by (simp_tac (simpset() addsplits [split_list_case]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   202
qed "tl_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   203
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   204
goal thy "!!xs. xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   205
by (asm_simp_tac (simpset() addsimps [tl_append]
4069
d6d06a03a2e9 expand_list_case -> split_list_case
nipkow
parents: 4032
diff changeset
   206
                           addsplits [split_list_case]) 1);
3571
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   207
qed "tl_append2";
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   208
Addsimps [tl_append2];
f1c8fa0f0bf9 List.ML: added lemmas by Stefan Merz.
nipkow
parents: 3468
diff changeset
   209
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   210
(** map **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   211
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   212
section "map";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   213
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   214
goal thy
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   215
  "(!x. x : set xs --> f x = g x) --> map f xs = map g xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   216
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   217
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   218
bind_thm("map_ext", impI RS (allI RS (result() RS mp)));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   219
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   220
goal thy "map (%x. x) = (%xs. xs)";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   221
by (rtac ext 1);
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   222
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   223
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   224
qed "map_ident";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   225
Addsimps[map_ident];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   226
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   227
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
   228
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   229
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   230
qed "map_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   231
Addsimps[map_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   232
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   233
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
   234
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   235
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   236
qed "map_compose";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   237
Addsimps[map_compose];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   238
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   239
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
   240
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   241
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   242
qed "rev_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   243
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   244
(* a congruence rule for map: *)
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   245
goal thy
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   246
 "(xs=ys) --> (!x. x : set ys --> f x = g x) --> map f xs = map g ys";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   247
by (rtac impI 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   248
by (hyp_subst_tac 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   249
by (induct_tac "ys" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   250
by (ALLGOALS Asm_simp_tac);
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   251
val lemma = result();
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   252
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
   253
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   254
goal List.thy "(map f xs = []) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   255
by (induct_tac "xs" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   256
by (ALLGOALS Asm_simp_tac);
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   257
qed "map_is_Nil_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   258
AddIffs [map_is_Nil_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   259
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   260
goal List.thy "([] = map f xs) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   261
by (induct_tac "xs" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   262
by (ALLGOALS Asm_simp_tac);
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   263
qed "Nil_is_map_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   264
AddIffs [Nil_is_map_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   265
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   266
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   267
(** rev **)
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   268
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   269
section "rev";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   270
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   271
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
   272
by (induct_tac "xs" 1);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   273
by (ALLGOALS Asm_simp_tac);
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   274
qed "rev_append";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   275
Addsimps[rev_append];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   276
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   277
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
   278
by (induct_tac "l" 1);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   279
by (ALLGOALS Asm_simp_tac);
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   280
qed "rev_rev_ident";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   281
Addsimps[rev_rev_ident];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   282
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   283
goal thy "(rev xs = []) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   284
by (induct_tac "xs" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   285
by (ALLGOALS Asm_simp_tac);
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   286
qed "rev_is_Nil_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   287
AddIffs [rev_is_Nil_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   288
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   289
goal thy "([] = rev xs) = (xs = [])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   290
by (induct_tac "xs" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   291
by (ALLGOALS Asm_simp_tac);
3860
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   292
qed "Nil_is_rev_conv";
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   293
AddIffs [Nil_is_rev_conv];
a29ab43f7174 More lemmas, esp. ~Bex and ~Ball conversions.
nipkow
parents: 3842
diff changeset
   294
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   295
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   296
(** mem **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   297
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   298
section "mem";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   299
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   300
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
   301
by (induct_tac "xs" 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   302
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if])));
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   303
qed "mem_append";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   304
Addsimps[mem_append];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   305
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   306
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
   307
by (induct_tac "xs" 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   308
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if])));
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   309
qed "mem_filter";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   310
Addsimps[mem_filter];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   311
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   312
(** set **)
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   313
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   314
section "set";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   315
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   316
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
   317
by (induct_tac "xs" 1);
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   318
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   319
qed "set_append";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   320
Addsimps[set_append];
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   321
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   322
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
   323
by (induct_tac "xs" 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   324
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if])));
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2739
diff changeset
   325
by (Blast_tac 1);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   326
qed "set_mem_eq";
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   327
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   328
goal thy "set l <= set (x#l)";
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
paulson
parents: 1908
diff changeset
   329
by (Simp_tac 1);
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2739
diff changeset
   330
by (Blast_tac 1);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   331
qed "set_subset_Cons";
1936
979e8b4f5fa5 Proved set_of_list_subset_Cons
paulson
parents: 1908
diff changeset
   332
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   333
goal thy "(set xs = {}) = (xs = [])";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   334
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   335
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   336
qed "set_empty";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   337
Addsimps [set_empty];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   338
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   339
goal thy "set(rev xs) = set(xs)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   340
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   341
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   342
qed "set_rev";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   343
Addsimps [set_rev];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   344
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   345
goal thy "set(map f xs) = f``(set xs)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   346
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   347
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   348
qed "set_map";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   349
Addsimps [set_map];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   350
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   351
goal thy "set(map f xs) = f``(set xs)";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   352
by (induct_tac "xs" 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   353
by (ALLGOALS Asm_simp_tac);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   354
qed "set_map";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   355
Addsimps [set_map];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   356
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   357
goal thy "(x : set(filter P xs)) = (x : set xs & P x)";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   358
by (induct_tac "xs" 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   359
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if])));
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   360
by(Blast_tac 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   361
qed "in_set_filter";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   362
Addsimps [in_set_filter];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   363
1812
debfc40b7756 Addition of setOfList
paulson
parents: 1760
diff changeset
   364
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   365
(** list_all **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   366
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   367
section "list_all";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   368
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   369
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
   370
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   371
by (ALLGOALS Asm_simp_tac);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   372
qed "list_all_True";
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   373
Addsimps [list_all_True];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   374
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   375
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
   376
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   377
by (ALLGOALS Asm_simp_tac);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   378
qed "list_all_append";
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   379
Addsimps [list_all_append];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   380
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   381
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
   382
by (induct_tac "xs" 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   383
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if])));
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2739
diff changeset
   384
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   385
qed "list_all_mem_conv";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   386
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   387
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   388
(** filter **)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   389
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   390
section "filter";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   391
3383
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   392
goal thy "filter P (xs@ys) = filter P xs @ filter P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   393
by (induct_tac "xs" 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   394
 by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if])));
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   395
qed "filter_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   396
Addsimps [filter_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   397
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   398
goal thy "filter (%x. True) xs = xs";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   399
by (induct_tac "xs" 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   400
by (ALLGOALS Asm_simp_tac);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   401
qed "filter_True";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   402
Addsimps [filter_True];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   403
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   404
goal thy "filter (%x. False) xs = []";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   405
by (induct_tac "xs" 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   406
by (ALLGOALS Asm_simp_tac);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   407
qed "filter_False";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   408
Addsimps [filter_False];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   409
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   410
goal thy "length (filter P xs) <= length xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   411
by (induct_tac "xs" 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   412
 by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if])));
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   413
qed "length_filter";
3383
7707cb7a5054 Corrected statement of filter_append; added filter_size
paulson
parents: 3342
diff changeset
   414
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   415
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   416
(** concat **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   417
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   418
section "concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   419
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   420
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
   421
by (induct_tac "xs" 1);
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 1202
diff changeset
   422
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   423
qed"concat_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   424
Addsimps [concat_append];
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   425
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   426
goal thy "(concat xss = []) = (!xs:set xss. xs=[])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   427
by (induct_tac "xss" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   428
by (ALLGOALS Asm_simp_tac);
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   429
qed "concat_eq_Nil_conv";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   430
AddIffs [concat_eq_Nil_conv];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   431
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   432
goal thy "([] = concat xss) = (!xs:set xss. xs=[])";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   433
by (induct_tac "xss" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   434
by (ALLGOALS Asm_simp_tac);
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   435
qed "Nil_eq_concat_conv";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   436
AddIffs [Nil_eq_concat_conv];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   437
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   438
goal thy  "set(concat xs) = Union(set `` set xs)";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   439
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   440
by (ALLGOALS Asm_simp_tac);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   441
qed"set_concat";
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   442
Addsimps [set_concat];
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   443
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   444
goal thy "map f (concat xs) = concat (map (map f) xs)"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   445
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   446
by (ALLGOALS Asm_simp_tac);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   447
qed "map_concat";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   448
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   449
goal thy "filter p (concat xs) = concat (map (filter p) xs)"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   450
by (induct_tac "xs" 1);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   451
by (ALLGOALS Asm_simp_tac);
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   452
qed"filter_concat"; 
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   453
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   454
goal thy "rev(concat xs) = concat (map rev (rev xs))";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   455
by (induct_tac "xs" 1);
2512
0231e4f467f2 Got rid of Alls in List.
nipkow
parents: 1985
diff changeset
   456
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   457
qed "rev_concat";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   458
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   459
(** nth **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   460
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   461
section "nth";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   462
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   463
goal thy
4502
337c073de95e nth -> !
nipkow
parents: 4423
diff changeset
   464
  "!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
   465
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   466
 by (Asm_simp_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   467
 by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   468
 by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   469
  by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   470
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   471
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   472
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   473
qed_spec_mp "nth_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   474
4502
337c073de95e nth -> !
nipkow
parents: 4423
diff changeset
   475
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
   476
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   477
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   478
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   479
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   480
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   481
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   482
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
   483
qed_spec_mp "nth_map";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   484
Addsimps [nth_map];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   485
4502
337c073de95e nth -> !
nipkow
parents: 4423
diff changeset
   486
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
   487
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   488
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   489
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   490
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   491
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   492
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   493
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
   494
qed_spec_mp "list_all_nth";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   495
4502
337c073de95e nth -> !
nipkow
parents: 4423
diff changeset
   496
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
   497
by (induct_tac "xs" 1);
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   498
(* case [] *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   499
by (Simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   500
(* case x#xl *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   501
by (rtac allI 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   502
by (nat_ind_tac "n" 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   503
(* case 0 *)
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   504
by (Asm_full_simp_tac 1);
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   505
(* case Suc x *)
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   506
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1);
1485
240cc98b94a7 Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents: 1465
diff changeset
   507
qed_spec_mp "nth_mem";
1301
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   508
Addsimps [nth_mem];
42782316d510 Added various thms and tactics.
nipkow
parents: 1264
diff changeset
   509
4643
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   510
(**  More case analysis and induction **)
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   511
section "More case analysis and induction";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   512
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   513
val [prem] = goal thy
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   514
  "(!!xs. (!ys. length ys < length xs --> P ys) ==> P xs) ==> P(xs)";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   515
by(rtac measure_induct 1 THEN etac prem 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   516
qed "length_induct";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   517
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   518
goal thy "xs ~= [] --> (? ys y. xs = ys@[y])";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   519
by(res_inst_tac [("xs","xs")] length_induct 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   520
by(Clarify_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   521
bd (neq_Nil_conv RS iffD1) 1;
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   522
by(Clarify_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   523
by(rename_tac "ys" 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   524
by(case_tac "ys = []" 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   525
 by(res_inst_tac [("x","[]")] exI 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   526
 by(Asm_full_simp_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   527
by(eres_inst_tac [("x","ys")] allE 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   528
by(Asm_full_simp_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   529
by(REPEAT(etac exE 1));
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   530
by(rename_tac "zs z" 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   531
by(hyp_subst_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   532
by(res_inst_tac [("x","y#zs")] exI 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   533
by(Simp_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   534
qed_spec_mp "neq_Nil_snocD";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   535
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   536
val prems = goal thy
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   537
  "[| 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
   538
by(case_tac "xs = []" 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   539
 by(Asm_simp_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   540
 bes prems 1;
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   541
bd neq_Nil_snocD 1;
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   542
by(REPEAT(etac exE 1));
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   543
by(Asm_simp_tac 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   544
bes prems 1;
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   545
qed "snoc_eq_cases";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   546
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   547
val prems = goal thy
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   548
  "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P(xs)";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   549
by(res_inst_tac [("xs","xs")] length_induct 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   550
by(res_inst_tac [("xs","xs")] snoc_eq_cases 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   551
 brs prems 1;
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   552
by(fast_tac (claset() addIs prems addss simpset()) 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   553
qed "snoc_induct";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   554
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   555
(** last & butlast **)
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   556
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   557
goal thy "last(xs@[x]) = x";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   558
by (induct_tac "xs" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   559
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if])));
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   560
qed "last_snoc";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   561
Addsimps [last_snoc];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   562
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   563
goal thy "butlast(xs@[x]) = xs";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   564
by (induct_tac "xs" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   565
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if])));
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   566
qed "butlast_snoc";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   567
Addsimps [butlast_snoc];
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   568
4643
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   569
goal thy "length(butlast xs) = length xs - 1";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   570
by(res_inst_tac [("xs","xs")] snoc_induct 1);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   571
by(ALLGOALS Asm_simp_tac);
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   572
qed "length_butlast";
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   573
Addsimps [length_butlast];
1b40fcac5a09 New induction schemas for lists (length and snoc).
nipkow
parents: 4628
diff changeset
   574
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   575
goal thy
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   576
  "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   577
by (induct_tac "xs" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   578
by (ALLGOALS(asm_simp_tac (simpset() addsplits [expand_if])));
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   579
qed_spec_mp "butlast_append";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   580
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   581
goal thy "x:set(butlast xs) --> x:set xs";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   582
by (induct_tac "xs" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   583
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if])));
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   584
qed_spec_mp "in_set_butlastD";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   585
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   586
goal thy "!!xs. x:set(butlast xs) ==> x:set(butlast(xs@ys))";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   587
by (asm_simp_tac (simpset() addsimps [butlast_append]
3919
c036caebfc75 setloop split_tac -> addsplits
nipkow
parents: 3902
diff changeset
   588
                          addsplits [expand_if]) 1);
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   589
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
   590
qed "in_set_butlast_appendI1";
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   591
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   592
goal thy "!!xs. x:set(butlast ys) ==> x:set(butlast(xs@ys))";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   593
by (asm_simp_tac (simpset() addsimps [butlast_append]
3919
c036caebfc75 setloop split_tac -> addsplits
nipkow
parents: 3902
diff changeset
   594
                          addsplits [expand_if]) 1);
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   595
by (Clarify_tac 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   596
by (Full_simp_tac 1);
3896
ee8ebb74ec00 Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents: 3860
diff changeset
   597
qed "in_set_butlast_appendI2";
3902
265a5d8ab88f Removed comment.
nipkow
parents: 3896
diff changeset
   598
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   599
(** take  & drop **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   600
section "take & drop";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   601
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   602
goal thy "take 0 xs = []";
3040
7d48671753da Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents: 3011
diff changeset
   603
by (induct_tac "xs" 1);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   604
by (ALLGOALS Asm_simp_tac);
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   605
qed "take_0";
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   606
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   607
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
   608
by (induct_tac "xs" 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   609
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   610
qed "drop_0";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   611
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   612
goal thy "take (Suc n) (x#xs) = x # take n xs";
1552
6f71b5d46700 Ran expandshort
paulson
parents: 1485
diff changeset
   613
by (Simp_tac 1);
1419
a6a034a47a71 defined take/drop by induction over list rather than nat.
nipkow
parents: 1327
diff changeset
   614
qed "take_Suc_Cons";
1327
6c29cfab679c added new arithmetic lemmas and the functions take and drop.
nipkow
parents: 1301
diff changeset
   615
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   616
goal thy "drop (Suc n) (x#xs) = drop n xs";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   617
by (Simp_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   618
qed "drop_Suc_Cons";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   619
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   620
Delsimps [take_Cons,drop_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   621
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   622
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   623
goal thy "!xs. length(take n xs) = min (length xs) n";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   624
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   625
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   626
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   627
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   628
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   629
qed_spec_mp "length_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   630
Addsimps [length_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   631
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   632
goal thy "!xs. length(drop n xs) = (length xs - n)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   633
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   634
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   635
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   636
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   637
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   638
qed_spec_mp "length_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   639
Addsimps [length_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   640
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   641
goal thy "!xs. length xs <= n --> take n xs = xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   642
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   643
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   644
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   645
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   646
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   647
qed_spec_mp "take_all";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   648
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   649
goal thy "!xs. length xs <= n --> drop n xs = []";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   650
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   651
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   652
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   653
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   654
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   655
qed_spec_mp "drop_all";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   656
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   657
goal thy 
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   658
  "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   659
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   660
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   661
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   662
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   663
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   664
qed_spec_mp "take_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   665
Addsimps [take_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   666
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   667
goal thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   668
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   669
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   670
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   671
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   672
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   673
qed_spec_mp "drop_append";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   674
Addsimps [drop_append];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   675
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   676
goal thy "!xs n. take n (take m xs) = take (min n m) xs"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   677
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   678
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   679
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   680
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   681
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   682
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   683
by (exhaust_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   684
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   685
qed_spec_mp "take_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   686
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   687
goal thy "!xs. drop n (drop m xs) = drop (n + m) xs"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   688
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   689
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   690
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   691
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   692
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   693
qed_spec_mp "drop_drop";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   694
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   695
goal thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   696
by (nat_ind_tac "m" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   697
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   698
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   699
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   700
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   701
qed_spec_mp "take_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   702
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   703
goal thy "!xs. take n (map f xs) = map f (take n xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   704
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   705
by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   706
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   707
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   708
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   709
qed_spec_mp "take_map"; 
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   710
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   711
goal thy "!xs. drop n (map f xs) = map f (drop n xs)"; 
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   712
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   713
by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   714
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   715
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   716
by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   717
qed_spec_mp "drop_map";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   718
4502
337c073de95e nth -> !
nipkow
parents: 4423
diff changeset
   719
goal thy "!n i. i < n --> (take n xs)!i = xs!i";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   720
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   721
 by (ALLGOALS Asm_simp_tac);
3708
56facaebf3e3 Changed some proofs to use Clarify_tac
paulson
parents: 3647
diff changeset
   722
by (Clarify_tac 1);
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   723
by (exhaust_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   724
 by (Blast_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   725
by (exhaust_tac "i" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   726
by (ALLGOALS Asm_full_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   727
qed_spec_mp "nth_take";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   728
Addsimps [nth_take];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   729
4502
337c073de95e nth -> !
nipkow
parents: 4423
diff changeset
   730
goal thy  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   731
by (nat_ind_tac "n" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   732
 by (ALLGOALS Asm_simp_tac);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   733
by (rtac allI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   734
by (exhaust_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   735
 by (ALLGOALS Asm_simp_tac);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   736
qed_spec_mp "nth_drop";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   737
Addsimps [nth_drop];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   738
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   739
(** takeWhile & dropWhile **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   740
3467
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   741
section "takeWhile & dropWhile";
a0797ba03dfe More concat lemmas.
nipkow
parents: 3465
diff changeset
   742
3586
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   743
goal thy "takeWhile P xs @ dropWhile P xs = xs";
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   744
by (induct_tac "xs" 1);
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   745
 by (Simp_tac 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   746
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1);
3586
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   747
qed "takeWhile_dropWhile_id";
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   748
Addsimps [takeWhile_dropWhile_id];
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   749
2ee1ed79c802 Added a take/dropWhile lemma.
nipkow
parents: 3585
diff changeset
   750
goal thy  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   751
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   752
 by (Simp_tac 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   753
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1);
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   754
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   755
bind_thm("takeWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   756
Addsimps [takeWhile_append1];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   757
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   758
goal thy
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   759
  "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   760
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   761
 by (Simp_tac 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   762
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   763
bind_thm("takeWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   764
Addsimps [takeWhile_append2];
1169
5873833cf37f Added function rev and its properties length_rev, etc.
lcp
parents: 995
diff changeset
   765
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   766
goal thy
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   767
  "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   768
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   769
 by (Simp_tac 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   770
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1);
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   771
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   772
bind_thm("dropWhile_append1", conjI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   773
Addsimps [dropWhile_append1];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   774
3011
a3b73ba44a11 Tidied up.
nipkow
parents: 2891
diff changeset
   775
goal thy
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3708
diff changeset
   776
  "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   777
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   778
 by (Simp_tac 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   779
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   780
bind_thm("dropWhile_append2", ballI RS (result() RS mp));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   781
Addsimps [dropWhile_append2];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   782
3465
e85c24717cad set_of_list -> set
nipkow
parents: 3457
diff changeset
   783
goal thy "x:set(takeWhile P xs) --> x:set xs & P x";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   784
by (induct_tac "xs" 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3383
diff changeset
   785
 by (Simp_tac 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4069
diff changeset
   786
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1);
3647
a64c8fbcd98f Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents: 3589
diff changeset
   787
qed_spec_mp"set_take_whileD";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents: 2512
diff changeset
   788
4132
daff3c9987cc added zip and nodup
oheimb
parents: 4089
diff changeset
   789
qed_goal "zip_Nil_Nil"   thy "zip []     []     = []" (K [Simp_tac 1]);
daff3c9987cc added zip and nodup
oheimb
parents: 4089
diff changeset
   790
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
   791
						      (K [Simp_tac 1]);
4605
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   792
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   793
(** nodups & remdups **)
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   794
section "nodups & remdups";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   795
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   796
goal thy "set(remdups xs) = set xs";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   797
by (induct_tac "xs" 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   798
 by (Simp_tac 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   799
by (asm_full_simp_tac (simpset() addsimps [insert_absorb]
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   800
                                 addsplits [expand_if]) 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   801
qed "set_remdups";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   802
Addsimps [set_remdups];
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   803
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   804
goal thy "nodups(remdups xs)";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   805
by (induct_tac "xs" 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   806
 by (Simp_tac 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   807
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   808
qed "nodups_remdups";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   809
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   810
goal thy "nodups xs --> nodups (filter P xs)";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   811
by (induct_tac "xs" 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   812
 by (Simp_tac 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   813
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1);
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   814
qed_spec_mp "nodups_filter";
579e0ef2df6b Added `remdups'
nipkow
parents: 4502
diff changeset
   815
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   816
(** replicate **)
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   817
section "replicate";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   818
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   819
goal thy "set(replicate (Suc n) x) = {x}";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   820
by (induct_tac "n" 1);
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   821
by (ALLGOALS Asm_full_simp_tac);
3589
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   822
val lemma = result();
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   823
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   824
goal thy "!!n. n ~= 0 ==> set(replicate n x) = {x}";
4423
a129b817b58a expandshort;
wenzelm
parents: 4132
diff changeset
   825
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
   826
qed "set_replicate";
244daa75f890 Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents: 3586
diff changeset
   827
Addsimps [set_replicate];