src/HOL/Finite_Set.thy
author berghofe
Wed Feb 07 17:28:09 2007 +0100 (2007-02-07)
changeset 22262 96ba62dff413
parent 21733 131dd2a27137
child 22316 f662831459de
permissions -rw-r--r--
Adapted to new inductive definition package.
wenzelm@12396
     1
(*  Title:      HOL/Finite_Set.thy
wenzelm@12396
     2
    ID:         $Id$
wenzelm@12396
     3
    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
avigad@16775
     4
                with contributions by Jeremy Avigad
wenzelm@12396
     5
*)
wenzelm@12396
     6
wenzelm@12396
     7
header {* Finite sets *}
wenzelm@12396
     8
nipkow@15131
     9
theory Finite_Set
haftmann@21409
    10
imports Power Divides Inductive Lattices
nipkow@15131
    11
begin
wenzelm@12396
    12
nipkow@15392
    13
subsection {* Definition and basic properties *}
wenzelm@12396
    14
berghofe@22262
    15
inductive2 finite :: "'a set => bool"
berghofe@22262
    16
  where
berghofe@22262
    17
    emptyI [simp, intro!]: "finite {}"
berghofe@22262
    18
  | insertI [simp, intro!]: "finite A ==> finite (insert a A)"
wenzelm@12396
    19
wenzelm@12396
    20
axclass finite \<subseteq> type
wenzelm@12396
    21
  finite: "finite UNIV"
wenzelm@12396
    22
nipkow@13737
    23
lemma ex_new_if_finite: -- "does not depend on def of finite at all"
wenzelm@14661
    24
  assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
wenzelm@14661
    25
  shows "\<exists>a::'a. a \<notin> A"
wenzelm@14661
    26
proof -
wenzelm@14661
    27
  from prems have "A \<noteq> UNIV" by blast
wenzelm@14661
    28
  thus ?thesis by blast
wenzelm@14661
    29
qed
wenzelm@12396
    30
berghofe@22262
    31
lemma finite_induct [case_names empty insert, induct set: finite]:
wenzelm@12396
    32
  "finite F ==>
nipkow@15327
    33
    P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
wenzelm@12396
    34
  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
wenzelm@12396
    35
proof -
wenzelm@13421
    36
  assume "P {}" and
nipkow@15327
    37
    insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
wenzelm@12396
    38
  assume "finite F"
wenzelm@12396
    39
  thus "P F"
wenzelm@12396
    40
  proof induct
wenzelm@12396
    41
    show "P {}" .
nipkow@15327
    42
    fix x F assume F: "finite F" and P: "P F"
wenzelm@12396
    43
    show "P (insert x F)"
wenzelm@12396
    44
    proof cases
wenzelm@12396
    45
      assume "x \<in> F"
wenzelm@12396
    46
      hence "insert x F = F" by (rule insert_absorb)
wenzelm@12396
    47
      with P show ?thesis by (simp only:)
wenzelm@12396
    48
    next
wenzelm@12396
    49
      assume "x \<notin> F"
wenzelm@12396
    50
      from F this P show ?thesis by (rule insert)
wenzelm@12396
    51
    qed
wenzelm@12396
    52
  qed
wenzelm@12396
    53
qed
wenzelm@12396
    54
nipkow@15484
    55
lemma finite_ne_induct[case_names singleton insert, consumes 2]:
nipkow@15484
    56
assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>
nipkow@15484
    57
 \<lbrakk> \<And>x. P{x};
nipkow@15484
    58
   \<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk>
nipkow@15484
    59
 \<Longrightarrow> P F"
nipkow@15484
    60
using fin
nipkow@15484
    61
proof induct
nipkow@15484
    62
  case empty thus ?case by simp
nipkow@15484
    63
next
nipkow@15484
    64
  case (insert x F)
nipkow@15484
    65
  show ?case
nipkow@15484
    66
  proof cases
nipkow@15484
    67
    assume "F = {}" thus ?thesis using insert(4) by simp
nipkow@15484
    68
  next
nipkow@15484
    69
    assume "F \<noteq> {}" thus ?thesis using insert by blast
nipkow@15484
    70
  qed
nipkow@15484
    71
qed
nipkow@15484
    72
wenzelm@12396
    73
lemma finite_subset_induct [consumes 2, case_names empty insert]:
wenzelm@12396
    74
  "finite F ==> F \<subseteq> A ==>
nipkow@15327
    75
    P {} ==> (!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==>
wenzelm@12396
    76
    P F"
wenzelm@12396
    77
proof -
wenzelm@13421
    78
  assume "P {}" and insert:
nipkow@15327
    79
    "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
wenzelm@12396
    80
  assume "finite F"
wenzelm@12396
    81
  thus "F \<subseteq> A ==> P F"
wenzelm@12396
    82
  proof induct
wenzelm@12396
    83
    show "P {}" .
nipkow@15327
    84
    fix x F assume "finite F" and "x \<notin> F"
wenzelm@12396
    85
      and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
wenzelm@12396
    86
    show "P (insert x F)"
wenzelm@12396
    87
    proof (rule insert)
wenzelm@12396
    88
      from i show "x \<in> A" by blast
wenzelm@12396
    89
      from i have "F \<subseteq> A" by blast
wenzelm@12396
    90
      with P show "P F" .
wenzelm@12396
    91
    qed
wenzelm@12396
    92
  qed
wenzelm@12396
    93
qed
wenzelm@12396
    94
nipkow@15392
    95
text{* Finite sets are the images of initial segments of natural numbers: *}
nipkow@15392
    96
paulson@15510
    97
lemma finite_imp_nat_seg_image_inj_on:
paulson@15510
    98
  assumes fin: "finite A" 
paulson@15510
    99
  shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}"
nipkow@15392
   100
using fin
nipkow@15392
   101
proof induct
nipkow@15392
   102
  case empty
paulson@15510
   103
  show ?case  
paulson@15510
   104
  proof show "\<exists>f. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp 
paulson@15510
   105
  qed
nipkow@15392
   106
next
nipkow@15392
   107
  case (insert a A)
paulson@15510
   108
  have notinA: "a \<notin> A" .
paulson@15510
   109
  from insert.hyps obtain n f
paulson@15510
   110
    where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
paulson@15510
   111
  hence "insert a A = f(n:=a) ` {i. i < Suc n}"
paulson@15510
   112
        "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
paulson@15510
   113
    by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
nipkow@15392
   114
  thus ?case by blast
nipkow@15392
   115
qed
nipkow@15392
   116
nipkow@15392
   117
lemma nat_seg_image_imp_finite:
nipkow@15392
   118
  "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
nipkow@15392
   119
proof (induct n)
nipkow@15392
   120
  case 0 thus ?case by simp
nipkow@15392
   121
next
nipkow@15392
   122
  case (Suc n)
nipkow@15392
   123
  let ?B = "f ` {i. i < n}"
nipkow@15392
   124
  have finB: "finite ?B" by(rule Suc.hyps[OF refl])
nipkow@15392
   125
  show ?case
nipkow@15392
   126
  proof cases
nipkow@15392
   127
    assume "\<exists>k<n. f n = f k"
nipkow@15392
   128
    hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
nipkow@15392
   129
    thus ?thesis using finB by simp
nipkow@15392
   130
  next
nipkow@15392
   131
    assume "\<not>(\<exists> k<n. f n = f k)"
nipkow@15392
   132
    hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
nipkow@15392
   133
    thus ?thesis using finB by simp
nipkow@15392
   134
  qed
nipkow@15392
   135
qed
nipkow@15392
   136
nipkow@15392
   137
lemma finite_conv_nat_seg_image:
nipkow@15392
   138
  "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
paulson@15510
   139
by(blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
nipkow@15392
   140
nipkow@15392
   141
subsubsection{* Finiteness and set theoretic constructions *}
nipkow@15392
   142
wenzelm@12396
   143
lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
wenzelm@12396
   144
  -- {* The union of two finite sets is finite. *}
berghofe@22262
   145
  by (induct set: finite) simp_all
wenzelm@12396
   146
wenzelm@12396
   147
lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
wenzelm@12396
   148
  -- {* Every subset of a finite set is finite. *}
wenzelm@12396
   149
proof -
wenzelm@12396
   150
  assume "finite B"
wenzelm@12396
   151
  thus "!!A. A \<subseteq> B ==> finite A"
wenzelm@12396
   152
  proof induct
wenzelm@12396
   153
    case empty
wenzelm@12396
   154
    thus ?case by simp
wenzelm@12396
   155
  next
nipkow@15327
   156
    case (insert x F A)
wenzelm@12396
   157
    have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" .
wenzelm@12396
   158
    show "finite A"
wenzelm@12396
   159
    proof cases
wenzelm@12396
   160
      assume x: "x \<in> A"
wenzelm@12396
   161
      with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
wenzelm@12396
   162
      with r have "finite (A - {x})" .
wenzelm@12396
   163
      hence "finite (insert x (A - {x}))" ..
wenzelm@12396
   164
      also have "insert x (A - {x}) = A" by (rule insert_Diff)
wenzelm@12396
   165
      finally show ?thesis .
wenzelm@12396
   166
    next
wenzelm@12396
   167
      show "A \<subseteq> F ==> ?thesis" .
wenzelm@12396
   168
      assume "x \<notin> A"
wenzelm@12396
   169
      with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
wenzelm@12396
   170
    qed
wenzelm@12396
   171
  qed
wenzelm@12396
   172
qed
wenzelm@12396
   173
nipkow@18423
   174
lemma finite_Collect_subset[simp]: "finite A \<Longrightarrow> finite{x \<in> A. P x}"
nipkow@17761
   175
using finite_subset[of "{x \<in> A. P x}" "A"] by blast
nipkow@17761
   176
wenzelm@12396
   177
lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
wenzelm@12396
   178
  by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
wenzelm@12396
   179
wenzelm@12396
   180
lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
wenzelm@12396
   181
  -- {* The converse obviously fails. *}
wenzelm@12396
   182
  by (blast intro: finite_subset)
wenzelm@12396
   183
wenzelm@12396
   184
lemma finite_insert [simp]: "finite (insert a A) = finite A"
wenzelm@12396
   185
  apply (subst insert_is_Un)
paulson@14208
   186
  apply (simp only: finite_Un, blast)
wenzelm@12396
   187
  done
wenzelm@12396
   188
nipkow@15281
   189
lemma finite_Union[simp, intro]:
nipkow@15281
   190
 "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
nipkow@15281
   191
by (induct rule:finite_induct) simp_all
nipkow@15281
   192
wenzelm@12396
   193
lemma finite_empty_induct:
wenzelm@12396
   194
  "finite A ==>
wenzelm@12396
   195
  P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
wenzelm@12396
   196
proof -
wenzelm@12396
   197
  assume "finite A"
wenzelm@12396
   198
    and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
wenzelm@12396
   199
  have "P (A - A)"
wenzelm@12396
   200
  proof -
wenzelm@12396
   201
    fix c b :: "'a set"
wenzelm@12396
   202
    presume c: "finite c" and b: "finite b"
wenzelm@12396
   203
      and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
wenzelm@12396
   204
    from c show "c \<subseteq> b ==> P (b - c)"
wenzelm@12396
   205
    proof induct
wenzelm@12396
   206
      case empty
wenzelm@12396
   207
      from P1 show ?case by simp
wenzelm@12396
   208
    next
nipkow@15327
   209
      case (insert x F)
wenzelm@12396
   210
      have "P (b - F - {x})"
wenzelm@12396
   211
      proof (rule P2)
wenzelm@12396
   212
        from _ b show "finite (b - F)" by (rule finite_subset) blast
wenzelm@12396
   213
        from insert show "x \<in> b - F" by simp
wenzelm@12396
   214
        from insert show "P (b - F)" by simp
wenzelm@12396
   215
      qed
wenzelm@12396
   216
      also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
wenzelm@12396
   217
      finally show ?case .
wenzelm@12396
   218
    qed
wenzelm@12396
   219
  next
wenzelm@12396
   220
    show "A \<subseteq> A" ..
wenzelm@12396
   221
  qed
wenzelm@12396
   222
  thus "P {}" by simp
wenzelm@12396
   223
qed
wenzelm@12396
   224
wenzelm@12396
   225
lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
wenzelm@12396
   226
  by (rule Diff_subset [THEN finite_subset])
wenzelm@12396
   227
wenzelm@12396
   228
lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
wenzelm@12396
   229
  apply (subst Diff_insert)
wenzelm@12396
   230
  apply (case_tac "a : A - B")
wenzelm@12396
   231
   apply (rule finite_insert [symmetric, THEN trans])
paulson@14208
   232
   apply (subst insert_Diff, simp_all)
wenzelm@12396
   233
  done
wenzelm@12396
   234
paulson@19870
   235
lemma finite_Diff_singleton [simp]: "finite (A - {a}) = finite A"
paulson@19870
   236
  by simp
paulson@19870
   237
wenzelm@12396
   238
nipkow@15392
   239
text {* Image and Inverse Image over Finite Sets *}
paulson@13825
   240
paulson@13825
   241
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
paulson@13825
   242
  -- {* The image of a finite set is finite. *}
berghofe@22262
   243
  by (induct set: finite) simp_all
paulson@13825
   244
paulson@14430
   245
lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
paulson@14430
   246
  apply (frule finite_imageI)
paulson@14430
   247
  apply (erule finite_subset, assumption)
paulson@14430
   248
  done
paulson@14430
   249
paulson@13825
   250
lemma finite_range_imageI:
paulson@13825
   251
    "finite (range g) ==> finite (range (%x. f (g x)))"
paulson@14208
   252
  apply (drule finite_imageI, simp)
paulson@13825
   253
  done
paulson@13825
   254
wenzelm@12396
   255
lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
wenzelm@12396
   256
proof -
wenzelm@12396
   257
  have aux: "!!A. finite (A - {}) = finite A" by simp
wenzelm@12396
   258
  fix B :: "'a set"
wenzelm@12396
   259
  assume "finite B"
wenzelm@12396
   260
  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
wenzelm@12396
   261
    apply induct
wenzelm@12396
   262
     apply simp
wenzelm@12396
   263
    apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
wenzelm@12396
   264
     apply clarify
wenzelm@12396
   265
     apply (simp (no_asm_use) add: inj_on_def)
paulson@14208
   266
     apply (blast dest!: aux [THEN iffD1], atomize)
wenzelm@12396
   267
    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
paulson@14208
   268
    apply (frule subsetD [OF equalityD2 insertI1], clarify)
wenzelm@12396
   269
    apply (rule_tac x = xa in bexI)
wenzelm@12396
   270
     apply (simp_all add: inj_on_image_set_diff)
wenzelm@12396
   271
    done
wenzelm@12396
   272
qed (rule refl)
wenzelm@12396
   273
wenzelm@12396
   274
paulson@13825
   275
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
paulson@13825
   276
  -- {* The inverse image of a singleton under an injective function
paulson@13825
   277
         is included in a singleton. *}
paulson@14430
   278
  apply (auto simp add: inj_on_def)
paulson@14430
   279
  apply (blast intro: the_equality [symmetric])
paulson@13825
   280
  done
paulson@13825
   281
paulson@13825
   282
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
paulson@13825
   283
  -- {* The inverse image of a finite set under an injective function
paulson@13825
   284
         is finite. *}
berghofe@22262
   285
  apply (induct set: finite)
wenzelm@21575
   286
   apply simp_all
paulson@14430
   287
  apply (subst vimage_insert)
paulson@14430
   288
  apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
paulson@13825
   289
  done
paulson@13825
   290
paulson@13825
   291
nipkow@15392
   292
text {* The finite UNION of finite sets *}
wenzelm@12396
   293
wenzelm@12396
   294
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
berghofe@22262
   295
  by (induct set: finite) simp_all
wenzelm@12396
   296
wenzelm@12396
   297
text {*
wenzelm@12396
   298
  Strengthen RHS to
paulson@14430
   299
  @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
wenzelm@12396
   300
wenzelm@12396
   301
  We'd need to prove
paulson@14430
   302
  @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
wenzelm@12396
   303
  by induction. *}
wenzelm@12396
   304
wenzelm@12396
   305
lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
wenzelm@12396
   306
  by (blast intro: finite_UN_I finite_subset)
wenzelm@12396
   307
wenzelm@12396
   308
nipkow@17022
   309
lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"
nipkow@17022
   310
by (simp add: Plus_def)
nipkow@17022
   311
nipkow@15392
   312
text {* Sigma of finite sets *}
wenzelm@12396
   313
wenzelm@12396
   314
lemma finite_SigmaI [simp]:
wenzelm@12396
   315
    "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
wenzelm@12396
   316
  by (unfold Sigma_def) (blast intro!: finite_UN_I)
wenzelm@12396
   317
nipkow@15402
   318
lemma finite_cartesian_product: "[| finite A; finite B |] ==>
nipkow@15402
   319
    finite (A <*> B)"
nipkow@15402
   320
  by (rule finite_SigmaI)
nipkow@15402
   321
wenzelm@12396
   322
lemma finite_Prod_UNIV:
wenzelm@12396
   323
    "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
wenzelm@12396
   324
  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
wenzelm@12396
   325
   apply (erule ssubst)
paulson@14208
   326
   apply (erule finite_SigmaI, auto)
wenzelm@12396
   327
  done
wenzelm@12396
   328
paulson@15409
   329
lemma finite_cartesian_productD1:
paulson@15409
   330
     "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
paulson@15409
   331
apply (auto simp add: finite_conv_nat_seg_image) 
paulson@15409
   332
apply (drule_tac x=n in spec) 
paulson@15409
   333
apply (drule_tac x="fst o f" in spec) 
paulson@15409
   334
apply (auto simp add: o_def) 
paulson@15409
   335
 prefer 2 apply (force dest!: equalityD2) 
paulson@15409
   336
apply (drule equalityD1) 
paulson@15409
   337
apply (rename_tac y x)
paulson@15409
   338
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
paulson@15409
   339
 prefer 2 apply force
paulson@15409
   340
apply clarify
paulson@15409
   341
apply (rule_tac x=k in image_eqI, auto)
paulson@15409
   342
done
paulson@15409
   343
paulson@15409
   344
lemma finite_cartesian_productD2:
paulson@15409
   345
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
paulson@15409
   346
apply (auto simp add: finite_conv_nat_seg_image) 
paulson@15409
   347
apply (drule_tac x=n in spec) 
paulson@15409
   348
apply (drule_tac x="snd o f" in spec) 
paulson@15409
   349
apply (auto simp add: o_def) 
paulson@15409
   350
 prefer 2 apply (force dest!: equalityD2) 
paulson@15409
   351
apply (drule equalityD1)
paulson@15409
   352
apply (rename_tac x y)
paulson@15409
   353
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
paulson@15409
   354
 prefer 2 apply force
paulson@15409
   355
apply clarify
paulson@15409
   356
apply (rule_tac x=k in image_eqI, auto)
paulson@15409
   357
done
paulson@15409
   358
paulson@15409
   359
nipkow@15392
   360
text {* The powerset of a finite set *}
wenzelm@12396
   361
wenzelm@12396
   362
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
wenzelm@12396
   363
proof
wenzelm@12396
   364
  assume "finite (Pow A)"
wenzelm@12396
   365
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
wenzelm@12396
   366
  thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
wenzelm@12396
   367
next
wenzelm@12396
   368
  assume "finite A"
wenzelm@12396
   369
  thus "finite (Pow A)"
wenzelm@12396
   370
    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
wenzelm@12396
   371
qed
wenzelm@12396
   372
nipkow@15392
   373
nipkow@15392
   374
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
nipkow@15392
   375
by(blast intro: finite_subset[OF subset_Pow_Union])
nipkow@15392
   376
nipkow@15392
   377
wenzelm@12396
   378
lemma finite_converse [iff]: "finite (r^-1) = finite r"
wenzelm@12396
   379
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
wenzelm@12396
   380
   apply simp
wenzelm@12396
   381
   apply (rule iffI)
wenzelm@12396
   382
    apply (erule finite_imageD [unfolded inj_on_def])
wenzelm@12396
   383
    apply (simp split add: split_split)
wenzelm@12396
   384
   apply (erule finite_imageI)
paulson@14208
   385
  apply (simp add: converse_def image_def, auto)
wenzelm@12396
   386
  apply (rule bexI)
wenzelm@12396
   387
   prefer 2 apply assumption
wenzelm@12396
   388
  apply simp
wenzelm@12396
   389
  done
wenzelm@12396
   390
paulson@14430
   391
nipkow@15392
   392
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
nipkow@15392
   393
Ehmety) *}
wenzelm@12396
   394
wenzelm@12396
   395
lemma finite_Field: "finite r ==> finite (Field r)"
wenzelm@12396
   396
  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
berghofe@22262
   397
  apply (induct set: finite)
wenzelm@12396
   398
   apply (auto simp add: Field_def Domain_insert Range_insert)
wenzelm@12396
   399
  done
wenzelm@12396
   400
wenzelm@12396
   401
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
wenzelm@12396
   402
  apply clarify
wenzelm@12396
   403
  apply (erule trancl_induct)
wenzelm@12396
   404
   apply (auto simp add: Field_def)
wenzelm@12396
   405
  done
wenzelm@12396
   406
wenzelm@12396
   407
lemma finite_trancl: "finite (r^+) = finite r"
wenzelm@12396
   408
  apply auto
wenzelm@12396
   409
   prefer 2
wenzelm@12396
   410
   apply (rule trancl_subset_Field2 [THEN finite_subset])
wenzelm@12396
   411
   apply (rule finite_SigmaI)
wenzelm@12396
   412
    prefer 3
berghofe@13704
   413
    apply (blast intro: r_into_trancl' finite_subset)
wenzelm@12396
   414
   apply (auto simp add: finite_Field)
wenzelm@12396
   415
  done
wenzelm@12396
   416
wenzelm@12396
   417
nipkow@15392
   418
subsection {* A fold functional for finite sets *}
nipkow@15392
   419
nipkow@15392
   420
text {* The intended behaviour is
nipkow@15480
   421
@{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<dots>)"}
nipkow@15392
   422
if @{text f} is associative-commutative. For an application of @{text fold}
nipkow@15392
   423
se the definitions of sums and products over finite sets.
nipkow@15392
   424
*}
nipkow@15392
   425
berghofe@22262
   426
inductive2
berghofe@22262
   427
  foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a => bool"
berghofe@22262
   428
  for f ::  "'a => 'a => 'a"
berghofe@22262
   429
  and g :: "'b => 'a"
berghofe@22262
   430
  and z :: 'a
berghofe@22262
   431
where
berghofe@22262
   432
  emptyI [intro]: "foldSet f g z {} z"
berghofe@22262
   433
| insertI [intro]:
berghofe@22262
   434
     "\<lbrakk> x \<notin> A; foldSet f g z A y \<rbrakk>
berghofe@22262
   435
      \<Longrightarrow> foldSet f g z (insert x A) (f (g x) y)"
berghofe@22262
   436
berghofe@22262
   437
inductive_cases2 empty_foldSetE [elim!]: "foldSet f g z {} x"
nipkow@15392
   438
nipkow@15392
   439
constdefs
nipkow@21733
   440
  fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
berghofe@22262
   441
  "fold f g z A == THE x. foldSet f g z A x"
nipkow@15392
   442
paulson@15498
   443
text{*A tempting alternative for the definiens is
berghofe@22262
   444
@{term "if finite A then THE x. foldSet f g e A x else e"}.
paulson@15498
   445
It allows the removal of finiteness assumptions from the theorems
paulson@15498
   446
@{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}.
paulson@15498
   447
The proofs become ugly, with @{text rule_format}. It is not worth the effort.*}
paulson@15498
   448
paulson@15498
   449
nipkow@15392
   450
lemma Diff1_foldSet:
berghofe@22262
   451
  "foldSet f g z (A - {x}) y ==> x: A ==> foldSet f g z A (f (g x) y)"
nipkow@15392
   452
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
nipkow@15392
   453
berghofe@22262
   454
lemma foldSet_imp_finite: "foldSet f g z A x==> finite A"
nipkow@15392
   455
  by (induct set: foldSet) auto
nipkow@15392
   456
berghofe@22262
   457
lemma finite_imp_foldSet: "finite A ==> EX x. foldSet f g z A x"
berghofe@22262
   458
  by (induct set: finite) auto
nipkow@15392
   459
nipkow@15392
   460
nipkow@15392
   461
subsubsection {* Commutative monoids *}
nipkow@15480
   462
nipkow@15392
   463
locale ACf =
nipkow@15392
   464
  fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
nipkow@15392
   465
  assumes commute: "x \<cdot> y = y \<cdot> x"
nipkow@15392
   466
    and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
nipkow@15392
   467
nipkow@15392
   468
locale ACe = ACf +
nipkow@15392
   469
  fixes e :: 'a
nipkow@15392
   470
  assumes ident [simp]: "x \<cdot> e = x"
nipkow@15392
   471
nipkow@15480
   472
locale ACIf = ACf +
nipkow@15480
   473
  assumes idem: "x \<cdot> x = x"
nipkow@15480
   474
nipkow@15392
   475
lemma (in ACf) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
nipkow@15392
   476
proof -
nipkow@15392
   477
  have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
nipkow@15392
   478
  also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
nipkow@15392
   479
  also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
nipkow@15392
   480
  finally show ?thesis .
nipkow@15392
   481
qed
nipkow@15392
   482
nipkow@15392
   483
lemmas (in ACf) AC = assoc commute left_commute
nipkow@15392
   484
nipkow@15392
   485
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
nipkow@15392
   486
proof -
nipkow@15392
   487
  have "x \<cdot> e = x" by (rule ident)
nipkow@15392
   488
  thus ?thesis by (subst commute)
nipkow@15392
   489
qed
nipkow@15392
   490
nipkow@15497
   491
lemma (in ACIf) idem2: "x \<cdot> (x \<cdot> y) = x \<cdot> y"
nipkow@15497
   492
proof -
nipkow@15497
   493
  have "x \<cdot> (x \<cdot> y) = (x \<cdot> x) \<cdot> y" by(simp add:assoc)
nipkow@15497
   494
  also have "\<dots> = x \<cdot> y" by(simp add:idem)
nipkow@15497
   495
  finally show ?thesis .
nipkow@15497
   496
qed
nipkow@15497
   497
nipkow@15497
   498
lemmas (in ACIf) ACI = AC idem idem2
nipkow@15497
   499
ballarin@15765
   500
text{* Interpretation of locales: *}
ballarin@15765
   501
ballarin@15765
   502
interpretation AC_add: ACe ["op +" "0::'a::comm_monoid_add"]
ballarin@19984
   503
  by unfold_locales (auto intro: add_assoc add_commute)
nipkow@15402
   504
ballarin@15765
   505
interpretation AC_mult: ACe ["op *" "1::'a::comm_monoid_mult"]
ballarin@19984
   506
  by unfold_locales (auto intro: mult_assoc mult_commute)
nipkow@15402
   507
nipkow@15392
   508
subsubsection{*From @{term foldSet} to @{term fold}*}
nipkow@15392
   509
paulson@15510
   510
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
wenzelm@19868
   511
  by (auto simp add: less_Suc_eq) 
paulson@15510
   512
paulson@15510
   513
lemma insert_image_inj_on_eq:
paulson@15510
   514
     "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; 
paulson@15510
   515
        inj_on h {i. i < Suc m}|] 
paulson@15510
   516
      ==> A = h ` {i. i < m}"
paulson@15510
   517
apply (auto simp add: image_less_Suc inj_on_def)
paulson@15510
   518
apply (blast intro: less_trans) 
paulson@15510
   519
done
paulson@15510
   520
paulson@15510
   521
lemma insert_inj_onE:
paulson@15510
   522
  assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" 
paulson@15510
   523
      and inj_on: "inj_on h {i::nat. i<n}"
paulson@15510
   524
  shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"
paulson@15510
   525
proof (cases n)
paulson@15510
   526
  case 0 thus ?thesis using aA by auto
paulson@15510
   527
next
paulson@15510
   528
  case (Suc m)
paulson@15510
   529
  have nSuc: "n = Suc m" . 
paulson@15510
   530
  have mlessn: "m<n" by (simp add: nSuc)
paulson@15532
   531
  from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
paulson@15520
   532
  let ?hm = "swap k m h"
paulson@15520
   533
  have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn 
paulson@15520
   534
    by (simp add: inj_on_swap_iff inj_on)
paulson@15510
   535
  show ?thesis
paulson@15520
   536
  proof (intro exI conjI)
paulson@15520
   537
    show "inj_on ?hm {i. i < m}" using inj_hm
paulson@15510
   538
      by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
paulson@15520
   539
    show "m<n" by (rule mlessn)
paulson@15520
   540
    show "A = ?hm ` {i. i < m}" 
paulson@15520
   541
    proof (rule insert_image_inj_on_eq)
paulson@15520
   542
      show "inj_on (swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
paulson@15520
   543
      show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) 
paulson@15520
   544
      show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
paulson@15520
   545
	using aA hkeq nSuc klessn
paulson@15520
   546
	by (auto simp add: swap_def image_less_Suc fun_upd_image 
paulson@15520
   547
			   less_Suc_eq inj_on_image_set_diff [OF inj_on])
nipkow@15479
   548
    qed
nipkow@15479
   549
  qed
nipkow@15479
   550
qed
nipkow@15479
   551
nipkow@15392
   552
lemma (in ACf) foldSet_determ_aux:
paulson@15510
   553
  "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n}; 
berghofe@22262
   554
                foldSet f g z A x; foldSet f g z A x' \<rbrakk>
nipkow@15392
   555
   \<Longrightarrow> x' = x"
paulson@15510
   556
proof (induct n rule: less_induct)
paulson@15510
   557
  case (less n)
paulson@15510
   558
    have IH: "!!m h A x x'. 
paulson@15510
   559
               \<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m}; 
berghofe@22262
   560
                foldSet f g z A x; foldSet f g z A x'\<rbrakk> \<Longrightarrow> x' = x" .
berghofe@22262
   561
    have Afoldx: "foldSet f g z A x" and Afoldx': "foldSet f g z A x'"
paulson@15510
   562
     and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" .
paulson@15510
   563
    show ?case
paulson@15510
   564
    proof (rule foldSet.cases [OF Afoldx])
berghofe@22262
   565
      assume "A = {}" and "x = z"
paulson@15510
   566
      with Afoldx' show "x' = x" by blast
nipkow@15392
   567
    next
paulson@15510
   568
      fix B b u
berghofe@22262
   569
      assume AbB: "A = insert b B" and x: "x = g b \<cdot> u"
berghofe@22262
   570
         and notinB: "b \<notin> B" and Bu: "foldSet f g z B u"
paulson@15510
   571
      show "x'=x" 
paulson@15510
   572
      proof (rule foldSet.cases [OF Afoldx'])
berghofe@22262
   573
        assume "A = {}" and "x' = z"
paulson@15510
   574
        with AbB show "x' = x" by blast
nipkow@15392
   575
      next
paulson@15510
   576
	fix C c v
berghofe@22262
   577
	assume AcC: "A = insert c C" and x': "x' = g c \<cdot> v"
berghofe@22262
   578
           and notinC: "c \<notin> C" and Cv: "foldSet f g z C v"
paulson@15510
   579
	from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
paulson@15510
   580
        from insert_inj_onE [OF Beq notinB injh]
paulson@15510
   581
        obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" 
paulson@15510
   582
                     and Beq: "B = hB ` {i. i < mB}"
paulson@15510
   583
                     and lessB: "mB < n" by auto 
paulson@15510
   584
	from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
paulson@15510
   585
        from insert_inj_onE [OF Ceq notinC injh]
paulson@15510
   586
        obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
paulson@15510
   587
                       and Ceq: "C = hC ` {i. i < mC}"
paulson@15510
   588
                       and lessC: "mC < n" by auto 
paulson@15510
   589
	show "x'=x"
nipkow@15392
   590
	proof cases
paulson@15510
   591
          assume "b=c"
paulson@15510
   592
	  then moreover have "B = C" using AbB AcC notinB notinC by auto
paulson@15510
   593
	  ultimately show ?thesis  using Bu Cv x x' IH[OF lessC Ceq inj_onC]
paulson@15510
   594
            by auto
nipkow@15392
   595
	next
nipkow@15392
   596
	  assume diff: "b \<noteq> c"
nipkow@15392
   597
	  let ?D = "B - {c}"
nipkow@15392
   598
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
paulson@15510
   599
	    using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
nipkow@15402
   600
	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
paulson@15510
   601
	  with AbB have "finite ?D" by simp
berghofe@22262
   602
	  then obtain d where Dfoldd: "foldSet f g z ?D d"
nipkow@17589
   603
	    using finite_imp_foldSet by iprover
paulson@15506
   604
	  moreover have cinB: "c \<in> B" using B by auto
berghofe@22262
   605
	  ultimately have "foldSet f g z B (g c \<cdot> d)"
nipkow@15392
   606
	    by(rule Diff1_foldSet)
paulson@15510
   607
	  hence "g c \<cdot> d = u" by (rule IH [OF lessB Beq inj_onB Bu]) 
paulson@15510
   608
          moreover have "g b \<cdot> d = v"
paulson@15510
   609
	  proof (rule IH[OF lessC Ceq inj_onC Cv])
berghofe@22262
   610
	    show "foldSet f g z C (g b \<cdot> d)" using C notinB Dfoldd
nipkow@15392
   611
	      by fastsimp
nipkow@15392
   612
	  qed
paulson@15510
   613
	  ultimately show ?thesis using x x' by (auto simp: AC)
nipkow@15392
   614
	qed
nipkow@15392
   615
      qed
nipkow@15392
   616
    qed
nipkow@15392
   617
  qed
nipkow@15392
   618
nipkow@15392
   619
nipkow@15392
   620
lemma (in ACf) foldSet_determ:
berghofe@22262
   621
  "foldSet f g z A x ==> foldSet f g z A y ==> y = x"
paulson@15510
   622
apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) 
paulson@15510
   623
apply (blast intro: foldSet_determ_aux [rule_format])
nipkow@15392
   624
done
nipkow@15392
   625
berghofe@22262
   626
lemma (in ACf) fold_equality: "foldSet f g z A y ==> fold f g z A = y"
nipkow@15392
   627
  by (unfold fold_def) (blast intro: foldSet_determ)
nipkow@15392
   628
nipkow@15392
   629
text{* The base case for @{text fold}: *}
nipkow@15392
   630
nipkow@15480
   631
lemma fold_empty [simp]: "fold f g z {} = z"
nipkow@15392
   632
  by (unfold fold_def) blast
nipkow@15392
   633
nipkow@15392
   634
lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
berghofe@22262
   635
    (foldSet f g z (insert x A) v) =
berghofe@22262
   636
    (EX y. foldSet f g z A y & v = f (g x) y)"
nipkow@15392
   637
  apply auto
nipkow@15392
   638
  apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
nipkow@15392
   639
   apply (fastsimp dest: foldSet_imp_finite)
nipkow@15392
   640
  apply (blast intro: foldSet_determ)
nipkow@15392
   641
  done
nipkow@15392
   642
nipkow@15392
   643
text{* The recursion equation for @{text fold}: *}
nipkow@15392
   644
nipkow@15392
   645
lemma (in ACf) fold_insert[simp]:
nipkow@15480
   646
    "finite A ==> x \<notin> A ==> fold f g z (insert x A) = f (g x) (fold f g z A)"
nipkow@15392
   647
  apply (unfold fold_def)
nipkow@15392
   648
  apply (simp add: fold_insert_aux)
nipkow@15392
   649
  apply (rule the_equality)
nipkow@15392
   650
  apply (auto intro: finite_imp_foldSet
nipkow@15392
   651
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
nipkow@15392
   652
  done
nipkow@15392
   653
nipkow@15535
   654
lemma (in ACf) fold_rec:
nipkow@15535
   655
assumes fin: "finite A" and a: "a:A"
nipkow@15535
   656
shows "fold f g z A = f (g a) (fold f g z (A - {a}))"
nipkow@15535
   657
proof-
nipkow@15535
   658
  have A: "A = insert a (A - {a})" using a by blast
nipkow@15535
   659
  hence "fold f g z A = fold f g z (insert a (A - {a}))" by simp
nipkow@15535
   660
  also have "\<dots> = f (g a) (fold f g z (A - {a}))"
nipkow@15535
   661
    by(rule fold_insert) (simp add:fin)+
nipkow@15535
   662
  finally show ?thesis .
nipkow@15535
   663
qed
nipkow@15535
   664
nipkow@15392
   665
nipkow@15480
   666
text{* A simplified version for idempotent functions: *}
nipkow@15480
   667
paulson@15509
   668
lemma (in ACIf) fold_insert_idem:
nipkow@15480
   669
assumes finA: "finite A"
paulson@15508
   670
shows "fold f g z (insert a A) = g a \<cdot> fold f g z A"
nipkow@15480
   671
proof cases
nipkow@15480
   672
  assume "a \<in> A"
nipkow@15480
   673
  then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
nipkow@15480
   674
    by(blast dest: mk_disjoint_insert)
nipkow@15480
   675
  show ?thesis
nipkow@15480
   676
  proof -
nipkow@15480
   677
    from finA A have finB: "finite B" by(blast intro: finite_subset)
nipkow@15480
   678
    have "fold f g z (insert a A) = fold f g z (insert a B)" using A by simp
nipkow@15480
   679
    also have "\<dots> = (g a) \<cdot> (fold f g z B)"
paulson@15506
   680
      using finB disj by simp
nipkow@15480
   681
    also have "\<dots> = g a \<cdot> fold f g z A"
nipkow@15480
   682
      using A finB disj by(simp add:idem assoc[symmetric])
nipkow@15480
   683
    finally show ?thesis .
nipkow@15480
   684
  qed
nipkow@15480
   685
next
nipkow@15480
   686
  assume "a \<notin> A"
nipkow@15480
   687
  with finA show ?thesis by simp
nipkow@15480
   688
qed
nipkow@15480
   689
nipkow@15484
   690
lemma (in ACIf) foldI_conv_id:
nipkow@15484
   691
  "finite A \<Longrightarrow> fold f g z A = fold f id z (g ` A)"
paulson@15509
   692
by(erule finite_induct)(simp_all add: fold_insert_idem del: fold_insert)
nipkow@15484
   693
nipkow@15392
   694
subsubsection{*Lemmas about @{text fold}*}
nipkow@15392
   695
nipkow@15392
   696
lemma (in ACf) fold_commute:
paulson@15487
   697
  "finite A ==> (!!z. f x (fold f g z A) = fold f g (f x z) A)"
berghofe@22262
   698
  apply (induct set: finite)
wenzelm@21575
   699
   apply simp
paulson@15487
   700
  apply (simp add: left_commute [of x])
nipkow@15392
   701
  done
nipkow@15392
   702
nipkow@15392
   703
lemma (in ACf) fold_nest_Un_Int:
nipkow@15392
   704
  "finite A ==> finite B
nipkow@15480
   705
    ==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)"
berghofe@22262
   706
  apply (induct set: finite)
wenzelm@21575
   707
   apply simp
nipkow@15392
   708
  apply (simp add: fold_commute Int_insert_left insert_absorb)
nipkow@15392
   709
  done
nipkow@15392
   710
nipkow@15392
   711
lemma (in ACf) fold_nest_Un_disjoint:
nipkow@15392
   712
  "finite A ==> finite B ==> A Int B = {}
nipkow@15480
   713
    ==> fold f g z (A Un B) = fold f g (fold f g z B) A"
nipkow@15392
   714
  by (simp add: fold_nest_Un_Int)
nipkow@15392
   715
nipkow@15392
   716
lemma (in ACf) fold_reindex:
paulson@15487
   717
assumes fin: "finite A"
paulson@15487
   718
shows "inj_on h A \<Longrightarrow> fold f g z (h ` A) = fold f (g \<circ> h) z A"
paulson@15506
   719
using fin apply induct
nipkow@15392
   720
 apply simp
nipkow@15392
   721
apply simp
nipkow@15392
   722
done
nipkow@15392
   723
nipkow@15392
   724
lemma (in ACe) fold_Un_Int:
nipkow@15392
   725
  "finite A ==> finite B ==>
nipkow@15392
   726
    fold f g e A \<cdot> fold f g e B =
nipkow@15392
   727
    fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
berghofe@22262
   728
  apply (induct set: finite, simp)
nipkow@15392
   729
  apply (simp add: AC insert_absorb Int_insert_left)
nipkow@15392
   730
  done
nipkow@15392
   731
nipkow@15392
   732
corollary (in ACe) fold_Un_disjoint:
nipkow@15392
   733
  "finite A ==> finite B ==> A Int B = {} ==>
nipkow@15392
   734
    fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B"
nipkow@15392
   735
  by (simp add: fold_Un_Int)
nipkow@15392
   736
nipkow@15392
   737
lemma (in ACe) fold_UN_disjoint:
nipkow@15392
   738
  "\<lbrakk> finite I; ALL i:I. finite (A i);
nipkow@15392
   739
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
nipkow@15392
   740
   \<Longrightarrow> fold f g e (UNION I A) =
nipkow@15392
   741
       fold f (%i. fold f g e (A i)) e I"
berghofe@22262
   742
  apply (induct set: finite, simp, atomize)
nipkow@15392
   743
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
nipkow@15392
   744
   prefer 2 apply blast
nipkow@15392
   745
  apply (subgoal_tac "A x Int UNION F A = {}")
nipkow@15392
   746
   prefer 2 apply blast
nipkow@15392
   747
  apply (simp add: fold_Un_disjoint)
nipkow@15392
   748
  done
nipkow@15392
   749
paulson@15506
   750
text{*Fusion theorem, as described in
paulson@15506
   751
Graham Hutton's paper,
paulson@15506
   752
A Tutorial on the Universality and Expressiveness of Fold,
paulson@15506
   753
JFP 9:4 (355-372), 1999.*}
paulson@15506
   754
lemma (in ACf) fold_fusion:
paulson@15506
   755
      includes ACf g
paulson@15506
   756
      shows
paulson@15506
   757
	"finite A ==> 
paulson@15506
   758
	 (!!x y. h (g x y) = f x (h y)) ==>
paulson@15506
   759
         h (fold g j w A) = fold f j (h w) A"
berghofe@22262
   760
  by (induct set: finite) simp_all
paulson@15506
   761
nipkow@15392
   762
lemma (in ACf) fold_cong:
nipkow@15480
   763
  "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A"
nipkow@15480
   764
  apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g z C = fold f h z C")
nipkow@15392
   765
   apply simp
nipkow@15392
   766
  apply (erule finite_induct, simp)
nipkow@15392
   767
  apply (simp add: subset_insert_iff, clarify)
nipkow@15392
   768
  apply (subgoal_tac "finite C")
nipkow@15392
   769
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
nipkow@15392
   770
  apply (subgoal_tac "C = insert x (C - {x})")
nipkow@15392
   771
   prefer 2 apply blast
nipkow@15392
   772
  apply (erule ssubst)
nipkow@15392
   773
  apply (drule spec)
nipkow@15392
   774
  apply (erule (1) notE impE)
nipkow@15392
   775
  apply (simp add: Ball_def del: insert_Diff_single)
nipkow@15392
   776
  done
nipkow@15392
   777
nipkow@15392
   778
lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@15392
   779
  fold f (%x. fold f (g x) e (B x)) e A =
nipkow@15392
   780
  fold f (split g) e (SIGMA x:A. B x)"
nipkow@15392
   781
apply (subst Sigma_def)
paulson@15506
   782
apply (subst fold_UN_disjoint, assumption, simp)
nipkow@15392
   783
 apply blast
nipkow@15392
   784
apply (erule fold_cong)
paulson@15506
   785
apply (subst fold_UN_disjoint, simp, simp)
nipkow@15392
   786
 apply blast
paulson@15506
   787
apply simp
nipkow@15392
   788
done
nipkow@15392
   789
nipkow@15392
   790
lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
nipkow@15392
   791
   fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
paulson@15506
   792
apply (erule finite_induct, simp)
nipkow@15392
   793
apply (simp add:AC)
nipkow@15392
   794
done
nipkow@15392
   795
nipkow@15392
   796
nipkow@15402
   797
subsection {* Generalized summation over a set *}
nipkow@15402
   798
nipkow@15402
   799
constdefs
nipkow@15402
   800
  setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
nipkow@15402
   801
  "setsum f A == if finite A then fold (op +) f 0 A else 0"
nipkow@15402
   802
wenzelm@19535
   803
abbreviation
wenzelm@21404
   804
  Setsum  ("\<Sum>_" [1000] 999) where
wenzelm@19535
   805
  "\<Sum>A == setsum (%x. x) A"
wenzelm@19535
   806
nipkow@15402
   807
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
nipkow@15402
   808
written @{text"\<Sum>x\<in>A. e"}. *}
nipkow@15402
   809
nipkow@15402
   810
syntax
paulson@17189
   811
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
nipkow@15402
   812
syntax (xsymbols)
paulson@17189
   813
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
   814
syntax (HTML output)
paulson@17189
   815
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
   816
nipkow@15402
   817
translations -- {* Beware of argument permutation! *}
nipkow@15402
   818
  "SUM i:A. b" == "setsum (%i. b) A"
nipkow@15402
   819
  "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
nipkow@15402
   820
nipkow@15402
   821
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
nipkow@15402
   822
 @{text"\<Sum>x|P. e"}. *}
nipkow@15402
   823
nipkow@15402
   824
syntax
paulson@17189
   825
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
nipkow@15402
   826
syntax (xsymbols)
paulson@17189
   827
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402
   828
syntax (HTML output)
paulson@17189
   829
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402
   830
nipkow@15402
   831
translations
nipkow@15402
   832
  "SUM x|P. t" => "setsum (%x. t) {x. P}"
nipkow@15402
   833
  "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
nipkow@15402
   834
nipkow@15402
   835
print_translation {*
nipkow@15402
   836
let
wenzelm@19535
   837
  fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
wenzelm@19535
   838
    if x<>y then raise Match
wenzelm@19535
   839
    else let val x' = Syntax.mark_bound x
wenzelm@19535
   840
             val t' = subst_bound(x',t)
wenzelm@19535
   841
             val P' = subst_bound(x',P)
wenzelm@19535
   842
         in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
wenzelm@19535
   843
in [("setsum", setsum_tr')] end
nipkow@15402
   844
*}
nipkow@15402
   845
wenzelm@19535
   846
nipkow@15402
   847
lemma setsum_empty [simp]: "setsum f {} = 0"
nipkow@15402
   848
  by (simp add: setsum_def)
nipkow@15402
   849
nipkow@15402
   850
lemma setsum_insert [simp]:
nipkow@15402
   851
    "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
ballarin@15765
   852
  by (simp add: setsum_def)
nipkow@15402
   853
paulson@15409
   854
lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
paulson@15409
   855
  by (simp add: setsum_def)
paulson@15409
   856
nipkow@15402
   857
lemma setsum_reindex:
nipkow@15402
   858
     "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
ballarin@15765
   859
by(auto simp add: setsum_def AC_add.fold_reindex dest!:finite_imageD)
nipkow@15402
   860
nipkow@15402
   861
lemma setsum_reindex_id:
nipkow@15402
   862
     "inj_on f B ==> setsum f B = setsum id (f ` B)"
nipkow@15402
   863
by (auto simp add: setsum_reindex)
nipkow@15402
   864
nipkow@15402
   865
lemma setsum_cong:
nipkow@15402
   866
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
ballarin@15765
   867
by(fastsimp simp: setsum_def intro: AC_add.fold_cong)
nipkow@15402
   868
nipkow@16733
   869
lemma strong_setsum_cong[cong]:
nipkow@16733
   870
  "A = B ==> (!!x. x:B =simp=> f x = g x)
nipkow@16733
   871
   ==> setsum (%x. f x) A = setsum (%x. g x) B"
berghofe@16632
   872
by(fastsimp simp: simp_implies_def setsum_def intro: AC_add.fold_cong)
berghofe@16632
   873
nipkow@15554
   874
lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A";
nipkow@15554
   875
  by (rule setsum_cong[OF refl], auto);
nipkow@15554
   876
nipkow@15402
   877
lemma setsum_reindex_cong:
nipkow@15554
   878
     "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] 
nipkow@15402
   879
      ==> setsum h B = setsum g A"
nipkow@15402
   880
  by (simp add: setsum_reindex cong: setsum_cong)
nipkow@15402
   881
nipkow@15542
   882
lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
nipkow@15402
   883
apply (clarsimp simp: setsum_def)
ballarin@15765
   884
apply (erule finite_induct, auto)
nipkow@15402
   885
done
nipkow@15402
   886
nipkow@15543
   887
lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
nipkow@15543
   888
by(simp add:setsum_cong)
nipkow@15402
   889
nipkow@15402
   890
lemma setsum_Un_Int: "finite A ==> finite B ==>
nipkow@15402
   891
  setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
nipkow@15402
   892
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
ballarin@15765
   893
by(simp add: setsum_def AC_add.fold_Un_Int [symmetric])
nipkow@15402
   894
nipkow@15402
   895
lemma setsum_Un_disjoint: "finite A ==> finite B
nipkow@15402
   896
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
nipkow@15402
   897
by (subst setsum_Un_Int [symmetric], auto)
nipkow@15402
   898
paulson@15409
   899
(*But we can't get rid of finite I. If infinite, although the rhs is 0, 
paulson@15409
   900
  the lhs need not be, since UNION I A could still be finite.*)
nipkow@15402
   901
lemma setsum_UN_disjoint:
nipkow@15402
   902
    "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402
   903
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402
   904
      setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
ballarin@15765
   905
by(simp add: setsum_def AC_add.fold_UN_disjoint cong: setsum_cong)
nipkow@15402
   906
paulson@15409
   907
text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
paulson@15409
   908
directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
nipkow@15402
   909
lemma setsum_Union_disjoint:
paulson@15409
   910
  "[| (ALL A:C. finite A);
paulson@15409
   911
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
paulson@15409
   912
   ==> setsum f (Union C) = setsum (setsum f) C"
paulson@15409
   913
apply (cases "finite C") 
paulson@15409
   914
 prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
nipkow@15402
   915
  apply (frule setsum_UN_disjoint [of C id f])
paulson@15409
   916
 apply (unfold Union_def id_def, assumption+)
paulson@15409
   917
done
nipkow@15402
   918
paulson@15409
   919
(*But we can't get rid of finite A. If infinite, although the lhs is 0, 
paulson@15409
   920
  the rhs need not be, since SIGMA A B could still be finite.*)
nipkow@15402
   921
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
paulson@17189
   922
    (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
ballarin@15765
   923
by(simp add:setsum_def AC_add.fold_Sigma split_def cong:setsum_cong)
nipkow@15402
   924
paulson@15409
   925
text{*Here we can eliminate the finiteness assumptions, by cases.*}
paulson@15409
   926
lemma setsum_cartesian_product: 
paulson@17189
   927
   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
paulson@15409
   928
apply (cases "finite A") 
paulson@15409
   929
 apply (cases "finite B") 
paulson@15409
   930
  apply (simp add: setsum_Sigma)
paulson@15409
   931
 apply (cases "A={}", simp)
nipkow@15543
   932
 apply (simp) 
paulson@15409
   933
apply (auto simp add: setsum_def
paulson@15409
   934
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
paulson@15409
   935
done
nipkow@15402
   936
nipkow@15402
   937
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
ballarin@15765
   938
by(simp add:setsum_def AC_add.fold_distrib)
nipkow@15402
   939
nipkow@15402
   940
nipkow@15402
   941
subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402
   942
nipkow@15402
   943
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
nipkow@15402
   944
  apply (case_tac "finite A")
nipkow@15402
   945
   prefer 2 apply (simp add: setsum_def)
nipkow@15402
   946
  apply (erule rev_mp)
nipkow@15402
   947
  apply (erule finite_induct, auto)
nipkow@15402
   948
  done
nipkow@15402
   949
nipkow@15402
   950
lemma setsum_eq_0_iff [simp]:
nipkow@15402
   951
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
berghofe@22262
   952
  by (induct set: finite) auto
nipkow@15402
   953
nipkow@15402
   954
lemma setsum_Un_nat: "finite A ==> finite B ==>
nipkow@15402
   955
    (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
nipkow@15402
   956
  -- {* For the natural numbers, we have subtraction. *}
nipkow@15402
   957
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
nipkow@15402
   958
nipkow@15402
   959
lemma setsum_Un: "finite A ==> finite B ==>
nipkow@15402
   960
    (setsum f (A Un B) :: 'a :: ab_group_add) =
nipkow@15402
   961
      setsum f A + setsum f B - setsum f (A Int B)"
nipkow@15402
   962
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
nipkow@15402
   963
nipkow@15402
   964
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
nipkow@15402
   965
    (if a:A then setsum f A - f a else setsum f A)"
nipkow@15402
   966
  apply (case_tac "finite A")
nipkow@15402
   967
   prefer 2 apply (simp add: setsum_def)
nipkow@15402
   968
  apply (erule finite_induct)
nipkow@15402
   969
   apply (auto simp add: insert_Diff_if)
nipkow@15402
   970
  apply (drule_tac a = a in mk_disjoint_insert, auto)
nipkow@15402
   971
  done
nipkow@15402
   972
nipkow@15402
   973
lemma setsum_diff1: "finite A \<Longrightarrow>
nipkow@15402
   974
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
nipkow@15402
   975
  (if a:A then setsum f A - f a else setsum f A)"
nipkow@15402
   976
  by (erule finite_induct) (auto simp add: insert_Diff_if)
nipkow@15402
   977
obua@15552
   978
lemma setsum_diff1'[rule_format]: "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
obua@15552
   979
  apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
obua@15552
   980
  apply (auto simp add: insert_Diff_if add_ac)
obua@15552
   981
  done
obua@15552
   982
nipkow@15402
   983
(* By Jeremy Siek: *)
nipkow@15402
   984
nipkow@15402
   985
lemma setsum_diff_nat: 
wenzelm@19535
   986
  assumes "finite B"
wenzelm@19535
   987
    and "B \<subseteq> A"
wenzelm@19535
   988
  shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
wenzelm@19535
   989
  using prems
wenzelm@19535
   990
proof induct
nipkow@15402
   991
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
nipkow@15402
   992
next
nipkow@15402
   993
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
nipkow@15402
   994
    and xFinA: "insert x F \<subseteq> A"
nipkow@15402
   995
    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
nipkow@15402
   996
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
nipkow@15402
   997
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
nipkow@15402
   998
    by (simp add: setsum_diff1_nat)
nipkow@15402
   999
  from xFinA have "F \<subseteq> A" by simp
nipkow@15402
  1000
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
nipkow@15402
  1001
  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
nipkow@15402
  1002
    by simp
nipkow@15402
  1003
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
nipkow@15402
  1004
  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
nipkow@15402
  1005
    by simp
nipkow@15402
  1006
  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
nipkow@15402
  1007
  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
nipkow@15402
  1008
    by simp
nipkow@15402
  1009
  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
nipkow@15402
  1010
qed
nipkow@15402
  1011
nipkow@15402
  1012
lemma setsum_diff:
nipkow@15402
  1013
  assumes le: "finite A" "B \<subseteq> A"
nipkow@15402
  1014
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
nipkow@15402
  1015
proof -
nipkow@15402
  1016
  from le have finiteB: "finite B" using finite_subset by auto
nipkow@15402
  1017
  show ?thesis using finiteB le
wenzelm@21575
  1018
  proof induct
wenzelm@19535
  1019
    case empty
wenzelm@19535
  1020
    thus ?case by auto
wenzelm@19535
  1021
  next
wenzelm@19535
  1022
    case (insert x F)
wenzelm@19535
  1023
    thus ?case using le finiteB 
wenzelm@19535
  1024
      by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
nipkow@15402
  1025
  qed
wenzelm@19535
  1026
qed
nipkow@15402
  1027
nipkow@15402
  1028
lemma setsum_mono:
nipkow@15402
  1029
  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
nipkow@15402
  1030
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
nipkow@15402
  1031
proof (cases "finite K")
nipkow@15402
  1032
  case True
nipkow@15402
  1033
  thus ?thesis using le
wenzelm@19535
  1034
  proof induct
nipkow@15402
  1035
    case empty
nipkow@15402
  1036
    thus ?case by simp
nipkow@15402
  1037
  next
nipkow@15402
  1038
    case insert
wenzelm@19535
  1039
    thus ?case using add_mono by fastsimp
nipkow@15402
  1040
  qed
nipkow@15402
  1041
next
nipkow@15402
  1042
  case False
nipkow@15402
  1043
  thus ?thesis
nipkow@15402
  1044
    by (simp add: setsum_def)
nipkow@15402
  1045
qed
nipkow@15402
  1046
nipkow@15554
  1047
lemma setsum_strict_mono:
wenzelm@19535
  1048
  fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
wenzelm@19535
  1049
  assumes "finite A"  "A \<noteq> {}"
wenzelm@19535
  1050
    and "!!x. x:A \<Longrightarrow> f x < g x"
wenzelm@19535
  1051
  shows "setsum f A < setsum g A"
wenzelm@19535
  1052
  using prems
nipkow@15554
  1053
proof (induct rule: finite_ne_induct)
nipkow@15554
  1054
  case singleton thus ?case by simp
nipkow@15554
  1055
next
nipkow@15554
  1056
  case insert thus ?case by (auto simp: add_strict_mono)
nipkow@15554
  1057
qed
nipkow@15554
  1058
nipkow@15535
  1059
lemma setsum_negf:
wenzelm@19535
  1060
  "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
nipkow@15535
  1061
proof (cases "finite A")
berghofe@22262
  1062
  case True thus ?thesis by (induct set: finite) auto
nipkow@15535
  1063
next
nipkow@15535
  1064
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15535
  1065
qed
nipkow@15402
  1066
nipkow@15535
  1067
lemma setsum_subtractf:
wenzelm@19535
  1068
  "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
wenzelm@19535
  1069
    setsum f A - setsum g A"
nipkow@15535
  1070
proof (cases "finite A")
nipkow@15535
  1071
  case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
nipkow@15535
  1072
next
nipkow@15535
  1073
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15535
  1074
qed
nipkow@15402
  1075
nipkow@15535
  1076
lemma setsum_nonneg:
wenzelm@19535
  1077
  assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
wenzelm@19535
  1078
  shows "0 \<le> setsum f A"
nipkow@15535
  1079
proof (cases "finite A")
nipkow@15535
  1080
  case True thus ?thesis using nn
wenzelm@21575
  1081
  proof induct
wenzelm@19535
  1082
    case empty then show ?case by simp
wenzelm@19535
  1083
  next
wenzelm@19535
  1084
    case (insert x F)
wenzelm@19535
  1085
    then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
wenzelm@19535
  1086
    with insert show ?case by simp
wenzelm@19535
  1087
  qed
nipkow@15535
  1088
next
nipkow@15535
  1089
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15535
  1090
qed
nipkow@15402
  1091
nipkow@15535
  1092
lemma setsum_nonpos:
wenzelm@19535
  1093
  assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
wenzelm@19535
  1094
  shows "setsum f A \<le> 0"
nipkow@15535
  1095
proof (cases "finite A")
nipkow@15535
  1096
  case True thus ?thesis using np
wenzelm@21575
  1097
  proof induct
wenzelm@19535
  1098
    case empty then show ?case by simp
wenzelm@19535
  1099
  next
wenzelm@19535
  1100
    case (insert x F)
wenzelm@19535
  1101
    then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
wenzelm@19535
  1102
    with insert show ?case by simp
wenzelm@19535
  1103
  qed
nipkow@15535
  1104
next
nipkow@15535
  1105
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15535
  1106
qed
nipkow@15402
  1107
nipkow@15539
  1108
lemma setsum_mono2:
nipkow@15539
  1109
fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}"
nipkow@15539
  1110
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
nipkow@15539
  1111
shows "setsum f A \<le> setsum f B"
nipkow@15539
  1112
proof -
nipkow@15539
  1113
  have "setsum f A \<le> setsum f A + setsum f (B-A)"
nipkow@15539
  1114
    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
nipkow@15539
  1115
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
nipkow@15539
  1116
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
nipkow@15539
  1117
  also have "A \<union> (B-A) = B" using sub by blast
nipkow@15539
  1118
  finally show ?thesis .
nipkow@15539
  1119
qed
nipkow@15542
  1120
avigad@16775
  1121
lemma setsum_mono3: "finite B ==> A <= B ==> 
avigad@16775
  1122
    ALL x: B - A. 
avigad@16775
  1123
      0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==>
avigad@16775
  1124
        setsum f A <= setsum f B"
avigad@16775
  1125
  apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
avigad@16775
  1126
  apply (erule ssubst)
avigad@16775
  1127
  apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
avigad@16775
  1128
  apply simp
avigad@16775
  1129
  apply (rule add_left_mono)
avigad@16775
  1130
  apply (erule setsum_nonneg)
avigad@16775
  1131
  apply (subst setsum_Un_disjoint [THEN sym])
avigad@16775
  1132
  apply (erule finite_subset, assumption)
avigad@16775
  1133
  apply (rule finite_subset)
avigad@16775
  1134
  prefer 2
avigad@16775
  1135
  apply assumption
avigad@16775
  1136
  apply auto
avigad@16775
  1137
  apply (rule setsum_cong)
avigad@16775
  1138
  apply auto
avigad@16775
  1139
done
avigad@16775
  1140
ballarin@19279
  1141
lemma setsum_right_distrib: 
nipkow@15402
  1142
  fixes f :: "'a => ('b::semiring_0_cancel)"
nipkow@15402
  1143
  shows "r * setsum f A = setsum (%n. r * f n) A"
nipkow@15402
  1144
proof (cases "finite A")
nipkow@15402
  1145
  case True
nipkow@15402
  1146
  thus ?thesis
wenzelm@21575
  1147
  proof induct
nipkow@15402
  1148
    case empty thus ?case by simp
nipkow@15402
  1149
  next
nipkow@15402
  1150
    case (insert x A) thus ?case by (simp add: right_distrib)
nipkow@15402
  1151
  qed
nipkow@15402
  1152
next
nipkow@15402
  1153
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15402
  1154
qed
nipkow@15402
  1155
ballarin@17149
  1156
lemma setsum_left_distrib:
ballarin@17149
  1157
  "setsum f A * (r::'a::semiring_0_cancel) = (\<Sum>n\<in>A. f n * r)"
ballarin@17149
  1158
proof (cases "finite A")
ballarin@17149
  1159
  case True
ballarin@17149
  1160
  then show ?thesis
ballarin@17149
  1161
  proof induct
ballarin@17149
  1162
    case empty thus ?case by simp
ballarin@17149
  1163
  next
ballarin@17149
  1164
    case (insert x A) thus ?case by (simp add: left_distrib)
ballarin@17149
  1165
  qed
ballarin@17149
  1166
next
ballarin@17149
  1167
  case False thus ?thesis by (simp add: setsum_def)
ballarin@17149
  1168
qed
ballarin@17149
  1169
ballarin@17149
  1170
lemma setsum_divide_distrib:
ballarin@17149
  1171
  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
ballarin@17149
  1172
proof (cases "finite A")
ballarin@17149
  1173
  case True
ballarin@17149
  1174
  then show ?thesis
ballarin@17149
  1175
  proof induct
ballarin@17149
  1176
    case empty thus ?case by simp
ballarin@17149
  1177
  next
ballarin@17149
  1178
    case (insert x A) thus ?case by (simp add: add_divide_distrib)
ballarin@17149
  1179
  qed
ballarin@17149
  1180
next
ballarin@17149
  1181
  case False thus ?thesis by (simp add: setsum_def)
ballarin@17149
  1182
qed
ballarin@17149
  1183
nipkow@15535
  1184
lemma setsum_abs[iff]: 
nipkow@15402
  1185
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
nipkow@15402
  1186
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
nipkow@15535
  1187
proof (cases "finite A")
nipkow@15535
  1188
  case True
nipkow@15535
  1189
  thus ?thesis
wenzelm@21575
  1190
  proof induct
nipkow@15535
  1191
    case empty thus ?case by simp
nipkow@15535
  1192
  next
nipkow@15535
  1193
    case (insert x A)
nipkow@15535
  1194
    thus ?case by (auto intro: abs_triangle_ineq order_trans)
nipkow@15535
  1195
  qed
nipkow@15402
  1196
next
nipkow@15535
  1197
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15402
  1198
qed
nipkow@15402
  1199
nipkow@15535
  1200
lemma setsum_abs_ge_zero[iff]: 
nipkow@15402
  1201
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
nipkow@15402
  1202
  shows "0 \<le> setsum (%i. abs(f i)) A"
nipkow@15535
  1203
proof (cases "finite A")
nipkow@15535
  1204
  case True
nipkow@15535
  1205
  thus ?thesis
wenzelm@21575
  1206
  proof induct
nipkow@15535
  1207
    case empty thus ?case by simp
nipkow@15535
  1208
  next
nipkow@21733
  1209
    case (insert x A) thus ?case by (auto simp: add_nonneg_nonneg)
nipkow@15535
  1210
  qed
nipkow@15402
  1211
next
nipkow@15535
  1212
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15402
  1213
qed
nipkow@15402
  1214
nipkow@15539
  1215
lemma abs_setsum_abs[simp]: 
nipkow@15539
  1216
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
nipkow@15539
  1217
  shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
nipkow@15539
  1218
proof (cases "finite A")
nipkow@15539
  1219
  case True
nipkow@15539
  1220
  thus ?thesis
wenzelm@21575
  1221
  proof induct
nipkow@15539
  1222
    case empty thus ?case by simp
nipkow@15539
  1223
  next
nipkow@15539
  1224
    case (insert a A)
nipkow@15539
  1225
    hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
nipkow@15539
  1226
    also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
avigad@16775
  1227
    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
avigad@16775
  1228
      by (simp del: abs_of_nonneg)
nipkow@15539
  1229
    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
nipkow@15539
  1230
    finally show ?case .
nipkow@15539
  1231
  qed
nipkow@15539
  1232
next
nipkow@15539
  1233
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15539
  1234
qed
nipkow@15539
  1235
nipkow@15402
  1236
ballarin@17149
  1237
text {* Commuting outer and inner summation *}
ballarin@17149
  1238
ballarin@17149
  1239
lemma swap_inj_on:
ballarin@17149
  1240
  "inj_on (%(i, j). (j, i)) (A \<times> B)"
ballarin@17149
  1241
  by (unfold inj_on_def) fast
ballarin@17149
  1242
ballarin@17149
  1243
lemma swap_product:
ballarin@17149
  1244
  "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
ballarin@17149
  1245
  by (simp add: split_def image_def) blast
ballarin@17149
  1246
ballarin@17149
  1247
lemma setsum_commute:
ballarin@17149
  1248
  "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
ballarin@17149
  1249
proof (simp add: setsum_cartesian_product)
paulson@17189
  1250
  have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
paulson@17189
  1251
    (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
ballarin@17149
  1252
    (is "?s = _")
ballarin@17149
  1253
    apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
ballarin@17149
  1254
    apply (simp add: split_def)
ballarin@17149
  1255
    done
paulson@17189
  1256
  also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
ballarin@17149
  1257
    (is "_ = ?t")
ballarin@17149
  1258
    apply (simp add: swap_product)
ballarin@17149
  1259
    done
ballarin@17149
  1260
  finally show "?s = ?t" .
ballarin@17149
  1261
qed
ballarin@17149
  1262
ballarin@19279
  1263
lemma setsum_product:
ballarin@19279
  1264
  fixes f :: "nat => ('a::semiring_0_cancel)"
ballarin@19279
  1265
  shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
ballarin@19279
  1266
  by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
ballarin@19279
  1267
ballarin@17149
  1268
nipkow@15402
  1269
subsection {* Generalized product over a set *}
nipkow@15402
  1270
nipkow@15402
  1271
constdefs
nipkow@15402
  1272
  setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
nipkow@15402
  1273
  "setprod f A == if finite A then fold (op *) f 1 A else 1"
nipkow@15402
  1274
wenzelm@19535
  1275
abbreviation
wenzelm@21404
  1276
  Setprod  ("\<Prod>_" [1000] 999) where
wenzelm@19535
  1277
  "\<Prod>A == setprod (%x. x) A"
wenzelm@19535
  1278
nipkow@15402
  1279
syntax
paulson@17189
  1280
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
nipkow@15402
  1281
syntax (xsymbols)
paulson@17189
  1282
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
  1283
syntax (HTML output)
paulson@17189
  1284
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@16550
  1285
nipkow@16550
  1286
translations -- {* Beware of argument permutation! *}
nipkow@16550
  1287
  "PROD i:A. b" == "setprod (%i. b) A" 
nipkow@16550
  1288
  "\<Prod>i\<in>A. b" == "setprod (%i. b) A" 
nipkow@16550
  1289
nipkow@16550
  1290
text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
nipkow@16550
  1291
 @{text"\<Prod>x|P. e"}. *}
nipkow@16550
  1292
nipkow@16550
  1293
syntax
paulson@17189
  1294
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
nipkow@16550
  1295
syntax (xsymbols)
paulson@17189
  1296
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
nipkow@16550
  1297
syntax (HTML output)
paulson@17189
  1298
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
nipkow@16550
  1299
nipkow@15402
  1300
translations
nipkow@16550
  1301
  "PROD x|P. t" => "setprod (%x. t) {x. P}"
nipkow@16550
  1302
  "\<Prod>x|P. t" => "setprod (%x. t) {x. P}"
nipkow@16550
  1303
nipkow@15402
  1304
nipkow@15402
  1305
lemma setprod_empty [simp]: "setprod f {} = 1"
nipkow@15402
  1306
  by (auto simp add: setprod_def)
nipkow@15402
  1307
nipkow@15402
  1308
lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
nipkow@15402
  1309
    setprod f (insert a A) = f a * setprod f A"
ballarin@19931
  1310
  by (simp add: setprod_def)
nipkow@15402
  1311
paulson@15409
  1312
lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
paulson@15409
  1313
  by (simp add: setprod_def)
paulson@15409
  1314
nipkow@15402
  1315
lemma setprod_reindex:
nipkow@15402
  1316
     "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
ballarin@15765
  1317
by(auto simp: setprod_def AC_mult.fold_reindex dest!:finite_imageD)
nipkow@15402
  1318
nipkow@15402
  1319
lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
nipkow@15402
  1320
by (auto simp add: setprod_reindex)
nipkow@15402
  1321
nipkow@15402
  1322
lemma setprod_cong:
nipkow@15402
  1323
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
ballarin@15765
  1324
by(fastsimp simp: setprod_def intro: AC_mult.fold_cong)
nipkow@15402
  1325
berghofe@16632
  1326
lemma strong_setprod_cong:
berghofe@16632
  1327
  "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
berghofe@16632
  1328
by(fastsimp simp: simp_implies_def setprod_def intro: AC_mult.fold_cong)
berghofe@16632
  1329
nipkow@15402
  1330
lemma setprod_reindex_cong: "inj_on f A ==>
nipkow@15402
  1331
    B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
nipkow@15402
  1332
  by (frule setprod_reindex, simp)
nipkow@15402
  1333
nipkow@15402
  1334
nipkow@15402
  1335
lemma setprod_1: "setprod (%i. 1) A = 1"
nipkow@15402
  1336
  apply (case_tac "finite A")
nipkow@15402
  1337
  apply (erule finite_induct, auto simp add: mult_ac)
nipkow@15402
  1338
  done
nipkow@15402
  1339
nipkow@15402
  1340
lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
nipkow@15402
  1341
  apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
nipkow@15402
  1342
  apply (erule ssubst, rule setprod_1)
nipkow@15402
  1343
  apply (rule setprod_cong, auto)
nipkow@15402
  1344
  done
nipkow@15402
  1345
nipkow@15402
  1346
lemma setprod_Un_Int: "finite A ==> finite B
nipkow@15402
  1347
    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
ballarin@15765
  1348
by(simp add: setprod_def AC_mult.fold_Un_Int[symmetric])
nipkow@15402
  1349
nipkow@15402
  1350
lemma setprod_Un_disjoint: "finite A ==> finite B
nipkow@15402
  1351
  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
nipkow@15402
  1352
by (subst setprod_Un_Int [symmetric], auto)
nipkow@15402
  1353
nipkow@15402
  1354
lemma setprod_UN_disjoint:
nipkow@15402
  1355
    "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402
  1356
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402
  1357
      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
ballarin@15765
  1358
by(simp add: setprod_def AC_mult.fold_UN_disjoint cong: setprod_cong)
nipkow@15402
  1359
nipkow@15402
  1360
lemma setprod_Union_disjoint:
paulson@15409
  1361
  "[| (ALL A:C. finite A);
paulson@15409
  1362
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |] 
paulson@15409
  1363
   ==> setprod f (Union C) = setprod (setprod f) C"
paulson@15409
  1364
apply (cases "finite C") 
paulson@15409
  1365
 prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
nipkow@15402
  1366
  apply (frule setprod_UN_disjoint [of C id f])
paulson@15409
  1367
 apply (unfold Union_def id_def, assumption+)
paulson@15409
  1368
done
nipkow@15402
  1369
nipkow@15402
  1370
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@16550
  1371
    (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
paulson@17189
  1372
    (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
ballarin@15765
  1373
by(simp add:setprod_def AC_mult.fold_Sigma split_def cong:setprod_cong)
nipkow@15402
  1374
paulson@15409
  1375
text{*Here we can eliminate the finiteness assumptions, by cases.*}
paulson@15409
  1376
lemma setprod_cartesian_product: 
paulson@17189
  1377
     "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
paulson@15409
  1378
apply (cases "finite A") 
paulson@15409
  1379
 apply (cases "finite B") 
paulson@15409
  1380
  apply (simp add: setprod_Sigma)
paulson@15409
  1381
 apply (cases "A={}", simp)
paulson@15409
  1382
 apply (simp add: setprod_1) 
paulson@15409
  1383
apply (auto simp add: setprod_def
paulson@15409
  1384
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
paulson@15409
  1385
done
nipkow@15402
  1386
nipkow@15402
  1387
lemma setprod_timesf:
paulson@15409
  1388
     "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
ballarin@15765
  1389
by(simp add:setprod_def AC_mult.fold_distrib)
nipkow@15402
  1390
nipkow@15402
  1391
nipkow@15402
  1392
subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402
  1393
nipkow@15402
  1394
lemma setprod_eq_1_iff [simp]:
nipkow@15402
  1395
    "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
berghofe@22262
  1396
  by (induct set: finite) auto
nipkow@15402
  1397
nipkow@15402
  1398
lemma setprod_zero:
nipkow@15402
  1399
     "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
berghofe@22262
  1400
  apply (induct set: finite, force, clarsimp)
nipkow@15402
  1401
  apply (erule disjE, auto)
nipkow@15402
  1402
  done
nipkow@15402
  1403
nipkow@15402
  1404
lemma setprod_nonneg [rule_format]:
nipkow@15402
  1405
     "(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
nipkow@15402
  1406
  apply (case_tac "finite A")
berghofe@22262
  1407
  apply (induct set: finite, force, clarsimp)
nipkow@15402
  1408
  apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
nipkow@15402
  1409
  apply (rule mult_mono, assumption+)
nipkow@15402
  1410
  apply (auto simp add: setprod_def)
nipkow@15402
  1411
  done
nipkow@15402
  1412
nipkow@15402
  1413
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
nipkow@15402
  1414
     --> 0 < setprod f A"
nipkow@15402
  1415
  apply (case_tac "finite A")
berghofe@22262
  1416
  apply (induct set: finite, force, clarsimp)
nipkow@15402
  1417
  apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
nipkow@15402
  1418
  apply (rule mult_strict_mono, assumption+)
nipkow@15402
  1419
  apply (auto simp add: setprod_def)
nipkow@15402
  1420
  done
nipkow@15402
  1421
nipkow@15402
  1422
lemma setprod_nonzero [rule_format]:
nipkow@15402
  1423
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
nipkow@15402
  1424
      finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
nipkow@15402
  1425
  apply (erule finite_induct, auto)
nipkow@15402
  1426
  done
nipkow@15402
  1427
nipkow@15402
  1428
lemma setprod_zero_eq:
nipkow@15402
  1429
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
nipkow@15402
  1430
     finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
nipkow@15402
  1431
  apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
nipkow@15402
  1432
  done
nipkow@15402
  1433
nipkow@15402
  1434
lemma setprod_nonzero_field:
nipkow@15402
  1435
    "finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0"
nipkow@15402
  1436
  apply (rule setprod_nonzero, auto)
nipkow@15402
  1437
  done
nipkow@15402
  1438
nipkow@15402
  1439
lemma setprod_zero_eq_field:
nipkow@15402
  1440
    "finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)"
nipkow@15402
  1441
  apply (rule setprod_zero_eq, auto)
nipkow@15402
  1442
  done
nipkow@15402
  1443
nipkow@15402
  1444
lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
nipkow@15402
  1445
    (setprod f (A Un B) :: 'a ::{field})
nipkow@15402
  1446
      = setprod f A * setprod f B / setprod f (A Int B)"
nipkow@15402
  1447
  apply (subst setprod_Un_Int [symmetric], auto)
nipkow@15402
  1448
  apply (subgoal_tac "finite (A Int B)")
nipkow@15402
  1449
  apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
nipkow@15402
  1450
  apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self)
nipkow@15402
  1451
  done
nipkow@15402
  1452
nipkow@15402
  1453
lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
nipkow@15402
  1454
    (setprod f (A - {a}) :: 'a :: {field}) =
nipkow@15402
  1455
      (if a:A then setprod f A / f a else setprod f A)"
nipkow@15402
  1456
  apply (erule finite_induct)
nipkow@15402
  1457
   apply (auto simp add: insert_Diff_if)
nipkow@15402
  1458
  apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a")
nipkow@15402
  1459
  apply (erule ssubst)
nipkow@15402
  1460
  apply (subst times_divide_eq_right [THEN sym])
nipkow@15402
  1461
  apply (auto simp add: mult_ac times_divide_eq_right divide_self)
nipkow@15402
  1462
  done
nipkow@15402
  1463
nipkow@15402
  1464
lemma setprod_inversef: "finite A ==>
nipkow@15402
  1465
    ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
nipkow@15402
  1466
      setprod (inverse \<circ> f) A = inverse (setprod f A)"
nipkow@15402
  1467
  apply (erule finite_induct)
nipkow@15402
  1468
  apply (simp, simp)
nipkow@15402
  1469
  done
nipkow@15402
  1470
nipkow@15402
  1471
lemma setprod_dividef:
nipkow@15402
  1472
     "[|finite A;
nipkow@15402
  1473
        \<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
nipkow@15402
  1474
      ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
nipkow@15402
  1475
  apply (subgoal_tac
nipkow@15402
  1476
         "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
nipkow@15402
  1477
  apply (erule ssubst)
nipkow@15402
  1478
  apply (subst divide_inverse)
nipkow@15402
  1479
  apply (subst setprod_timesf)
nipkow@15402
  1480
  apply (subst setprod_inversef, assumption+, rule refl)
nipkow@15402
  1481
  apply (rule setprod_cong, rule refl)
nipkow@15402
  1482
  apply (subst divide_inverse, auto)
nipkow@15402
  1483
  done
nipkow@15402
  1484
wenzelm@12396
  1485
subsection {* Finite cardinality *}
wenzelm@12396
  1486
nipkow@15402
  1487
text {* This definition, although traditional, is ugly to work with:
nipkow@15402
  1488
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
nipkow@15402
  1489
But now that we have @{text setsum} things are easy:
wenzelm@12396
  1490
*}
wenzelm@12396
  1491
wenzelm@12396
  1492
constdefs
wenzelm@12396
  1493
  card :: "'a set => nat"
nipkow@15402
  1494
  "card A == setsum (%x. 1::nat) A"
wenzelm@12396
  1495
wenzelm@12396
  1496
lemma card_empty [simp]: "card {} = 0"
nipkow@15402
  1497
  by (simp add: card_def)
nipkow@15402
  1498
paulson@15409
  1499
lemma card_infinite [simp]: "~ finite A ==> card A = 0"
paulson@15409
  1500
  by (simp add: card_def)
paulson@15409
  1501
nipkow@15402
  1502
lemma card_eq_setsum: "card A = setsum (%x. 1) A"
nipkow@15402
  1503
by (simp add: card_def)
wenzelm@12396
  1504
wenzelm@12396
  1505
lemma card_insert_disjoint [simp]:
wenzelm@12396
  1506
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
ballarin@15765
  1507
by(simp add: card_def)
nipkow@15402
  1508
nipkow@15402
  1509
lemma card_insert_if:
nipkow@15402
  1510
    "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
nipkow@15402
  1511
  by (simp add: insert_absorb)
wenzelm@12396
  1512
wenzelm@12396
  1513
lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
wenzelm@12396
  1514
  apply auto
paulson@15506
  1515
  apply (drule_tac a = x in mk_disjoint_insert, clarify, auto)
wenzelm@12396
  1516
  done
wenzelm@12396
  1517
paulson@15409
  1518
lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
paulson@15409
  1519
by auto
paulson@15409
  1520
wenzelm@12396
  1521
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
nipkow@14302
  1522
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
nipkow@14302
  1523
apply(simp del:insert_Diff_single)
nipkow@14302
  1524
done
wenzelm@12396
  1525
wenzelm@12396
  1526
lemma card_Diff_singleton:
wenzelm@12396
  1527
    "finite A ==> x: A ==> card (A - {x}) = card A - 1"
wenzelm@12396
  1528
  by (simp add: card_Suc_Diff1 [symmetric])
wenzelm@12396
  1529
wenzelm@12396
  1530
lemma card_Diff_singleton_if:
wenzelm@12396
  1531
    "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
wenzelm@12396
  1532
  by (simp add: card_Diff_singleton)
wenzelm@12396
  1533
wenzelm@12396
  1534
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
wenzelm@12396
  1535
  by (simp add: card_insert_if card_Suc_Diff1)
wenzelm@12396
  1536
wenzelm@12396
  1537
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
wenzelm@12396
  1538
  by (simp add: card_insert_if)
wenzelm@12396
  1539
nipkow@15402
  1540
lemma card_mono: "\<lbrakk> finite B; A \<subseteq> B \<rbrakk> \<Longrightarrow> card A \<le> card B"
nipkow@15539
  1541
by (simp add: card_def setsum_mono2)
nipkow@15402
  1542
wenzelm@12396
  1543
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
berghofe@22262
  1544
  apply (induct set: finite, simp, clarify)
wenzelm@12396
  1545
  apply (subgoal_tac "finite A & A - {x} <= F")
paulson@14208
  1546
   prefer 2 apply (blast intro: finite_subset, atomize)
wenzelm@12396
  1547
  apply (drule_tac x = "A - {x}" in spec)
wenzelm@12396
  1548
  apply (simp add: card_Diff_singleton_if split add: split_if_asm)
paulson@14208
  1549
  apply (case_tac "card A", auto)
wenzelm@12396
  1550
  done
wenzelm@12396
  1551
wenzelm@12396
  1552
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
wenzelm@12396
  1553
  apply (simp add: psubset_def linorder_not_le [symmetric])
wenzelm@12396
  1554
  apply (blast dest: card_seteq)
wenzelm@12396
  1555
  done
wenzelm@12396
  1556
wenzelm@12396
  1557
lemma card_Un_Int: "finite A ==> finite B
wenzelm@12396
  1558
    ==> card A + card B = card (A Un B) + card (A Int B)"
nipkow@15402
  1559
by(simp add:card_def setsum_Un_Int)
wenzelm@12396
  1560
wenzelm@12396
  1561
lemma card_Un_disjoint: "finite A ==> finite B
wenzelm@12396
  1562
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
wenzelm@12396
  1563
  by (simp add: card_Un_Int)
wenzelm@12396
  1564
wenzelm@12396
  1565
lemma card_Diff_subset:
nipkow@15402
  1566
  "finite B ==> B <= A ==> card (A - B) = card A - card B"
nipkow@15402
  1567
by(simp add:card_def setsum_diff_nat)
wenzelm@12396
  1568
wenzelm@12396
  1569
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
wenzelm@12396
  1570
  apply (rule Suc_less_SucD)
wenzelm@12396
  1571
  apply (simp add: card_Suc_Diff1)
wenzelm@12396
  1572
  done
wenzelm@12396
  1573
wenzelm@12396
  1574
lemma card_Diff2_less:
wenzelm@12396
  1575
    "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
wenzelm@12396
  1576
  apply (case_tac "x = y")
wenzelm@12396
  1577
   apply (simp add: card_Diff1_less)
wenzelm@12396
  1578
  apply (rule less_trans)
wenzelm@12396
  1579
   prefer 2 apply (auto intro!: card_Diff1_less)
wenzelm@12396
  1580
  done
wenzelm@12396
  1581
wenzelm@12396
  1582
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
wenzelm@12396
  1583
  apply (case_tac "x : A")
wenzelm@12396
  1584
   apply (simp_all add: card_Diff1_less less_imp_le)
wenzelm@12396
  1585
  done
wenzelm@12396
  1586
wenzelm@12396
  1587
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
paulson@14208
  1588
by (erule psubsetI, blast)
wenzelm@12396
  1589
paulson@14889
  1590
lemma insert_partition:
nipkow@15402
  1591
  "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
nipkow@15402
  1592
  \<Longrightarrow> x \<inter> \<Union> F = {}"
paulson@14889
  1593
by auto
paulson@14889
  1594
paulson@19793
  1595
text{* main cardinality theorem *}
paulson@14889
  1596
lemma card_partition [rule_format]:
paulson@14889
  1597
     "finite C ==>  
paulson@14889
  1598
        finite (\<Union> C) -->  
paulson@14889
  1599
        (\<forall>c\<in>C. card c = k) -->   
paulson@14889
  1600
        (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->  
paulson@14889
  1601
        k * card(C) = card (\<Union> C)"
paulson@14889
  1602
apply (erule finite_induct, simp)
paulson@14889
  1603
apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition 
paulson@14889
  1604
       finite_subset [of _ "\<Union> (insert x F)"])
paulson@14889
  1605
done
paulson@14889
  1606
wenzelm@12396
  1607
paulson@19793
  1608
text{*The form of a finite set of given cardinality*}
paulson@19793
  1609
paulson@19793
  1610
lemma card_eq_SucD:
paulson@19793
  1611
  assumes cardeq: "card A = Suc k" and fin: "finite A" 
paulson@19793
  1612
  shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k"
paulson@19793
  1613
proof -
paulson@19793
  1614
  have "card A \<noteq> 0" using cardeq by auto
paulson@19793
  1615
  then obtain b where b: "b \<in> A" using fin by auto
paulson@19793
  1616
  show ?thesis
paulson@19793
  1617
  proof (intro exI conjI)
paulson@19793
  1618
    show "A = insert b (A-{b})" using b by blast
paulson@19793
  1619
    show "b \<notin> A - {b}" by blast
paulson@19793
  1620
    show "card (A - {b}) = k" by (simp add: fin cardeq b card_Diff_singleton) 
paulson@19793
  1621
  qed
paulson@19793
  1622
qed
paulson@19793
  1623
paulson@19793
  1624
paulson@19793
  1625
lemma card_Suc_eq:
paulson@19793
  1626
  "finite A ==>
paulson@19793
  1627
   (card A = Suc k) = (\<exists>b B. A = insert b B & b \<notin> B & card B = k)"
paulson@19793
  1628
by (auto dest!: card_eq_SucD) 
paulson@19793
  1629
paulson@19793
  1630
lemma card_1_eq:
paulson@19793
  1631
  "finite A ==> (card A = Suc 0) = (\<exists>x. A = {x})"
paulson@19793
  1632
by (auto dest!: card_eq_SucD) 
paulson@19793
  1633
paulson@19793
  1634
lemma card_2_eq:
paulson@19793
  1635
  "finite A ==> (card A = Suc(Suc 0)) = (\<exists>x y. x\<noteq>y & A = {x,y})" 
paulson@19793
  1636
by (auto dest!: card_eq_SucD, blast) 
paulson@19793
  1637
paulson@19793
  1638
nipkow@15539
  1639
lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
nipkow@15539
  1640
apply (cases "finite A")
nipkow@15539
  1641
apply (erule finite_induct)
nipkow@15539
  1642
apply (auto simp add: ring_distrib add_ac)
paulson@15409
  1643
done
nipkow@15402
  1644
nipkow@15539
  1645
krauss@21199
  1646
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{recpower, comm_monoid_mult})) = y^(card A)"
nipkow@15402
  1647
  apply (erule finite_induct)
nipkow@15402
  1648
  apply (auto simp add: power_Suc)
nipkow@15402
  1649
  done
nipkow@15402
  1650
nipkow@15542
  1651
lemma setsum_bounded:
nipkow@15542
  1652
  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{comm_semiring_1_cancel, pordered_ab_semigroup_add})"
nipkow@15542
  1653
  shows "setsum f A \<le> of_nat(card A) * K"
nipkow@15542
  1654
proof (cases "finite A")
nipkow@15542
  1655
  case True
nipkow@15542
  1656
  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
nipkow@15542
  1657
next
nipkow@15542
  1658
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15542
  1659
qed
nipkow@15542
  1660
nipkow@15402
  1661
nipkow@15402
  1662
subsubsection {* Cardinality of unions *}
nipkow@15402
  1663
nipkow@15539
  1664
lemma of_nat_id[simp]: "(of_nat n :: nat) = n"
wenzelm@21575
  1665
by(induct n) auto
nipkow@15539
  1666
nipkow@15402
  1667
lemma card_UN_disjoint:
nipkow@15402
  1668
    "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402
  1669
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402
  1670
      card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
nipkow@15539
  1671
  apply (simp add: card_def del: setsum_constant)
nipkow@15402
  1672
  apply (subgoal_tac
nipkow@15402
  1673
           "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
nipkow@15539
  1674
  apply (simp add: setsum_UN_disjoint del: setsum_constant)
nipkow@15539
  1675
  apply (simp cong: setsum_cong)
nipkow@15402
  1676
  done
nipkow@15402
  1677
nipkow@15402
  1678
lemma card_Union_disjoint:
nipkow@15402
  1679
  "finite C ==> (ALL A:C. finite A) ==>
nipkow@15402
  1680
        (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
nipkow@15402
  1681
      card (Union C) = setsum card C"
nipkow@15402
  1682
  apply (frule card_UN_disjoint [of C id])
nipkow@15402
  1683
  apply (unfold Union_def id_def, assumption+)
nipkow@15402
  1684
  done
nipkow@15402
  1685
wenzelm@12396
  1686
subsubsection {* Cardinality of image *}
wenzelm@12396
  1687
paulson@15447
  1688
text{*The image of a finite set can be expressed using @{term fold}.*}
paulson@15447
  1689
lemma image_eq_fold: "finite A ==> f ` A = fold (op Un) (%x. {f x}) {} A"
paulson@15447
  1690
  apply (erule finite_induct, simp)
paulson@15447
  1691
  apply (subst ACf.fold_insert) 
paulson@15447
  1692
  apply (auto simp add: ACf_def) 
paulson@15447
  1693
  done
paulson@15447
  1694
wenzelm@12396
  1695
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
berghofe@22262
  1696
  apply (induct set: finite)
wenzelm@21575
  1697
   apply simp
wenzelm@12396
  1698
  apply (simp add: le_SucI finite_imageI card_insert_if)
wenzelm@12396
  1699
  done
wenzelm@12396
  1700
nipkow@15402
  1701
lemma card_image: "inj_on f A ==> card (f ` A) = card A"
nipkow@15539
  1702
by(simp add:card_def setsum_reindex o_def del:setsum_constant)
wenzelm@12396
  1703
wenzelm@12396
  1704
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
wenzelm@12396
  1705
  by (simp add: card_seteq card_image)
wenzelm@12396
  1706
nipkow@15111
  1707
lemma eq_card_imp_inj_on:
nipkow@15111
  1708
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
wenzelm@21575
  1709
apply (induct rule:finite_induct)
wenzelm@21575
  1710
apply simp
nipkow@15111
  1711
apply(frule card_image_le[where f = f])
nipkow@15111
  1712
apply(simp add:card_insert_if split:if_splits)
nipkow@15111
  1713
done
nipkow@15111
  1714
nipkow@15111
  1715
lemma inj_on_iff_eq_card:
nipkow@15111
  1716
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
nipkow@15111
  1717
by(blast intro: card_image eq_card_imp_inj_on)
nipkow@15111
  1718
wenzelm@12396
  1719
nipkow@15402
  1720
lemma card_inj_on_le:
nipkow@15402
  1721
    "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
nipkow@15402
  1722
apply (subgoal_tac "finite A") 
nipkow@15402
  1723
 apply (force intro: card_mono simp add: card_image [symmetric])
nipkow@15402
  1724
apply (blast intro: finite_imageD dest: finite_subset) 
nipkow@15402
  1725
done
nipkow@15402
  1726
nipkow@15402
  1727
lemma card_bij_eq:
nipkow@15402
  1728
    "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
nipkow@15402
  1729
       finite A; finite B |] ==> card A = card B"
nipkow@15402
  1730
  by (auto intro: le_anti_sym card_inj_on_le)
nipkow@15402
  1731
nipkow@15402
  1732
nipkow@15402
  1733
subsubsection {* Cardinality of products *}
nipkow@15402
  1734
nipkow@15402
  1735
(*
nipkow@15402
  1736
lemma SigmaI_insert: "y \<notin> A ==>
nipkow@15402
  1737
  (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
nipkow@15402
  1738
  by auto
nipkow@15402
  1739
*)
nipkow@15402
  1740
nipkow@15402
  1741
lemma card_SigmaI [simp]:
nipkow@15402
  1742
  "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
nipkow@15402
  1743
  \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
nipkow@15539
  1744
by(simp add:card_def setsum_Sigma del:setsum_constant)
nipkow@15402
  1745
paulson@15409
  1746
lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
paulson@15409
  1747
apply (cases "finite A") 
paulson@15409
  1748
apply (cases "finite B") 
paulson@15409
  1749
apply (auto simp add: card_eq_0_iff
nipkow@15539
  1750
            dest: finite_cartesian_productD1 finite_cartesian_productD2)
paulson@15409
  1751
done
nipkow@15402
  1752
nipkow@15402
  1753
lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
nipkow@15539
  1754
by (simp add: card_cartesian_product)
paulson@15409
  1755
nipkow@15402
  1756
nipkow@15402
  1757
wenzelm@12396
  1758
subsubsection {* Cardinality of the Powerset *}
wenzelm@12396
  1759
wenzelm@12396
  1760
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
berghofe@22262
  1761
  apply (induct set: finite)
wenzelm@12396
  1762
   apply (simp_all add: Pow_insert)
paulson@14208
  1763
  apply (subst card_Un_disjoint, blast)
paulson@14208
  1764
    apply (blast intro: finite_imageI, blast)
wenzelm@12396
  1765
  apply (subgoal_tac "inj_on (insert x) (Pow F)")
wenzelm@12396
  1766
   apply (simp add: card_image Pow_insert)
wenzelm@12396
  1767
  apply (unfold inj_on_def)
wenzelm@12396
  1768
  apply (blast elim!: equalityE)
wenzelm@12396
  1769
  done
wenzelm@12396
  1770
nipkow@15392
  1771
text {* Relates to equivalence classes.  Based on a theorem of
nipkow@15392
  1772
F. Kammüller's.  *}
wenzelm@12396
  1773
wenzelm@12396
  1774
lemma dvd_partition:
nipkow@15392
  1775
  "finite (Union C) ==>
wenzelm@12396
  1776
    ALL c : C. k dvd card c ==>
paulson@14430
  1777
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
wenzelm@12396
  1778
  k dvd card (Union C)"
nipkow@15392
  1779
apply(frule finite_UnionD)
nipkow@15392
  1780
apply(rotate_tac -1)
berghofe@22262
  1781
  apply (induct set: finite, simp_all, clarify)
wenzelm@12396
  1782
  apply (subst card_Un_disjoint)
wenzelm@12396
  1783
  apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
wenzelm@12396
  1784
  done
wenzelm@12396
  1785
wenzelm@12396
  1786
nipkow@15392
  1787
subsection{* A fold functional for non-empty sets *}
nipkow@15392
  1788
nipkow@15392
  1789
text{* Does not require start value. *}
wenzelm@12396
  1790
berghofe@22262
  1791
inductive2
berghofe@22262
  1792
  fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
berghofe@22262
  1793
  for f :: "'a => 'a => 'a"
berghofe@22262
  1794
where
paulson@15506
  1795
  fold1Set_insertI [intro]:
berghofe@22262
  1796
   "\<lbrakk> foldSet f id a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
wenzelm@12396
  1797
nipkow@15392
  1798
constdefs
nipkow@15392
  1799
  fold1 :: "('a => 'a => 'a) => 'a set => 'a"
berghofe@22262
  1800
  "fold1 f A == THE x. fold1Set f A x"
paulson@15506
  1801
paulson@15506
  1802
lemma fold1Set_nonempty:
berghofe@22262
  1803
 "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
paulson@15506
  1804
by(erule fold1Set.cases, simp_all) 
paulson@15506
  1805
nipkow@15392
  1806
berghofe@22262
  1807
inductive_cases2 empty_fold1SetE [elim!]: "fold1Set f {} x"
berghofe@22262
  1808
berghofe@22262
  1809
inductive_cases2 insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
berghofe@22262
  1810
berghofe@22262
  1811
berghofe@22262
  1812
lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
paulson@15506
  1813
  by (blast intro: foldSet.intros elim: foldSet.cases)
nipkow@15392
  1814
paulson@15508
  1815
lemma fold1_singleton[simp]: "fold1 f {a} = a"
paulson@15508
  1816
  by (unfold fold1_def) blast
wenzelm@12396
  1817
paulson@15508
  1818
lemma finite_nonempty_imp_fold1Set:
berghofe@22262
  1819
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
paulson@15508
  1820
apply (induct A rule: finite_induct)
paulson@15508
  1821
apply (auto dest: finite_imp_foldSet [of _ f id])  
paulson@15508
  1822
done
paulson@15506
  1823
paulson@15506
  1824
text{*First, some lemmas about @{term foldSet}.*}
nipkow@15392
  1825
paulson@15508
  1826
lemma (in ACf) foldSet_insert_swap:
berghofe@22262
  1827
assumes fold: "foldSet f id b A y"
berghofe@22262
  1828
shows "b \<notin> A \<Longrightarrow> foldSet f id z (insert b A) (z \<cdot> y)"
paulson@15508
  1829
using fold
paulson@15508
  1830
proof (induct rule: foldSet.induct)
paulson@15508
  1831
  case emptyI thus ?case by (force simp add: fold_insert_aux commute)
paulson@15508
  1832
next
berghofe@22262
  1833
  case (insertI x A y)
berghofe@22262
  1834
    have "foldSet f (\<lambda>u. u) z (insert x (insert b A)) (x \<cdot> (z \<cdot> y))"
paulson@15521
  1835
      using insertI by force  --{*how does @{term id} get unfolded?*}
paulson@15508
  1836
    thus ?case by (simp add: insert_commute AC)
paulson@15508
  1837
qed
paulson@15508
  1838
paulson@15508
  1839
lemma (in ACf) foldSet_permute_diff:
berghofe@22262
  1840
assumes fold: "foldSet f id b A x"
berghofe@22262
  1841
shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> foldSet f id a (insert b (A-{a})) x"
paulson@15508
  1842
using fold
paulson@15508
  1843
proof (induct rule: foldSet.induct)
paulson@15508
  1844
  case emptyI thus ?case by simp
paulson@15508
  1845
next
berghofe@22262
  1846
  case (insertI x A y)
paulson@15521
  1847
  have "a = x \<or> a \<in> A" using insertI by simp
paulson@15521
  1848
  thus ?case
paulson@15521
  1849
  proof
paulson@15521
  1850
    assume "a = x"
paulson@15521
  1851
    with insertI show ?thesis
paulson@15521
  1852
      by (simp add: id_def [symmetric], blast intro: foldSet_insert_swap) 
paulson@15521
  1853
  next
paulson@15521
  1854
    assume ainA: "a \<in> A"
berghofe@22262
  1855
    hence "foldSet f id a (insert x (insert b (A - {a}))) (x \<cdot> y)"
paulson@15521
  1856
      using insertI by (force simp: id_def)
paulson@15521
  1857
    moreover
paulson@15521
  1858
    have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
paulson@15521
  1859
      using ainA insertI by blast
paulson@15521
  1860
    ultimately show ?thesis by (simp add: id_def)
paulson@15508
  1861
  qed
paulson@15508
  1862
qed
paulson@15508
  1863
paulson@15508
  1864
lemma (in ACf) fold1_eq_fold:
paulson@15508
  1865
     "[|finite A; a \<notin> A|] ==> fold1 f (insert a A) = fold f id a A"
paulson@15508
  1866
apply (simp add: fold1_def fold_def) 
paulson@15508
  1867
apply (rule the_equality)
paulson@15508
  1868
apply (best intro: foldSet_determ theI dest: finite_imp_foldSet [of _ f id]) 
paulson@15508
  1869
apply (rule sym, clarify)
paulson@15508
  1870
apply (case_tac "Aa=A")
paulson@15508
  1871
 apply (best intro: the_equality foldSet_determ)  
berghofe@22262
  1872
apply (subgoal_tac "foldSet f id a A x")
paulson@15508
  1873
 apply (best intro: the_equality foldSet_determ)  
paulson@15508
  1874
apply (subgoal_tac "insert aa (Aa - {a}) = A") 
paulson@15508
  1875
 prefer 2 apply (blast elim: equalityE) 
paulson@15508
  1876
apply (auto dest: foldSet_permute_diff [where a=a]) 
paulson@15508
  1877
done
paulson@15508
  1878
paulson@15521
  1879
lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
paulson@15521
  1880
apply safe
paulson@15521
  1881
apply simp 
paulson@15521
  1882
apply (drule_tac x=x in spec)
paulson@15521
  1883
apply (drule_tac x="A-{x}" in spec, auto) 
paulson@15508
  1884
done
paulson@15508
  1885
paulson@15521
  1886
lemma (in ACf) fold1_insert:
paulson@15521
  1887
  assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
paulson@15521
  1888
  shows "fold1 f (insert x A) = f x (fold1 f A)"
paulson@15521
  1889
proof -
paulson@15521
  1890
  from nonempty obtain a A' where "A = insert a A' & a ~: A'" 
paulson@15521
  1891
    by (auto simp add: nonempty_iff)
paulson@15521
  1892
  with A show ?thesis
paulson@15521
  1893
    by (simp add: insert_commute [of x] fold1_eq_fold eq_commute) 
paulson@15521
  1894
qed
paulson@15521
  1895
paulson@15509
  1896
lemma (in ACIf) fold1_insert_idem [simp]:
paulson@15521
  1897
  assumes nonempty: "A \<noteq> {}" and A: "finite A" 
paulson@15521
  1898
  shows "fold1 f (insert x A) = f x (fold1 f A)"
paulson@15521
  1899
proof -
paulson@15521
  1900
  from nonempty obtain a A' where A': "A = insert a A' & a ~: A'" 
paulson@15521
  1901
    by (auto simp add: nonempty_iff)
paulson@15521
  1902
  show ?thesis
paulson@15521
  1903
  proof cases
paulson@15521
  1904
    assume "a = x"
paulson@15521
  1905
    thus ?thesis 
paulson@15521
  1906
    proof cases
paulson@15521
  1907
      assume "A' = {}"
paulson@15521
  1908
      with prems show ?thesis by (simp add: idem) 
paulson@15521
  1909
    next
paulson@15521
  1910
      assume "A' \<noteq> {}"
paulson@15521
  1911
      with prems show ?thesis
paulson@15521
  1912
	by (simp add: fold1_insert assoc [symmetric] idem) 
paulson@15521
  1913
    qed
paulson@15521
  1914
  next
paulson@15521
  1915
    assume "a \<noteq> x"
paulson@15521
  1916
    with prems show ?thesis
paulson@15521
  1917
      by (simp add: insert_commute fold1_eq_fold fold_insert_idem)
paulson@15521
  1918
  qed
paulson@15521
  1919
qed
paulson@15506
  1920
paulson@15506
  1921
paulson@15508
  1922
text{* Now the recursion rules for definitions: *}
paulson@15508
  1923
paulson@15508
  1924
lemma fold1_singleton_def: "g \<equiv> fold1 f \<Longrightarrow> g {a} = a"
paulson@15508
  1925
by(simp add:fold1_singleton)
paulson@15508
  1926
paulson@15508
  1927
lemma (in ACf) fold1_insert_def:
paulson@15508
  1928
  "\<lbrakk> g \<equiv> fold1 f; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
paulson@15508
  1929
by(simp add:fold1_insert)
paulson@15508
  1930
paulson@15509
  1931
lemma (in ACIf) fold1_insert_idem_def:
paulson@15508
  1932
  "\<lbrakk> g \<equiv> fold1 f; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
paulson@15509
  1933
by(simp add:fold1_insert_idem)
paulson@15508
  1934
paulson@15508
  1935
subsubsection{* Determinacy for @{term fold1Set} *}
paulson@15508
  1936
paulson@15508
  1937
text{*Not actually used!!*}
wenzelm@12396
  1938
paulson@15506
  1939
lemma (in ACf) foldSet_permute:
berghofe@22262
  1940
  "[|foldSet f id b (insert a A) x; a \<notin> A; b \<notin> A|]
berghofe@22262
  1941
   ==> foldSet f id a (insert b A) x"
paulson@15506
  1942
apply (case_tac "a=b") 
paulson@15506
  1943
apply (auto dest: foldSet_permute_diff) 
paulson@15506
  1944
done
nipkow@15376
  1945
paulson@15506
  1946
lemma (in ACf) fold1Set_determ:
berghofe@22262
  1947
  "fold1Set f A x ==> fold1Set f A y ==> y = x"
paulson@15506
  1948
proof (clarify elim!: fold1Set.cases)
paulson@15506
  1949
  fix A x B y a b
berghofe@22262
  1950
  assume Ax: "foldSet f id a A x"
berghofe@22262
  1951
  assume By: "foldSet f id b B y"
paulson@15506
  1952
  assume anotA:  "a \<notin> A"
paulson@15506
  1953
  assume bnotB:  "b \<notin> B"
paulson@15506
  1954
  assume eq: "insert a A = insert b B"
paulson@15506
  1955
  show "y=x"
paulson@15506
  1956
  proof cases
paulson@15506
  1957
    assume same: "a=b"
paulson@15506
  1958
    hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
paulson@15506
  1959
    thus ?thesis using Ax By same by (blast intro: foldSet_determ)
nipkow@15392
  1960
  next
paulson@15506
  1961
    assume diff: "a\<noteq>b"
paulson@15506
  1962
    let ?D = "B - {a}"
paulson@15506
  1963
    have B: "B = insert a ?D" and A: "A = insert b ?D"
paulson@15506
  1964
     and aB: "a \<in> B" and bA: "b \<in> A"
paulson@15506
  1965
      using eq anotA bnotB diff by (blast elim!:equalityE)+
paulson@15506
  1966
    with aB bnotB By
berghofe@22262
  1967
    have "foldSet f id a (insert b ?D) y" 
paulson@15506
  1968
      by (auto intro: foldSet_permute simp add: insert_absorb)
paulson@15506
  1969
    moreover
berghofe@22262
  1970
    have "foldSet f id a (insert b ?D) x"
paulson@15506
  1971
      by (simp add: A [symmetric] Ax) 
paulson@15506
  1972
    ultimately show ?thesis by (blast intro: foldSet_determ) 
nipkow@15392
  1973
  qed
wenzelm@12396
  1974
qed
wenzelm@12396
  1975
berghofe@22262
  1976
lemma (in ACf) fold1Set_equality: "fold1Set f A y ==> fold1 f A = y"
paulson@15506
  1977
  by (unfold fold1_def) (blast intro: fold1Set_determ)
paulson@15506
  1978
paulson@15506
  1979
declare
paulson@15506
  1980
  empty_foldSetE [rule del]   foldSet.intros [rule del]
paulson@15506
  1981
  empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
ballarin@19931
  1982
  -- {* No more proofs involve these relations. *}
nipkow@15376
  1983
nipkow@15497
  1984
subsubsection{* Semi-Lattices *}
nipkow@15497
  1985
nipkow@15497
  1986
locale ACIfSL = ACIf +
nipkow@15500
  1987
  fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubseteq>" 50)
nipkow@18493
  1988
  and strict_below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubset>" 50)
nipkow@15500
  1989
  assumes below_def: "(x \<sqsubseteq> y) = (x\<cdot>y = x)"
nipkow@18493
  1990
  defines strict_below_def:  "(x \<sqsubset> y) \<equiv> (x \<sqsubseteq> y \<and> x \<noteq> y)"
nipkow@15497
  1991
nipkow@15497
  1992
locale ACIfSLlin = ACIfSL +
nipkow@15497
  1993
  assumes lin: "x\<cdot>y \<in> {x,y}"
nipkow@15497
  1994
nipkow@15500
  1995
lemma (in ACIfSL) below_refl[simp]: "x \<sqsubseteq> x"
nipkow@15497
  1996
by(simp add: below_def idem)
nipkow@15497
  1997
nipkow@15500
  1998
lemma (in ACIfSL) below_f_conv[simp]: "x \<sqsubseteq> y \<cdot> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
nipkow@15497
  1999
proof
nipkow@15500
  2000
  assume "x \<sqsubseteq> y \<cdot> z"
nipkow@15497
  2001
  hence xyzx: "x \<cdot> (y \<cdot> z) = x"  by(simp add: below_def)
nipkow@15497
  2002
  have "x \<cdot> y = x"
nipkow@15497
  2003
  proof -
nipkow@15497
  2004
    have "x \<cdot> y = (x \<cdot> (y \<cdot> z)) \<cdot> y" by(rule subst[OF xyzx], rule refl)
nipkow@15497
  2005
    also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
nipkow@15497
  2006
    also have "\<dots> = x" by(rule xyzx)
nipkow@15497
  2007
    finally show ?thesis .
nipkow@15497
  2008
  qed
nipkow@15497
  2009
  moreover have "x \<cdot> z = x"
nipkow@15497
  2010
  proof -
nipkow@15497
  2011
    have "x \<cdot> z = (x \<cdot> (y \<cdot> z)) \<cdot> z" by(rule subst[OF xyzx], rule refl)
nipkow@15497
  2012
    also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
nipkow@15497
  2013
    also have "\<dots> = x" by(rule xyzx)
nipkow@15497
  2014
    finally show ?thesis .
nipkow@15497
  2015
  qed
nipkow@15500
  2016
  ultimately show "x \<sqsubseteq> y \<and> x \<sqsubseteq> z" by(simp add: below_def)
nipkow@15497
  2017
next
nipkow@15500
  2018
  assume a: "x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
nipkow@15497
  2019
  hence y: "x \<cdot> y = x" and z: "x \<cdot> z = x" by(simp_all add: below_def)
nipkow@15497
  2020
  have "x \<cdot> (y \<cdot> z) = (x \<cdot> y) \<cdot> z" by(simp add:assoc)
nipkow@15497
  2021
  also have "x \<cdot> y = x" using a by(simp_all add: below_def)
nipkow@15497
  2022
  also have "x \<cdot> z = x" using a by(simp_all add: below_def)
nipkow@15500
  2023
  finally show "x \<sqsubseteq> y \<cdot> z" by(simp_all add: below_def)
nipkow@15497
  2024
qed
nipkow@15497
  2025
nipkow@15497
  2026
lemma (in ACIfSLlin) above_f_conv:
nipkow@15500
  2027
 "x \<cdot> y \<sqsubseteq> z = (x \<sqsubseteq> z \<or> y \<sqsubseteq> z)"
nipkow@15497
  2028
proof
nipkow@15500
  2029
  assume a: "x \<cdot> y \<sqsubseteq> z"
nipkow@15497
  2030
  have "x \<cdot> y = x \<or> x \<cdot> y = y" using lin[of x y] by simp
nipkow@15500
  2031
  thus "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
nipkow@15497
  2032
  proof
nipkow@15500
  2033
    assume "x \<cdot> y = x" hence "x \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
nipkow@15497
  2034
  next
nipkow@15500
  2035
    assume "x \<cdot> y = y" hence "y \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
nipkow@15497
  2036
  qed
nipkow@15497
  2037
next
nipkow@15500
  2038
  assume "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
nipkow@15500
  2039
  thus "x \<cdot> y \<sqsubseteq> z"
nipkow@15497
  2040
  proof
nipkow@15500
  2041
    assume a: "x \<sqsubseteq> z"
nipkow@15497
  2042
    have "(x \<cdot> y) \<cdot> z = (x \<cdot> z) \<cdot> y" by(simp add:ACI)
nipkow@15497
  2043
    also have "x \<cdot> z = x" using a by(simp add:below_def)
nipkow@15500
  2044
    finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
nipkow@15497
  2045
  next
nipkow@15500
  2046
    assume a: "y \<sqsubseteq> z"
nipkow@15497
  2047
    have "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
nipkow@15497
  2048
    also have "y \<cdot> z = y" using a by(simp add:below_def)
nipkow@15500
  2049
    finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
nipkow@15497
  2050
  qed
nipkow@15497
  2051
qed
nipkow@15497
  2052
nipkow@15497
  2053
nipkow@18493
  2054
lemma (in ACIfSLlin) strict_below_f_conv[simp]: "x \<sqsubset> y \<cdot> z = (x \<sqsubset> y \<and> x \<sqsubset> z)"
nipkow@18493
  2055
apply(simp add: strict_below_def)
nipkow@18493
  2056
using lin[of y z] by (auto simp:below_def ACI)
nipkow@18493
  2057
nipkow@18493
  2058
nipkow@18493
  2059
lemma (in ACIfSLlin) strict_above_f_conv:
ballarin@19931
  2060
  "x \<cdot> y \<sqsubset> z = (x \<sqsubset> z \<or> y \<sqsubset> z)"
nipkow@18493
  2061
apply(simp add: strict_below_def above_f_conv)
nipkow@18493
  2062
using lin[of y z] lin[of x z] by (auto simp:below_def ACI)
nipkow@18493
  2063
nipkow@18493
  2064
nipkow@15502
  2065
subsubsection{* Lemmas about @{text fold1} *}
nipkow@15484
  2066
nipkow@15484
  2067
lemma (in ACf) fold1_Un:
nipkow@15484
  2068
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  2069
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
nipkow@15484
  2070
       fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
nipkow@15484
  2071
using A
nipkow@15484
  2072
proof(induct rule:finite_ne_induct)
nipkow@15484
  2073
  case singleton thus ?case by(simp add:fold1_insert)
nipkow@15484
  2074
next
nipkow@15484
  2075
  case insert thus ?case by (simp add:fold1_insert assoc)
nipkow@15484
  2076
qed
nipkow@15484
  2077
nipkow@15484
  2078
lemma (in ACIf) fold1_Un2:
nipkow@15484
  2079
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  2080
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
nipkow@15484
  2081
       fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
nipkow@15484
  2082
using A
nipkow@15484
  2083
proof(induct rule:finite_ne_induct)
paulson@15509
  2084
  case singleton thus ?case by(simp add:fold1_insert_idem)
nipkow@15484
  2085
next
paulson@15509
  2086
  case insert thus ?case by (simp add:fold1_insert_idem assoc)
nipkow@15484
  2087
qed
nipkow@15484
  2088
nipkow@15484
  2089
lemma (in ACf) fold1_in:
nipkow@15484
  2090
  assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x\<cdot>y \<in> {x,y}"
nipkow@15484
  2091
  shows "fold1 f A \<in> A"
nipkow@15484
  2092
using A
nipkow@15484
  2093
proof (induct rule:finite_ne_induct)
paulson@15506
  2094
  case singleton thus ?case by simp
nipkow@15484
  2095
next
nipkow@15484
  2096
  case insert thus ?case using elem by (force simp add:fold1_insert)
nipkow@15484
  2097
qed
nipkow@15484
  2098
nipkow@15497
  2099
lemma (in ACIfSL) below_fold1_iff:
nipkow@15497
  2100
assumes A: "finite A" "A \<noteq> {}"
nipkow@15500
  2101
shows "x \<sqsubseteq> fold1 f A = (\<forall>a\<in>A. x \<sqsubseteq> a)"
nipkow@15497
  2102
using A
nipkow@15497
  2103
by(induct rule:finite_ne_induct) simp_all
nipkow@15497
  2104
nipkow@18493
  2105
lemma (in ACIfSLlin) strict_below_fold1_iff:
nipkow@18493
  2106
  "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> x \<sqsubset> fold1 f A = (\<forall>a\<in>A. x \<sqsubset> a)"
nipkow@18493
  2107
by(induct rule:finite_ne_induct) simp_all
nipkow@18493
  2108
nipkow@18493
  2109
nipkow@15497
  2110
lemma (in ACIfSL) fold1_belowI:
nipkow@15497
  2111
assumes A: "finite A" "A \<noteq> {}"
nipkow@15500
  2112
shows "a \<in> A \<Longrightarrow> fold1 f A \<sqsubseteq> a"
nipkow@15484
  2113
using A
nipkow@15484
  2114
proof (induct rule:finite_ne_induct)
nipkow@15497
  2115
  case singleton thus ?case by simp
nipkow@15484
  2116
next
nipkow@15497
  2117
  case (insert x F)
berghofe@15517
  2118
  from insert(5) have "a = x \<or> a \<in> F" by simp
nipkow@15497
  2119
  thus ?case
nipkow@15497
  2120
  proof
nipkow@15497
  2121
    assume "a = x" thus ?thesis using insert by(simp add:below_def ACI)
nipkow@15497
  2122
  next
nipkow@15497
  2123
    assume "a \<in> F"
paulson@15508
  2124
    hence bel: "fold1 f F \<sqsubseteq> a" by(rule insert)
paulson@15508
  2125
    have "fold1 f (insert x F) \<cdot> a = x \<cdot> (fold1 f F \<cdot> a)"
nipkow@15497
  2126
      using insert by(simp add:below_def ACI)
paulson@15508
  2127
    also have "fold1 f F \<cdot> a = fold1 f F"
nipkow@15497
  2128
      using bel  by(simp add:below_def ACI)
paulson@15508
  2129
    also have "x \<cdot> \<dots> = fold1 f (insert x F)"
nipkow@15497
  2130
      using insert by(simp add:below_def ACI)
nipkow@15497
  2131
    finally show ?thesis  by(simp add:below_def)
nipkow@15497
  2132
  qed
nipkow@15484
  2133
qed
nipkow@15484
  2134
nipkow@18493
  2135
nipkow@15497
  2136
lemma (in ACIfSLlin) fold1_below_iff:
nipkow@15497
  2137
assumes A: "finite A" "A \<noteq> {}"
nipkow@15500
  2138
shows "fold1 f A \<sqsubseteq> x = (\<exists>a\<in>A. a \<sqsubseteq> x)"
nipkow@15484
  2139
using A
nipkow@15497
  2140
by(induct rule:finite_ne_induct)(simp_all add:above_f_conv)
nipkow@15484
  2141
nipkow@18493
  2142
lemma (in ACIfSLlin) fold1_strict_below_iff:
nipkow@18493
  2143
assumes A: "finite A" "A \<noteq> {}"
nipkow@18493
  2144
shows "fold1 f A \<sqsubset> x = (\<exists>a\<in>A. a \<sqsubset> x)"
nipkow@18493
  2145
using A
nipkow@18493
  2146
by(induct rule:finite_ne_induct)(simp_all add:strict_above_f_conv)
nipkow@18493
  2147
nipkow@15512
  2148
nipkow@18423
  2149
lemma (in ACIfSLlin) fold1_antimono:
nipkow@18423
  2150
assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B"
nipkow@18423
  2151
shows "fold1 f B \<sqsubseteq> fold1 f A"
nipkow@18423
  2152
proof(cases)
nipkow@18423
  2153
  assume "A = B" thus ?thesis by simp
nipkow@18423
  2154
next
nipkow@18423
  2155
  assume "A \<noteq> B"
nipkow@18423
  2156
  have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast
nipkow@18423
  2157
  have "fold1 f B = fold1 f (A \<union> (B-A))" by(subst B)(rule refl)
nipkow@18423
  2158
  also have "\<dots> = f (fold1 f A) (fold1 f (B-A))"
nipkow@18423
  2159
  proof -
nipkow@18423
  2160
    have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])
nipkow@18493
  2161
    moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`]) (* by(blast intro:finite_Diff prems) fails *)
nipkow@18423
  2162
    moreover have "(B-A) \<noteq> {}" using prems by blast
nipkow@18423
  2163
    moreover have "A Int (B-A) = {}" using prems by blast
nipkow@18423
  2164
    ultimately show ?thesis using `A \<noteq> {}` by(rule_tac fold1_Un)
nipkow@18423
  2165
  qed
nipkow@18423
  2166
  also have "\<dots> \<sqsubseteq> fold1 f A" by(simp add: above_f_conv)
nipkow@18423
  2167
  finally show ?thesis .
nipkow@18423
  2168
qed
nipkow@18423
  2169
nipkow@18423
  2170
nipkow@18493
  2171
nipkow@15500
  2172
subsubsection{* Lattices *}
nipkow@15500
  2173
nipkow@15512
  2174
locale Lattice = lattice +
nipkow@15512
  2175
  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
nipkow@15500
  2176
  and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
nipkow@15500
  2177
  defines "Inf == fold1 inf"  and "Sup == fold1 sup"
nipkow@15500
  2178
nipkow@15512
  2179
locale Distrib_Lattice = distrib_lattice + Lattice
nipkow@15504
  2180
nipkow@15500
  2181
text{* Lattices are semilattices *}
nipkow@15500
  2182
nipkow@15500
  2183
lemma (in Lattice) ACf_inf: "ACf inf"
nipkow@15512
  2184
by(blast intro: ACf.intro inf_commute inf_assoc)
nipkow@15500
  2185
nipkow@15500
  2186
lemma (in Lattice) ACf_sup: "ACf sup"
nipkow@15512
  2187
by(blast intro: ACf.intro sup_commute sup_assoc)
nipkow@15500
  2188
nipkow@15500
  2189
lemma (in Lattice) ACIf_inf: "ACIf inf"
nipkow@15500
  2190
apply(rule ACIf.intro)
nipkow@15500
  2191
apply(rule ACf_inf)
nipkow@15500
  2192
apply(rule ACIf_axioms.intro)
nipkow@15500
  2193
apply(rule inf_idem)
nipkow@15500
  2194
done
nipkow@15500
  2195
nipkow@15500
  2196
lemma (in Lattice) ACIf_sup: "ACIf sup"
nipkow@15500
  2197
apply(rule ACIf.intro)
nipkow@15500
  2198
apply(rule ACf_sup)
nipkow@15500
  2199
apply(rule ACIf_axioms.intro)
nipkow@15500
  2200
apply(rule sup_idem)
nipkow@15500
  2201
done
nipkow@15500
  2202
nipkow@15500
  2203
lemma (in Lattice) ACIfSL_inf: "ACIfSL inf (op \<sqsubseteq>)"
nipkow@15500
  2204
apply(rule ACIfSL.intro)
ballarin@19931
  2205
apply(rule ACIf.intro)
nipkow@15500
  2206
apply(rule ACf_inf)
nipkow@15500
  2207
apply(rule ACIf.axioms[OF ACIf_inf])
nipkow@15500
  2208
apply(rule ACIfSL_axioms.intro)
nipkow@15500
  2209
apply(rule iffI)
nipkow@21733
  2210
 apply(blast intro: antisym inf_le1 inf_le2 inf_greatest refl)
nipkow@15500
  2211
apply(erule subst)
nipkow@15500
  2212
apply(rule inf_le2)
nipkow@15500
  2213
done
nipkow@15500
  2214
nipkow@15500
  2215
lemma (in Lattice) ACIfSL_sup: "ACIfSL sup (%x y. y \<sqsubseteq> x)"
ballarin@19984
  2216
(* FIXME: insert ACf_sup and use unfold_locales *)
nipkow@15500
  2217
apply(rule ACIfSL.intro)
ballarin@19931
  2218
apply(rule ACIf.intro)
nipkow@15500
  2219
apply(rule ACf_sup)
nipkow@15500
  2220
apply(rule ACIf.axioms[OF ACIf_sup])
nipkow@15500
  2221
apply(rule ACIfSL_axioms.intro)
nipkow@15500
  2222
apply(rule iffI)
nipkow@21733
  2223
 apply(blast intro: antisym sup_ge1 sup_ge2 sup_least refl)
nipkow@15500
  2224
apply(erule subst)
nipkow@15500
  2225
apply(rule sup_ge2)
nipkow@15500
  2226
done
nipkow@15500
  2227
nipkow@15505
  2228
nipkow@15505
  2229
subsubsection{* Fold laws in lattices *}
nipkow@15500
  2230
nipkow@15780
  2231
lemma (in Lattice) Inf_le_Sup[simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Squnion>A"
nipkow@15500
  2232
apply(unfold Sup_def Inf_def)
nipkow@15500
  2233
apply(subgoal_tac "EX a. a:A")
nipkow@15500
  2234
prefer 2 apply blast
nipkow@15500
  2235
apply(erule exE)
nipkow@15500
  2236
apply(rule trans)
nipkow@15500
  2237
apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_inf])
nipkow@15500
  2238
apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_sup])
nipkow@15500
  2239
done
nipkow@15500
  2240
nipkow@15780
  2241
lemma (in Lattice) sup_Inf_absorb[simp]:
nipkow@15504
  2242
  "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<squnion> \<Sqinter>A) = a"
nipkow@15512
  2243
apply(subst sup_commute)
nipkow@21733
  2244
apply(simp add:Inf_def sup_absorb2 ACIfSL.fold1_belowI[OF ACIfSL_inf])
nipkow@15504
  2245
done
nipkow@15504
  2246
nipkow@15780
  2247
lemma (in Lattice) inf_Sup_absorb[simp]:
nipkow@15504
  2248
  "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<sqinter> \<Squnion>A) = a"
nipkow@21733
  2249
by(simp add:Sup_def inf_absorb1 ACIfSL.fold1_belowI[OF ACIfSL_sup])
nipkow@15504
  2250
nipkow@15504
  2251
nipkow@18423
  2252
lemma (in ACIf) hom_fold1_commute:
nipkow@18423
  2253
assumes hom: "!!x y. h(f x y) = f (h x) (h y)"
nipkow@18423
  2254
and N: "finite N" "N \<noteq> {}" shows "h(fold1 f N) = fold1 f (h ` N)"
nipkow@18423
  2255
using N proof (induct rule: finite_ne_induct)
nipkow@18423
  2256
  case singleton thus ?case by simp
nipkow@15500
  2257
next
nipkow@18423
  2258
  case (insert n N)
nipkow@18423
  2259
  have "h(fold1 f (insert n N)) = h(f n (fold1 f N))" using insert by(simp)
nipkow@18423
  2260
  also have "\<dots> = f (h n) (h(fold1 f N))" by(rule hom)
nipkow@18423
  2261
  also have "h(fold1 f N) = fold1 f (h ` N)" by(rule insert)
nipkow@18423
  2262
  also have "f (h n) \<dots> = fold1 f (insert (h n) (h ` N))"
nipkow@18423
  2263
    using insert by(simp)
nipkow@18423
  2264
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
nipkow@15500
  2265
  finally show ?case .
nipkow@15500
  2266
qed
nipkow@15500
  2267
nipkow@18423
  2268
lemma (in Distrib_Lattice) sup_Inf1_distrib:
nipkow@18423
  2269
 "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> (x \<squnion> \<Sqinter>A) = \<Sqinter>{x \<squnion> a|a. a \<in> A}"
nipkow@18423
  2270
apply(simp add:Inf_def image_def
nipkow@18423
  2271
  ACIf.hom_fold1_commute[OF ACIf_inf, where h="sup x", OF sup_inf_distrib1])
nipkow@18423
  2272
apply(rule arg_cong, blast)
nipkow@18423
  2273
done
nipkow@18423
  2274
nipkow@18423
  2275
nipkow@15512
  2276
lemma (in Distrib_Lattice) sup_Inf2_distrib:
nipkow@15500
  2277
assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
nipkow@15500
  2278
shows "(\<Sqinter>A \<squnion> \<Sqinter>B) = \<Sqinter>{a \<squnion> b|a b. a \<in> A \<and> b \<in> B}"
nipkow@15500
  2279
using A
nipkow@15500
  2280
proof (induct rule: finite_ne_induct)
nipkow@15500
  2281
  case singleton thus ?case
nipkow@15500
  2282
    by(simp add: sup_Inf1_distrib[OF B] fold1_singleton_def[OF Inf_def])
nipkow@15500
  2283
next
nipkow@15500
  2284
  case (insert x A)
nipkow@15500
  2285
  have finB: "finite {x \<squnion> b |b. b \<in> B}"
nipkow@21733
  2286
    by(rule finite_surj[where f = "%b. x \<squnion> b", OF B(1)], auto)
nipkow@15500
  2287
  have finAB: "finite {a \<squnion> b |a b. a \<in> A \<and> b \<in> B}"
nipkow@15500
  2288
  proof -
nipkow@15500
  2289
    have "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {a \<squnion> b})"
nipkow@15500
  2290
      by blast
berghofe@15517
  2291
    thus ?thesis by(simp add: insert(1) B(1))
nipkow@15500
  2292
  qed
nipkow@15500
  2293
  have ne: "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
nipkow@15500
  2294
  have "\<Sqinter>(insert x A) \<squnion> \<Sqinter>B = (x \<sqinter> \<Sqinter>A) \<squnion> \<Sqinter>B"
paulson@15509
  2295
    using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def])
nipkow@15500
  2296
  also have "\<dots> = (x \<squnion> \<Sqinter>B) \<sqinter> (\<Sqinter>A \<squnion> \<Sqinter>B)" by(rule sup_inf_distrib2)
nipkow@15500
  2297
  also have "\<dots> = \<Sqinter>{x \<squnion> b|b. b \<in> B} \<sqinter> \<Sqinter>{a \<squnion> b|a b. a \<in> A \<and> b \<in> B}"
nipkow@15500
  2298
    using insert by(simp add:sup_Inf1_distrib[OF B])
nipkow@15500
  2299
  also have "\<dots> = \<Sqinter>({x\<squnion>b |b. b \<in> B} \<union> {a\<squnion>b |a b. a \<in> A \<and> b \<in> B})"
nipkow@15500
  2300
    (is "_ = \<Sqinter>?M")
nipkow@15500
  2301
    using B insert
nipkow@15500
  2302
    by(simp add:Inf_def ACIf.fold1_Un2[OF ACIf_inf finB _ finAB ne])
nipkow@15500
  2303
  also have "?M = {a \<squnion> b |a b. a \<in> insert x A \<and> b \<in> B}"
nipkow@15500
  2304
    by blast
nipkow@15500
  2305
  finally show ?case .
nipkow@15500
  2306
qed
nipkow@15500
  2307
nipkow@15484
  2308
nipkow@18423
  2309
lemma (in Distrib_Lattice) inf_Sup1_distrib:
nipkow@18423
  2310
 "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> (x \<sqinter> \<Squnion>A) = \<Squnion>{x \<sqinter> a|a. a \<in> A}"
nipkow@18423
  2311
apply(simp add:Sup_def image_def
nipkow@18423
  2312
  ACIf.hom_fold1_commute[OF ACIf_sup, where h="inf x", OF inf_sup_distrib1])
nipkow@18423
  2313
apply(rule arg_cong, blast)
nipkow@18423
  2314
done
nipkow@18423
  2315
nipkow@18423
  2316
nipkow@18423
  2317
lemma (in Distrib_Lattice) inf_Sup2_distrib:
nipkow@18423
  2318
assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
nipkow@18423
  2319
shows "(\<Squnion>A \<sqinter> \<Squnion>B) = \<Squnion>{a \<sqinter> b|a b. a \<in> A \<and> b \<in> B}"
nipkow@18423
  2320
using A
nipkow@18423
  2321
proof (induct rule: finite_ne_induct)
nipkow@18423
  2322
  case singleton thus ?case
nipkow@18423
  2323
    by(simp add: inf_Sup1_distrib[OF B] fold1_singleton_def[OF Sup_def])
nipkow@18423
  2324
next
nipkow@18423
  2325
  case (insert x A)
nipkow@18423
  2326
  have finB: "finite {x \<sqinter> b |b. b \<in> B}"
nipkow@21733
  2327
    by(rule finite_surj[where f = "%b. x \<sqinter> b", OF B(1)], auto)
nipkow@18423
  2328
  have finAB: "finite {a \<sqinter> b |a b. a \<in> A \<and> b \<in> B}"
nipkow@18423
  2329
  proof -
nipkow@18423
  2330
    have "{a \<sqinter> b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {a \<sqinter> b})"
nipkow@18423
  2331
      by blast
nipkow@18423
  2332
    thus ?thesis by(simp add: insert(1) B(1))
nipkow@18423
  2333
  qed
nipkow@18423
  2334
  have ne: "{a \<sqinter> b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
nipkow@18423
  2335
  have "\<Squnion>(insert x A) \<sqinter> \<Squnion>B = (x \<squnion> \<Squnion>A) \<sqinter> \<Squnion>B"
nipkow@18423
  2336
    using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_sup Sup_def])
nipkow@18423
  2337
  also have "\<dots> = (x \<sqinter> \<Squnion>B) \<squnion> (\<Squnion>A \<sqinter> \<Squnion>B)" by(rule inf_sup_distrib2)
nipkow@18423
  2338
  also have "\<dots> = \<Squnion>{x \<sqinter> b|b. b \<in> B} \<squnion> \<Squnion>{a \<sqinter> b|a b. a \<in> A \<and> b \<in> B}"
nipkow@18423
  2339
    using insert by(simp add:inf_Sup1_distrib[OF B])
nipkow@18423
  2340
  also have "\<dots> = \<Squnion>({x\<sqinter>b |b. b \<in> B} \<union> {a\<sqinter>b |a b. a \<in> A \<and> b \<in> B})"
nipkow@18423
  2341
    (is "_ = \<Squnion>?M")
nipkow@18423
  2342
    using B insert
nipkow@18423
  2343
    by(simp add:Sup_def ACIf.fold1_Un2[OF ACIf_sup finB _ finAB ne])
nipkow@18423
  2344
  also have "?M = {a \<sqinter> b |a b. a \<in> insert x A \<and> b \<in> B}"
nipkow@18423
  2345
    by blast
nipkow@18423
  2346
  finally show ?case .
nipkow@18423
  2347
qed
nipkow@18423
  2348
nipkow@18423
  2349
nipkow@15392
  2350
subsection{*Min and Max*}
nipkow@15392
  2351
nipkow@15392
  2352
text{* As an application of @{text fold1} we define the minimal and
nipkow@15497
  2353
maximal element of a (non-empty) set over a linear order. *}
nipkow@15497
  2354
nipkow@15497
  2355
constdefs
nipkow@15497
  2356
  Min :: "('a::linorder)set => 'a"
nipkow@15497
  2357
  "Min  ==  fold1 min"
nipkow@15497
  2358
nipkow@15497
  2359
  Max :: "('a::linorder)set => 'a"
nipkow@15497
  2360
  "Max  ==  fold1 max"
nipkow@15497
  2361
nipkow@15497
  2362
nipkow@15497
  2363
text{* Before we can do anything, we need to show that @{text min} and
nipkow@15497
  2364
@{text max} are ACI and the ordering is linear: *}
nipkow@15392
  2365
ballarin@15837
  2366
interpretation min: ACf ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
nipkow@15392
  2367
apply(rule ACf.intro)
nipkow@15392
  2368
apply(auto simp:min_def)
nipkow@15392
  2369
done
nipkow@15392
  2370
ballarin@15837
  2371
interpretation min: ACIf ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
ballarin@19984
  2372
apply unfold_locales
nipkow@15392
  2373
apply(auto simp:min_def)
nipkow@15376
  2374
done
nipkow@15376
  2375
ballarin@15837
  2376
interpretation max: ACf ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
ballarin@19984
  2377
apply unfold_locales
nipkow@15392
  2378
apply(auto simp:max_def)
nipkow@15392
  2379
done
nipkow@15392
  2380
ballarin@15837
  2381
interpretation max: ACIf ["max:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
ballarin@19984
  2382
apply unfold_locales
nipkow@15392
  2383
apply(auto simp:max_def)
nipkow@15376
  2384
done
wenzelm@12396
  2385
ballarin@15837
  2386
interpretation min:
nipkow@18493
  2387
  ACIfSL ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "op \<le>" "op <"]
nipkow@18493
  2388
apply(simp add:order_less_le)
ballarin@19984
  2389
apply unfold_locales
nipkow@15497
  2390
apply(auto simp:min_def)
nipkow@15497
  2391
done
nipkow@15497
  2392
ballarin@15837
  2393
interpretation min:
nipkow@18493
  2394
  ACIfSLlin ["min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "op \<le>" "op <"]
ballarin@19984
  2395
apply unfold_locales
nipkow@15497
  2396
apply(auto simp:min_def)
nipkow@15497
  2397
done
nipkow@15392
  2398
ballarin@15837
  2399
interpretation max:
nipkow@18493
  2400
  ACIfSL ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "%x y. y\<le>x" "%x y. y<x"]
nipkow@18493
  2401
apply(simp add:order_less_le eq_sym_conv)
ballarin@19984
  2402
apply unfold_locales
nipkow@15497
  2403
apply(auto simp:max_def)
nipkow@15497
  2404
done
nipkow@15392
  2405
ballarin@15837
  2406
interpretation max:
nipkow@18493
  2407
  ACIfSLlin ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "%x y. y\<le>x" "%x y. y<x"]
ballarin@19984
  2408
apply unfold_locales
nipkow@15497
  2409
apply(auto simp:max_def)
nipkow@15497
  2410
done
nipkow@15392
  2411
ballarin@15837
  2412
interpretation min_max:
haftmann@21215
  2413
  Lattice ["op \<le>" "op <" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max" "Min" "Max"]
nipkow@15780
  2414
apply -
nipkow@15780
  2415
apply(rule Min_def)
nipkow@15780
  2416
apply(rule Max_def)
ballarin@19984
  2417
apply unfold_locales
nipkow@15507
  2418
done
nipkow@15500
  2419
nipkow@15500
  2420
ballarin@15837
  2421
interpretation min_max:
haftmann@21215
  2422
  Distrib_Lattice ["op \<le>" "op <" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max" "Min" "Max"]
ballarin@19984
  2423
  by unfold_locales
ballarin@19931
  2424
ballarin@15765
  2425
nipkow@15402
  2426
text{* Now we instantiate the recursion equations and declare them
nipkow@15392
  2427
simplification rules: *}
nipkow@15392
  2428
paulson@17085
  2429
(* Making Min or Max a defined parameter of a locale, suitably
paulson@17085
  2430
  extending ACIf, could make the following interpretations more automatic. *)
ballarin@15765
  2431
paulson@17085
  2432
lemmas Min_singleton = fold1_singleton_def [OF Min_def]
paulson@17085
  2433
lemmas Max_singleton = fold1_singleton_def [OF Max_def]
paulson@17085
  2434
lemmas Min_insert = min.fold1_insert_idem_def [OF Min_def]
paulson@17085
  2435
lemmas Max_insert = max.fold1_insert_idem_def [OF Max_def]
paulson@17085
  2436
paulson@17085
  2437
declare Min_singleton [simp]  Max_singleton [simp]
paulson@17085
  2438
declare Min_insert [simp]  Max_insert [simp]
paulson@17085
  2439
nipkow@15392
  2440
nipkow@15484
  2441
text{* Now we instantiate some @{text fold1} properties: *}
nipkow@15392
  2442
nipkow@15392
  2443
lemma Min_in [simp]:
nipkow@15484
  2444
  shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Min A \<in> A"
nipkow@15791
  2445
using min.fold1_in
nipkow@15484
  2446
by(fastsimp simp: Min_def min_def)
nipkow@15392
  2447
nipkow@15392
  2448
lemma Max_in [simp]:
nipkow@15484
  2449
  shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Max A \<in> A"
nipkow@15791
  2450
using max.fold1_in
nipkow@15484
  2451
by(fastsimp simp: Max_def max_def)
nipkow@15392
  2452
nipkow@18423
  2453
lemma Min_antimono: "\<lbrakk> M \<subseteq> N; M \<noteq> {}; finite N \<rbrakk> \<Longrightarrow> Min N \<le> Min M"
nipkow@18423
  2454
by(simp add:Finite_Set.Min_def min.fold1_antimono)
nipkow@18423
  2455
nipkow@18423
  2456
lemma Max_mono: "\<lbrakk> M \<subseteq> N; M \<noteq> {}; finite N \<rbrakk> \<Longrightarrow> Max M \<le> Max N"
nipkow@18423
  2457
by(simp add:Max_def max.fold1_antimono)
nipkow@18423
  2458
nipkow@15484
  2459
lemma Min_le [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> Min A \<le> x"
nipkow@15791
  2460
by(simp add: Min_def min.fold1_belowI)
nipkow@15392
  2461
nipkow@15484
  2462
lemma Max_ge [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> x \<le> Max A"
nipkow@15791
  2463
by(simp add: Max_def max.fold1_belowI)
nipkow@15497
  2464
nipkow@15497
  2465
lemma Min_ge_iff[simp]:
nipkow@15497
  2466
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Min A) = (\<forall>a\<in>A. x \<le> a)"
nipkow@15791
  2467
by(simp add: Min_def min.below_fold1_iff)
nipkow@15497
  2468
nipkow@15497
  2469
lemma Max_le_iff[simp]:
nipkow@15497
  2470
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Max A \<le> x) = (\<forall>a\<in>A. a \<le> x)"
nipkow@15791
  2471
by(simp add: Max_def max.below_fold1_iff)
nipkow@15497
  2472
nipkow@18493
  2473
lemma Min_gr_iff[simp]:
nipkow@18493
  2474
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x < Min A) = (\<forall>a\<in>A. x < a)"
nipkow@18493
  2475
by(simp add: Min_def min.strict_below_fold1_iff)
nipkow@18493
  2476
nipkow@18493
  2477
lemma Max_less_iff[simp]:
nipkow@18493
  2478
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Max A < x) = (\<forall>a\<in>A. a < x)"
nipkow@18493
  2479
by(simp add: Max_def max.strict_below_fold1_iff)
nipkow@18493
  2480
nipkow@15497
  2481
lemma Min_le_iff:
nipkow@15497
  2482
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Min A \<le> x) = (\<exists>a\<in>A. a \<le> x)"
nipkow@15791
  2483
by(simp add: Min_def min.fold1_below_iff)
nipkow@15497
  2484
nipkow@15497
  2485
lemma Max_ge_iff:
nipkow@15497
  2486
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Max A) = (\<exists>a\<in>A. x \<le> a)"
nipkow@15791
  2487
by(simp add: Max_def max.fold1_below_iff)
wenzelm@12396
  2488
nipkow@18493
  2489
lemma Min_le_iff:
nipkow@18493
  2490
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Min A < x) = (\<exists>a\<in>A. a < x)"
nipkow@18493
  2491
by(simp add: Min_def min.fold1_strict_below_iff)
nipkow@18493
  2492
nipkow@18493
  2493
lemma Max_ge_iff:
nipkow@18493
  2494
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x < Max A) = (\<exists>a\<in>A. x < a)"
nipkow@18493
  2495
by(simp add: Max_def max.fold1_strict_below_iff)
nipkow@18493
  2496
nipkow@18423
  2497
lemma Min_Un: "\<lbrakk>finite A; A \<noteq> {}; finite B; B \<noteq> {}\<rbrakk>
nipkow@18423
  2498
  \<Longrightarrow> Min (A \<union> B) = min (Min A) (Min B)"
nipkow@18423
  2499
by(simp add:Min_def min.f.fold1_Un2)
nipkow@18423
  2500
nipkow@18423
  2501
lemma Max_Un: "\<lbrakk>finite A; A \<noteq> {}; finite B; B \<noteq> {}\<rbrakk>
nipkow@18423
  2502
  \<Longrightarrow> Max (A \<union> B) = max (Max A) (Max B)"
nipkow@18423
  2503
by(simp add:Max_def max.f.fold1_Un2)
nipkow@18423
  2504
nipkow@18423
  2505
nipkow@18423
  2506
lemma hom_Min_commute:
nipkow@18423
  2507
 "(!!x y::'a::linorder. h(min x y) = min (h x) (h y::'a))
nipkow@18423
  2508
  \<Longrightarrow> finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> h(Min N) = Min(h ` N)"
nipkow@18423
  2509
by(simp add:Finite_Set.Min_def min.hom_fold1_commute)
nipkow@18423
  2510
nipkow@18423
  2511
lemma hom_Max_commute:
nipkow@18423
  2512
 "(!!x y::'a::linorder. h(max x y) = max (h x) (h y::'a))
nipkow@18423
  2513
  \<Longrightarrow> finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> h(Max N) = Max(h ` N)"
nipkow@18423
  2514
by(simp add:Max_def max.hom_fold1_commute)
nipkow@18423
  2515
nipkow@18423
  2516
nipkow@18423
  2517
lemma add_Min_commute: fixes k::"'a::{pordered_ab_semigroup_add,linorder}"
nipkow@18423
  2518
 shows "finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> k + Min N = Min {k+m|m. m \<in> N}"
nipkow@18423
  2519
apply(subgoal_tac "!!x y. k + min x y = min (k + x) (k + y)")
nipkow@18423
  2520
using hom_Min_commute[of "op + k" N]
nipkow@18423
  2521
apply simp apply(rule arg_cong[where f = Min]) apply blast
nipkow@18423
  2522
apply(simp add:min_def linorder_not_le)
nipkow@18423
  2523
apply(blast intro:order.antisym order_less_imp_le add_left_mono)
nipkow@18423
  2524
done
nipkow@18423
  2525
nipkow@18423
  2526
lemma add_Max_commute: fixes k::"'a::{pordered_ab_semigroup_add,linorder}"
nipkow@18423
  2527
 shows "finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> k + Max N = Max {k+m|m. m \<in> N}"
nipkow@18423
  2528
apply(subgoal_tac "!!x y. k + max x y = max (k + x) (k + y)")
nipkow@18423
  2529
using hom_Max_commute[of "op + k" N]
nipkow@18423
  2530
apply simp apply(rule arg_cong[where f = Max]) apply blast
nipkow@18423
  2531
apply(simp add:max_def linorder_not_le)
nipkow@18423
  2532
apply(blast intro:order.antisym order_less_imp_le add_left_mono)
nipkow@18423
  2533
done
nipkow@18423
  2534
nipkow@18423
  2535
nipkow@18423
  2536
nipkow@17022
  2537
subsection {* Properties of axclass @{text finite} *}
nipkow@17022
  2538
nipkow@17022
  2539
text{* Many of these are by Brian Huffman. *}
nipkow@17022
  2540
nipkow@17022
  2541
lemma finite_set: "finite (A::'a::finite set)"
nipkow@17022
  2542
by (rule finite_subset [OF subset_UNIV finite])
nipkow@17022
  2543
nipkow@17022
  2544
nipkow@17022
  2545
instance unit :: finite
nipkow@17022
  2546
proof
nipkow@17022
  2547
  have "finite {()}" by simp
nipkow@17022
  2548
  also have "{()} = UNIV" by auto
nipkow@17022
  2549
  finally show "finite (UNIV :: unit set)" .
nipkow@17022
  2550
qed
nipkow@17022
  2551
nipkow@17022
  2552
instance bool :: finite
nipkow@17022
  2553
proof
nipkow@17022
  2554
  have "finite {True, False}" by simp
nipkow@17022
  2555
  also have "{True, False} = UNIV" by auto
nipkow@17022
  2556
  finally show "finite (UNIV :: bool set)" .
nipkow@17022
  2557
qed
nipkow@17022
  2558
nipkow@17022
  2559
nipkow@17022
  2560
instance * :: (finite, finite) finite
nipkow@17022
  2561
proof
nipkow@17022
  2562
  show "finite (UNIV :: ('a \<times> 'b) set)"
nipkow@17022
  2563
  proof (rule finite_Prod_UNIV)
nipkow@17022
  2564
    show "finite (UNIV :: 'a set)" by (rule finite)
nipkow@17022
  2565
    show "finite (UNIV :: 'b set)" by (rule finite)
nipkow@17022
  2566
  qed
nipkow@17022
  2567
qed
nipkow@17022
  2568
nipkow@17022
  2569
instance "+" :: (finite, finite) finite
nipkow@17022
  2570
proof
nipkow@17022
  2571
  have a: "finite (UNIV :: 'a set)" by (rule finite)
nipkow@17022
  2572
  have b: "finite (UNIV :: 'b set)" by (rule finite)
nipkow@17022
  2573
  from a b have "finite ((UNIV :: 'a set) <+> (UNIV :: 'b set))"
nipkow@17022
  2574
    by (rule finite_Plus)
nipkow@17022
  2575
  thus "finite (UNIV :: ('a + 'b) set)" by simp
nipkow@17022
  2576
qed
nipkow@17022
  2577
nipkow@17022
  2578
nipkow@17022
  2579
instance set :: (finite) finite
nipkow@17022
  2580
proof
nipkow@17022
  2581
  have "finite (UNIV :: 'a set)" by (rule finite)
nipkow@17022
  2582
  hence "finite (Pow (UNIV :: 'a set))"
nipkow@17022
  2583
    by (rule finite_Pow_iff [THEN iffD2])
nipkow@17022
  2584
  thus "finite (UNIV :: 'a set set)" by simp
nipkow@17022
  2585
qed
nipkow@17022
  2586
nipkow@17022
  2587
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
nipkow@17022
  2588
by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
nipkow@17022
  2589
haftmann@21215
  2590
instance "fun" :: (finite, finite) finite
nipkow@17022
  2591
proof
nipkow@17022
  2592
  show "finite (UNIV :: ('a => 'b) set)"
nipkow@17022
  2593
  proof (rule finite_imageD)
nipkow@17022
  2594
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
nipkow@17022
  2595
    show "finite (range ?graph)" by (rule finite_set)
nipkow@17022
  2596
    show "inj ?graph" by (rule inj_graph)
nipkow@17022
  2597
  qed
nipkow@17022
  2598
qed
nipkow@17022
  2599
nipkow@15042
  2600
end