src/HOL/Finite_Set.thy
author berghofe
Thu, 10 Feb 2005 10:43:57 +0100
changeset 15517 3bc57d428ec1
parent 15512 ed1fa4617f52
child 15520 0ed33cd8f238
permissions -rw-r--r--
Subscripts for theorem lists now start at 1.
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
     1
(*  Title:      HOL/Finite_Set.thy
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
     2
    ID:         $Id$
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
     3
    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
     4
                Additions by Jeremy Avigad in Feb 2004
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
     5
*)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
     6
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
     7
header {* Finite sets *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
     8
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15124
diff changeset
     9
theory Finite_Set
15512
ed1fa4617f52 Extracted generic lattice stuff to new Lattice_Locales.thy
nipkow
parents: 15510
diff changeset
    10
imports Divides Power Inductive Lattice_Locales
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15124
diff changeset
    11
begin
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    12
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
    13
subsection {* Definition and basic properties *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    14
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    15
consts Finites :: "'a set set"
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13735
diff changeset
    16
syntax
e564c3d2d174 added a few lemmas
nipkow
parents: 13735
diff changeset
    17
  finite :: "'a set => bool"
e564c3d2d174 added a few lemmas
nipkow
parents: 13735
diff changeset
    18
translations
e564c3d2d174 added a few lemmas
nipkow
parents: 13735
diff changeset
    19
  "finite A" == "A : Finites"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    20
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    21
inductive Finites
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    22
  intros
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    23
    emptyI [simp, intro!]: "{} : Finites"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    24
    insertI [simp, intro!]: "A : Finites ==> insert a A : Finites"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    25
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    26
axclass finite \<subseteq> type
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    27
  finite: "finite UNIV"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    28
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13735
diff changeset
    29
lemma ex_new_if_finite: -- "does not depend on def of finite at all"
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
    30
  assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
    31
  shows "\<exists>a::'a. a \<notin> A"
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
    32
proof -
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
    33
  from prems have "A \<noteq> UNIV" by blast
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
    34
  thus ?thesis by blast
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
    35
qed
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    36
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    37
lemma finite_induct [case_names empty insert, induct set: Finites]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    38
  "finite F ==>
15327
0230a10582d3 changed the order of !!-quantifiers in finite set induction.
nipkow
parents: 15318
diff changeset
    39
    P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    40
  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    41
proof -
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13400
diff changeset
    42
  assume "P {}" and
15327
0230a10582d3 changed the order of !!-quantifiers in finite set induction.
nipkow
parents: 15318
diff changeset
    43
    insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    44
  assume "finite F"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    45
  thus "P F"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    46
  proof induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    47
    show "P {}" .
15327
0230a10582d3 changed the order of !!-quantifiers in finite set induction.
nipkow
parents: 15318
diff changeset
    48
    fix x F assume F: "finite F" and P: "P F"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    49
    show "P (insert x F)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    50
    proof cases
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    51
      assume "x \<in> F"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    52
      hence "insert x F = F" by (rule insert_absorb)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    53
      with P show ?thesis by (simp only:)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    54
    next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    55
      assume "x \<notin> F"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    56
      from F this P show ?thesis by (rule insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    57
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    58
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    59
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    60
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    61
lemma finite_ne_induct[case_names singleton insert, consumes 2]:
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    62
assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    63
 \<lbrakk> \<And>x. P{x};
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    64
   \<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk>
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    65
 \<Longrightarrow> P F"
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    66
using fin
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    67
proof induct
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    68
  case empty thus ?case by simp
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    69
next
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    70
  case (insert x F)
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    71
  show ?case
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    72
  proof cases
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    73
    assume "F = {}" thus ?thesis using insert(4) by simp
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    74
  next
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    75
    assume "F \<noteq> {}" thus ?thesis using insert by blast
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    76
  qed
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    77
qed
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    78
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    79
lemma finite_subset_induct [consumes 2, case_names empty insert]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    80
  "finite F ==> F \<subseteq> A ==>
15327
0230a10582d3 changed the order of !!-quantifiers in finite set induction.
nipkow
parents: 15318
diff changeset
    81
    P {} ==> (!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==>
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    82
    P F"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    83
proof -
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13400
diff changeset
    84
  assume "P {}" and insert:
15327
0230a10582d3 changed the order of !!-quantifiers in finite set induction.
nipkow
parents: 15318
diff changeset
    85
    "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    86
  assume "finite F"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    87
  thus "F \<subseteq> A ==> P F"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    88
  proof induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    89
    show "P {}" .
15327
0230a10582d3 changed the order of !!-quantifiers in finite set induction.
nipkow
parents: 15318
diff changeset
    90
    fix x F assume "finite F" and "x \<notin> F"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    91
      and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    92
    show "P (insert x F)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    93
    proof (rule insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    94
      from i show "x \<in> A" by blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    95
      from i have "F \<subseteq> A" by blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    96
      with P show "P F" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    97
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    98
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    99
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   100
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   101
text{* Finite sets are the images of initial segments of natural numbers: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   102
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   103
lemma finite_imp_nat_seg_image_inj_on:
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   104
  assumes fin: "finite A" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   105
  shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   106
using fin
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   107
proof induct
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   108
  case empty
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   109
  show ?case  
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   110
  proof show "\<exists>f. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   111
  qed
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   112
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   113
  case (insert a A)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   114
  have notinA: "a \<notin> A" .
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   115
  from insert.hyps obtain n f
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   116
    where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   117
  hence "insert a A = f(n:=a) ` {i. i < Suc n}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   118
        "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   119
    by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   120
  thus ?case by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   121
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   122
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   123
lemma nat_seg_image_imp_finite:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   124
  "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   125
proof (induct n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   126
  case 0 thus ?case by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   127
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   128
  case (Suc n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   129
  let ?B = "f ` {i. i < n}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   130
  have finB: "finite ?B" by(rule Suc.hyps[OF refl])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   131
  show ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   132
  proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   133
    assume "\<exists>k<n. f n = f k"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   134
    hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   135
    thus ?thesis using finB by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   136
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   137
    assume "\<not>(\<exists> k<n. f n = f k)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   138
    hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   139
    thus ?thesis using finB by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   140
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   141
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   142
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   143
lemma finite_conv_nat_seg_image:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   144
  "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   145
by(blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   146
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   147
subsubsection{* Finiteness and set theoretic constructions *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   148
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   149
lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   150
  -- {* The union of two finite sets is finite. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   151
  by (induct set: Finites) simp_all
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   152
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   153
lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   154
  -- {* Every subset of a finite set is finite. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   155
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   156
  assume "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   157
  thus "!!A. A \<subseteq> B ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   158
  proof induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   159
    case empty
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   160
    thus ?case by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   161
  next
15327
0230a10582d3 changed the order of !!-quantifiers in finite set induction.
nipkow
parents: 15318
diff changeset
   162
    case (insert x F A)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   163
    have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   164
    show "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   165
    proof cases
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   166
      assume x: "x \<in> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   167
      with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   168
      with r have "finite (A - {x})" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   169
      hence "finite (insert x (A - {x}))" ..
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   170
      also have "insert x (A - {x}) = A" by (rule insert_Diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   171
      finally show ?thesis .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   172
    next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   173
      show "A \<subseteq> F ==> ?thesis" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   174
      assume "x \<notin> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   175
      with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   176
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   177
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   178
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   179
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   180
lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   181
  by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   182
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   183
lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   184
  -- {* The converse obviously fails. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   185
  by (blast intro: finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   186
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   187
lemma finite_insert [simp]: "finite (insert a A) = finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   188
  apply (subst insert_is_Un)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   189
  apply (simp only: finite_Un, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   190
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   191
15281
bd4611956c7b More lemmas
nipkow
parents: 15234
diff changeset
   192
lemma finite_Union[simp, intro]:
bd4611956c7b More lemmas
nipkow
parents: 15234
diff changeset
   193
 "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
bd4611956c7b More lemmas
nipkow
parents: 15234
diff changeset
   194
by (induct rule:finite_induct) simp_all
bd4611956c7b More lemmas
nipkow
parents: 15234
diff changeset
   195
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   196
lemma finite_empty_induct:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   197
  "finite A ==>
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   198
  P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   199
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   200
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   201
    and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   202
  have "P (A - A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   203
  proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   204
    fix c b :: "'a set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   205
    presume c: "finite c" and b: "finite b"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   206
      and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   207
    from c show "c \<subseteq> b ==> P (b - c)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   208
    proof induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   209
      case empty
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   210
      from P1 show ?case by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   211
    next
15327
0230a10582d3 changed the order of !!-quantifiers in finite set induction.
nipkow
parents: 15318
diff changeset
   212
      case (insert x F)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   213
      have "P (b - F - {x})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   214
      proof (rule P2)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   215
        from _ b show "finite (b - F)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   216
        from insert show "x \<in> b - F" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   217
        from insert show "P (b - F)" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   218
      qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   219
      also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   220
      finally show ?case .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   221
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   222
  next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   223
    show "A \<subseteq> A" ..
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   224
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   225
  thus "P {}" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   226
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   227
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   228
lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   229
  by (rule Diff_subset [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   230
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   231
lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   232
  apply (subst Diff_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   233
  apply (case_tac "a : A - B")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   234
   apply (rule finite_insert [symmetric, THEN trans])
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   235
   apply (subst insert_Diff, simp_all)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   236
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   237
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   238
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   239
text {* Image and Inverse Image over Finite Sets *}
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   240
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   241
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   242
  -- {* The image of a finite set is finite. *}
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   243
  by (induct set: Finites) simp_all
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   244
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   245
lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   246
  apply (frule finite_imageI)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   247
  apply (erule finite_subset, assumption)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   248
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   249
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   250
lemma finite_range_imageI:
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   251
    "finite (range g) ==> finite (range (%x. f (g x)))"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   252
  apply (drule finite_imageI, simp)
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   253
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   254
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   255
lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   256
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   257
  have aux: "!!A. finite (A - {}) = finite A" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   258
  fix B :: "'a set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   259
  assume "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   260
  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   261
    apply induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   262
     apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   263
    apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   264
     apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   265
     apply (simp (no_asm_use) add: inj_on_def)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   266
     apply (blast dest!: aux [THEN iffD1], atomize)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   267
    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   268
    apply (frule subsetD [OF equalityD2 insertI1], clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   269
    apply (rule_tac x = xa in bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   270
     apply (simp_all add: inj_on_image_set_diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   271
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   272
qed (rule refl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   273
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   274
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   275
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   276
  -- {* The inverse image of a singleton under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   277
         is included in a singleton. *}
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   278
  apply (auto simp add: inj_on_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   279
  apply (blast intro: the_equality [symmetric])
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   280
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   281
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   282
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   283
  -- {* The inverse image of a finite set under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   284
         is finite. *}
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   285
  apply (induct set: Finites, simp_all)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   286
  apply (subst vimage_insert)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   287
  apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   288
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   289
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   290
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   291
text {* The finite UNION of finite sets *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   292
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   293
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   294
  by (induct set: Finites) simp_all
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   295
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   296
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   297
  Strengthen RHS to
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   298
  @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   299
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   300
  We'd need to prove
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   301
  @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   302
  by induction. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   303
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   304
lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   305
  by (blast intro: finite_UN_I finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   306
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   307
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   308
text {* Sigma of finite sets *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   309
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   310
lemma finite_SigmaI [simp]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   311
    "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   312
  by (unfold Sigma_def) (blast intro!: finite_UN_I)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   313
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   314
lemma finite_cartesian_product: "[| finite A; finite B |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   315
    finite (A <*> B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   316
  by (rule finite_SigmaI)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   317
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   318
lemma finite_Prod_UNIV:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   319
    "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   320
  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   321
   apply (erule ssubst)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   322
   apply (erule finite_SigmaI, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   323
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   324
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   325
lemma finite_cartesian_productD1:
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   326
     "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   327
apply (auto simp add: finite_conv_nat_seg_image) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   328
apply (drule_tac x=n in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   329
apply (drule_tac x="fst o f" in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   330
apply (auto simp add: o_def) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   331
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   332
apply (drule equalityD1) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   333
apply (rename_tac y x)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   334
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   335
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   336
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   337
apply (rule_tac x=k in image_eqI, auto)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   338
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   339
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   340
lemma finite_cartesian_productD2:
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   341
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   342
apply (auto simp add: finite_conv_nat_seg_image) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   343
apply (drule_tac x=n in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   344
apply (drule_tac x="snd o f" in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   345
apply (auto simp add: o_def) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   346
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   347
apply (drule equalityD1)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   348
apply (rename_tac x y)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   349
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   350
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   351
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   352
apply (rule_tac x=k in image_eqI, auto)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   353
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   354
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   355
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   356
instance unit :: finite
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   357
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   358
  have "finite {()}" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   359
  also have "{()} = UNIV" by auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   360
  finally show "finite (UNIV :: unit set)" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   361
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   362
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   363
instance * :: (finite, finite) finite
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   364
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   365
  show "finite (UNIV :: ('a \<times> 'b) set)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   366
  proof (rule finite_Prod_UNIV)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   367
    show "finite (UNIV :: 'a set)" by (rule finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   368
    show "finite (UNIV :: 'b set)" by (rule finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   369
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   370
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   371
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   372
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   373
text {* The powerset of a finite set *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   374
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   375
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   376
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   377
  assume "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   378
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   379
  thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   380
next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   381
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   382
  thus "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   383
    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   384
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   385
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   386
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   387
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   388
by(blast intro: finite_subset[OF subset_Pow_Union])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   389
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   390
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   391
lemma finite_converse [iff]: "finite (r^-1) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   392
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   393
   apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   394
   apply (rule iffI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   395
    apply (erule finite_imageD [unfolded inj_on_def])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   396
    apply (simp split add: split_split)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   397
   apply (erule finite_imageI)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   398
  apply (simp add: converse_def image_def, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   399
  apply (rule bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   400
   prefer 2 apply assumption
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   401
  apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   402
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   403
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   404
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   405
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   406
Ehmety) *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   407
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   408
lemma finite_Field: "finite r ==> finite (Field r)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   409
  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   410
  apply (induct set: Finites)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   411
   apply (auto simp add: Field_def Domain_insert Range_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   412
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   413
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   414
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   415
  apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   416
  apply (erule trancl_induct)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   417
   apply (auto simp add: Field_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   418
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   419
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   420
lemma finite_trancl: "finite (r^+) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   421
  apply auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   422
   prefer 2
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   423
   apply (rule trancl_subset_Field2 [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   424
   apply (rule finite_SigmaI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   425
    prefer 3
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 13595
diff changeset
   426
    apply (blast intro: r_into_trancl' finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   427
   apply (auto simp add: finite_Field)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   428
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   429
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   430
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   431
subsection {* A fold functional for finite sets *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   432
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   433
text {* The intended behaviour is
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   434
@{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<dots>)"}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   435
if @{text f} is associative-commutative. For an application of @{text fold}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   436
se the definitions of sums and products over finite sets.
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   437
*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   438
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   439
consts
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   440
  foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   441
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   442
inductive "foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   443
intros
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   444
emptyI [intro]: "({}, z) : foldSet f g z"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   445
insertI [intro]:
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   446
     "\<lbrakk> x \<notin> A; (A, y) : foldSet f g z \<rbrakk>
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   447
      \<Longrightarrow> (insert x A, f (g x) y) : foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   448
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   449
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   450
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   451
constdefs
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   452
  fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   453
  "fold f g z A == THE x. (A, x) : foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   454
15498
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   455
text{*A tempting alternative for the definiens is
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   456
@{term "if finite A then THE x. (A, x) : foldSet f g e else e"}.
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   457
It allows the removal of finiteness assumptions from the theorems
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   458
@{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}.
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   459
The proofs become ugly, with @{text rule_format}. It is not worth the effort.*}
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   460
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   461
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   462
lemma Diff1_foldSet:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   463
  "(A - {x}, y) : foldSet f g z ==> x: A ==> (A, f (g x) y) : foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   464
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   465
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   466
lemma foldSet_imp_finite: "(A, x) : foldSet f g z ==> finite A"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   467
  by (induct set: foldSet) auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   468
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   469
lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   470
  by (induct set: Finites) auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   471
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   472
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   473
subsubsection {* Commutative monoids *}
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   474
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   475
locale ACf =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   476
  fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   477
  assumes commute: "x \<cdot> y = y \<cdot> x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   478
    and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   479
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   480
locale ACe = ACf +
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   481
  fixes e :: 'a
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   482
  assumes ident [simp]: "x \<cdot> e = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   483
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   484
locale ACIf = ACf +
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   485
  assumes idem: "x \<cdot> x = x"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   486
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   487
lemma (in ACf) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   488
proof -
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   489
  have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   490
  also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   491
  also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   492
  finally show ?thesis .
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   493
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   494
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   495
lemmas (in ACf) AC = assoc commute left_commute
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   496
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   497
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   498
proof -
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   499
  have "x \<cdot> e = x" by (rule ident)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   500
  thus ?thesis by (subst commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   501
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   502
15497
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   503
lemma (in ACIf) idem2: "x \<cdot> (x \<cdot> y) = x \<cdot> y"
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   504
proof -
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   505
  have "x \<cdot> (x \<cdot> y) = (x \<cdot> x) \<cdot> y" by(simp add:assoc)
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   506
  also have "\<dots> = x \<cdot> y" by(simp add:idem)
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   507
  finally show ?thesis .
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   508
qed
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   509
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   510
lemmas (in ACIf) ACI = AC idem idem2
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   511
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   512
text{* Instantiation of locales: *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   513
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   514
lemma ACf_add: "ACf (op + :: 'a::comm_monoid_add \<Rightarrow> 'a \<Rightarrow> 'a)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   515
by(fastsimp intro: ACf.intro add_assoc add_commute)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   516
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   517
lemma ACe_add: "ACe (op +) (0::'a::comm_monoid_add)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   518
by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_add)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   519
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   520
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   521
lemma ACf_mult: "ACf (op * :: 'a::comm_monoid_mult \<Rightarrow> 'a \<Rightarrow> 'a)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   522
by(fast intro: ACf.intro mult_assoc ab_semigroup_mult.mult_commute)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   523
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   524
lemma ACe_mult: "ACe (op *) (1::'a::comm_monoid_mult)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   525
by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_mult)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   526
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   527
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   528
subsubsection{*From @{term foldSet} to @{term fold}*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   529
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   530
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   531
by (auto simp add: less_Suc_eq) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   532
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   533
lemma insert_image_inj_on_eq:
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   534
     "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   535
        inj_on h {i. i < Suc m}|] 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   536
      ==> A = h ` {i. i < m}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   537
apply (auto simp add: image_less_Suc inj_on_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   538
apply (blast intro: less_trans) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   539
done
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   540
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   541
lemma insert_inj_onE:
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   542
  assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   543
      and inj_on: "inj_on h {i::nat. i<n}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   544
  shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   545
proof (cases n)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   546
  case 0 thus ?thesis using aA by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   547
next
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   548
  case (Suc m)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   549
  have nSuc: "n = Suc m" . 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   550
  have mlessn: "m<n" by (simp add: nSuc)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   551
  have "a \<in> h ` {i. i < n}" using aA by blast
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   552
  then obtain k where hkeq: "h k = a" and klessn: "k<n" by blast
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   553
  show ?thesis
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   554
  proof cases
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   555
    assume eq: "k=m"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   556
    show ?thesis
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   557
    proof (intro exI conjI)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   558
      show "inj_on h {i::nat. i<m}" using inj_on
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   559
	by (simp add: nSuc inj_on_def) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   560
      show "m<n" by (rule mlessn)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   561
      show "A = h ` {i. i < m}" using aA anot nSuc hkeq eq inj_on
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   562
	by (rules intro: insert_image_inj_on_eq) 
15479
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   563
    qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   564
  next
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   565
    assume diff: "k~=m"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   566
    hence klessm: "k<m" using nSuc klessn by arith
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   567
    have hdiff: "h k ~= h m" using diff inj_on klessn mlessn
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   568
	by (auto simp add: inj_on_def) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   569
    let ?hm = "swap k m h"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   570
    have inj_onhm_n: "inj_on ?hm {i. i < n}" using klessn mlessn 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   571
      by (simp add: inj_on_swap_iff inj_on)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   572
    hence inj_onhm_m: "inj_on ?hm {i. i < m}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   573
      by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   574
    show ?thesis
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   575
    proof (intro exI conjI)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   576
      show "inj_on ?hm {i. i < m}" by (rule inj_onhm_m)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   577
      show "m<n" by (simp add: nSuc)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   578
      show "A = ?hm ` {i. i < m}" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   579
      proof (rule insert_image_inj_on_eq)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   580
	show "inj_on (swap k m h) {i. i < Suc m}" using inj_onhm_n
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   581
	  by (simp add: nSuc)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   582
        show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   583
        show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   584
          using aA hkeq diff hdiff nSuc
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   585
	  by (auto simp add: swap_def image_less_Suc fun_upd_image klessm 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   586
                             inj_on_image_set_diff [OF inj_on])
15479
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   587
      qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   588
    qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   589
  qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   590
qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   591
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   592
lemma (in ACf) foldSet_determ_aux:
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   593
  "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n}; 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   594
                (A,x) : foldSet f g z; (A,x') : foldSet f g z \<rbrakk>
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   595
   \<Longrightarrow> x' = x"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   596
proof (induct n rule: less_induct)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   597
  case (less n)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   598
    have IH: "!!m h A x x'. 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   599
               \<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m}; 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   600
                (A,x) \<in> foldSet f g z; (A, x') \<in> foldSet f g z\<rbrakk> \<Longrightarrow> x' = x" .
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   601
    have Afoldx: "(A,x) \<in> foldSet f g z" and Afoldx': "(A,x') \<in> foldSet f g z"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   602
     and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" .
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   603
    show ?case
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   604
    proof (rule foldSet.cases [OF Afoldx])
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   605
      assume "(A, x) = ({}, z)"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   606
      with Afoldx' show "x' = x" by blast
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   607
    next
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   608
      fix B b u
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   609
      assume "(A,x) = (insert b B, g b \<cdot> u)" and notinB: "b \<notin> B"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   610
         and Bu: "(B,u) \<in> foldSet f g z"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   611
      hence AbB: "A = insert b B" and x: "x = g b \<cdot> u" by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   612
      show "x'=x" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   613
      proof (rule foldSet.cases [OF Afoldx'])
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   614
        assume "(A, x') = ({}, z)"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   615
        with AbB show "x' = x" by blast
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   616
      next
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   617
	fix C c v
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   618
	assume "(A,x') = (insert c C, g c \<cdot> v)" and notinC: "c \<notin> C"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   619
	   and Cv: "(C,v) \<in> foldSet f g z"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   620
	hence AcC: "A = insert c C" and x': "x' = g c \<cdot> v" by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   621
	from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   622
        from insert_inj_onE [OF Beq notinB injh]
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   623
        obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   624
                     and Beq: "B = hB ` {i. i < mB}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   625
                     and lessB: "mB < n" by auto 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   626
	from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   627
        from insert_inj_onE [OF Ceq notinC injh]
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   628
        obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   629
                       and Ceq: "C = hC ` {i. i < mC}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   630
                       and lessC: "mC < n" by auto 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   631
	show "x'=x"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   632
	proof cases
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   633
          assume "b=c"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   634
	  then moreover have "B = C" using AbB AcC notinB notinC by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   635
	  ultimately show ?thesis  using Bu Cv x x' IH[OF lessC Ceq inj_onC]
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   636
            by auto
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   637
	next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   638
	  assume diff: "b \<noteq> c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   639
	  let ?D = "B - {c}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   640
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   641
	    using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   642
	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   643
	  with AbB have "finite ?D" by simp
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   644
	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   645
	    using finite_imp_foldSet by rules
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   646
	  moreover have cinB: "c \<in> B" using B by auto
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   647
	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   648
	    by(rule Diff1_foldSet)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   649
	  hence "g c \<cdot> d = u" by (rule IH [OF lessB Beq inj_onB Bu]) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   650
          moreover have "g b \<cdot> d = v"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   651
	  proof (rule IH[OF lessC Ceq inj_onC Cv])
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   652
	    show "(C, g b \<cdot> d) \<in> foldSet f g z" using C notinB Dfoldd
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   653
	      by fastsimp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   654
	  qed
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   655
	  ultimately show ?thesis using x x' by (auto simp: AC)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   656
	qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   657
      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   658
    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   659
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   660
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   661
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   662
lemma (in ACf) foldSet_determ:
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   663
  "(A,x) : foldSet f g z ==> (A,y) : foldSet f g z ==> y = x"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   664
apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   665
apply (blast intro: foldSet_determ_aux [rule_format])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   666
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   667
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   668
lemma (in ACf) fold_equality: "(A, y) : foldSet f g z ==> fold f g z A = y"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   669
  by (unfold fold_def) (blast intro: foldSet_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   670
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   671
text{* The base case for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   672
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   673
lemma fold_empty [simp]: "fold f g z {} = z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   674
  by (unfold fold_def) blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   675
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   676
lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   677
    ((insert x A, v) : foldSet f g z) =
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   678
    (EX y. (A, y) : foldSet f g z & v = f (g x) y)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   679
  apply auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   680
  apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   681
   apply (fastsimp dest: foldSet_imp_finite)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   682
  apply (blast intro: foldSet_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   683
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   684
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   685
text{* The recursion equation for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   686
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   687
lemma (in ACf) fold_insert[simp]:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   688
    "finite A ==> x \<notin> A ==> fold f g z (insert x A) = f (g x) (fold f g z A)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   689
  apply (unfold fold_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   690
  apply (simp add: fold_insert_aux)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   691
  apply (rule the_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   692
  apply (auto intro: finite_imp_foldSet
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   693
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   694
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   695
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   696
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   697
text{* A simplified version for idempotent functions: *}
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   698
15509
c54970704285 revised fold1 proofs
paulson
parents: 15508
diff changeset
   699
lemma (in ACIf) fold_insert_idem:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   700
assumes finA: "finite A"
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
   701
shows "fold f g z (insert a A) = g a \<cdot> fold f g z A"
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   702
proof cases
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   703
  assume "a \<in> A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   704
  then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   705
    by(blast dest: mk_disjoint_insert)
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   706
  show ?thesis
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   707
  proof -
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   708
    from finA A have finB: "finite B" by(blast intro: finite_subset)
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   709
    have "fold f g z (insert a A) = fold f g z (insert a B)" using A by simp
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   710
    also have "\<dots> = (g a) \<cdot> (fold f g z B)"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   711
      using finB disj by simp
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   712
    also have "\<dots> = g a \<cdot> fold f g z A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   713
      using A finB disj by(simp add:idem assoc[symmetric])
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   714
    finally show ?thesis .
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   715
  qed
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   716
next
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   717
  assume "a \<notin> A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   718
  with finA show ?thesis by simp
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   719
qed
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   720
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   721
lemma (in ACIf) foldI_conv_id:
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   722
  "finite A \<Longrightarrow> fold f g z A = fold f id z (g ` A)"
15509
c54970704285 revised fold1 proofs
paulson
parents: 15508
diff changeset
   723
by(erule finite_induct)(simp_all add: fold_insert_idem del: fold_insert)
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   724
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   725
subsubsection{*Lemmas about @{text fold}*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   726
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   727
lemma (in ACf) fold_commute:
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   728
  "finite A ==> (!!z. f x (fold f g z A) = fold f g (f x z) A)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   729
  apply (induct set: Finites, simp)
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   730
  apply (simp add: left_commute [of x])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   731
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   732
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   733
lemma (in ACf) fold_nest_Un_Int:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   734
  "finite A ==> finite B
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   735
    ==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   736
  apply (induct set: Finites, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   737
  apply (simp add: fold_commute Int_insert_left insert_absorb)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   738
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   739
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   740
lemma (in ACf) fold_nest_Un_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   741
  "finite A ==> finite B ==> A Int B = {}
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   742
    ==> fold f g z (A Un B) = fold f g (fold f g z B) A"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   743
  by (simp add: fold_nest_Un_Int)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   744
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   745
lemma (in ACf) fold_reindex:
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   746
assumes fin: "finite A"
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   747
shows "inj_on h A \<Longrightarrow> fold f g z (h ` A) = fold f (g \<circ> h) z A"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   748
using fin apply induct
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   749
 apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   750
apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   751
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   752
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   753
lemma (in ACe) fold_Un_Int:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   754
  "finite A ==> finite B ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   755
    fold f g e A \<cdot> fold f g e B =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   756
    fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   757
  apply (induct set: Finites, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   758
  apply (simp add: AC insert_absorb Int_insert_left)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   759
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   760
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   761
corollary (in ACe) fold_Un_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   762
  "finite A ==> finite B ==> A Int B = {} ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   763
    fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   764
  by (simp add: fold_Un_Int)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   765
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   766
lemma (in ACe) fold_UN_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   767
  "\<lbrakk> finite I; ALL i:I. finite (A i);
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   768
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   769
   \<Longrightarrow> fold f g e (UNION I A) =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   770
       fold f (%i. fold f g e (A i)) e I"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   771
  apply (induct set: Finites, simp, atomize)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   772
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   773
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   774
  apply (subgoal_tac "A x Int UNION F A = {}")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   775
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   776
  apply (simp add: fold_Un_disjoint)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   777
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   778
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   779
text{*Fusion theorem, as described in
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   780
Graham Hutton's paper,
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   781
A Tutorial on the Universality and Expressiveness of Fold,
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   782
JFP 9:4 (355-372), 1999.*}
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   783
lemma (in ACf) fold_fusion:
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   784
      includes ACf g
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   785
      shows
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   786
	"finite A ==> 
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   787
	 (!!x y. h (g x y) = f x (h y)) ==>
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   788
         h (fold g j w A) = fold f j (h w) A"
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   789
  by (induct set: Finites, simp_all)
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   790
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   791
lemma (in ACf) fold_cong:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   792
  "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   793
  apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g z C = fold f h z C")
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   794
   apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   795
  apply (erule finite_induct, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   796
  apply (simp add: subset_insert_iff, clarify)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   797
  apply (subgoal_tac "finite C")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   798
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   799
  apply (subgoal_tac "C = insert x (C - {x})")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   800
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   801
  apply (erule ssubst)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   802
  apply (drule spec)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   803
  apply (erule (1) notE impE)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   804
  apply (simp add: Ball_def del: insert_Diff_single)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   805
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   806
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   807
lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   808
  fold f (%x. fold f (g x) e (B x)) e A =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   809
  fold f (split g) e (SIGMA x:A. B x)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   810
apply (subst Sigma_def)
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   811
apply (subst fold_UN_disjoint, assumption, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   812
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   813
apply (erule fold_cong)
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   814
apply (subst fold_UN_disjoint, simp, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   815
 apply blast
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   816
apply simp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   817
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   818
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   819
lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   820
   fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   821
apply (erule finite_induct, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   822
apply (simp add:AC)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   823
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   824
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   825
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   826
subsection {* Generalized summation over a set *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   827
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   828
constdefs
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   829
  setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   830
  "setsum f A == if finite A then fold (op +) f 0 A else 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   831
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   832
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   833
written @{text"\<Sum>x\<in>A. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   834
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   835
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   836
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   837
syntax (xsymbols)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   838
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   839
syntax (HTML output)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   840
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   841
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   842
translations -- {* Beware of argument permutation! *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   843
  "SUM i:A. b" == "setsum (%i. b) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   844
  "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   845
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   846
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   847
 @{text"\<Sum>x|P. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   848
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   849
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   850
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   851
syntax (xsymbols)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   852
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   853
syntax (HTML output)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   854
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   855
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   856
translations
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   857
  "SUM x|P. t" => "setsum (%x. t) {x. P}"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   858
  "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   859
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   860
text{* Finally we abbreviate @{term"\<Sum>x\<in>A. x"} by @{text"\<Sum>A"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   861
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   862
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   863
  "_Setsum" :: "'a set => 'a::comm_monoid_mult"  ("\<Sum>_" [1000] 999)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   864
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   865
parse_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   866
  let
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   867
    fun Setsum_tr [A] = Syntax.const "setsum" $ Abs ("", dummyT, Bound 0) $ A
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   868
  in [("_Setsum", Setsum_tr)] end;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   869
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   870
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   871
print_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   872
let
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   873
  fun setsum_tr' [Abs(_,_,Bound 0), A] = Syntax.const "_Setsum" $ A
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   874
    | setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   875
       if x<>y then raise Match
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   876
       else let val x' = Syntax.mark_bound x
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   877
                val t' = subst_bound(x',t)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   878
                val P' = subst_bound(x',P)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   879
            in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   880
in
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   881
[("setsum", setsum_tr')]
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   882
end
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   883
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   884
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   885
lemma setsum_empty [simp]: "setsum f {} = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   886
  by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   887
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   888
lemma setsum_insert [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   889
    "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   890
  by (simp add: setsum_def ACf.fold_insert [OF ACf_add])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   891
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   892
lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   893
  by (simp add: setsum_def)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   894
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   895
lemma setsum_reindex:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   896
     "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   897
by(auto simp add: setsum_def ACf.fold_reindex[OF ACf_add] dest!:finite_imageD)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   898
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   899
lemma setsum_reindex_id:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   900
     "inj_on f B ==> setsum f B = setsum id (f ` B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   901
by (auto simp add: setsum_reindex)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   902
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   903
lemma setsum_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   904
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   905
by(fastsimp simp: setsum_def intro: ACf.fold_cong[OF ACf_add])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   906
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   907
lemma setsum_reindex_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   908
     "[|inj_on f A; B = f ` A; !!a. g a = h (f a)|] 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   909
      ==> setsum h B = setsum g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   910
  by (simp add: setsum_reindex cong: setsum_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   911
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   912
lemma setsum_0: "setsum (%i. 0) A = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   913
apply (clarsimp simp: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   914
apply (erule finite_induct, auto simp:ACf.fold_insert [OF ACf_add])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   915
done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   916
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   917
lemma setsum_0': "ALL a:F. f a = 0 ==> setsum f F = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   918
  apply (subgoal_tac "setsum f F = setsum (%x. 0) F")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   919
  apply (erule ssubst, rule setsum_0)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   920
  apply (rule setsum_cong, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   921
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   922
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   923
lemma setsum_Un_Int: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   924
  setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   925
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   926
by(simp add: setsum_def ACe.fold_Un_Int[OF ACe_add,symmetric])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   927
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   928
lemma setsum_Un_disjoint: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   929
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   930
by (subst setsum_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   931
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   932
(*But we can't get rid of finite I. If infinite, although the rhs is 0, 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   933
  the lhs need not be, since UNION I A could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   934
lemma setsum_UN_disjoint:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   935
    "finite I ==> (ALL i:I. finite (A i)) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   936
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   937
      setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   938
by(simp add: setsum_def ACe.fold_UN_disjoint[OF ACe_add] cong: setsum_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   939
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   940
text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   941
directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   942
lemma setsum_Union_disjoint:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   943
  "[| (ALL A:C. finite A);
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   944
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   945
   ==> setsum f (Union C) = setsum (setsum f) C"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   946
apply (cases "finite C") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   947
 prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   948
  apply (frule setsum_UN_disjoint [of C id f])
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   949
 apply (unfold Union_def id_def, assumption+)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   950
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   951
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   952
(*But we can't get rid of finite A. If infinite, although the lhs is 0, 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   953
  the rhs need not be, since SIGMA A B could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   954
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   955
    (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   956
    (\<Sum>z\<in>(SIGMA x:A. B x). f (fst z) (snd z))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   957
by(simp add:setsum_def ACe.fold_Sigma[OF ACe_add] split_def cong:setsum_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   958
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   959
text{*Here we can eliminate the finiteness assumptions, by cases.*}
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   960
lemma setsum_cartesian_product: 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   961
   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>z\<in>A <*> B. f (fst z) (snd z))"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   962
apply (cases "finite A") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   963
 apply (cases "finite B") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   964
  apply (simp add: setsum_Sigma)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   965
 apply (cases "A={}", simp)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   966
 apply (simp add: setsum_0) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   967
apply (auto simp add: setsum_def
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   968
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   969
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   970
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   971
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   972
by(simp add:setsum_def ACe.fold_distrib[OF ACe_add])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   973
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   974
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   975
subsubsection {* Properties in more restricted classes of structures *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   976
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   977
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   978
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   979
   prefer 2 apply (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   980
  apply (erule rev_mp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   981
  apply (erule finite_induct, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   982
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   983
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   984
lemma setsum_eq_0_iff [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   985
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   986
  by (induct set: Finites) auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   987
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   988
lemma setsum_Un_nat: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   989
    (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   990
  -- {* For the natural numbers, we have subtraction. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   991
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   992
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   993
lemma setsum_Un: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   994
    (setsum f (A Un B) :: 'a :: ab_group_add) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   995
      setsum f A + setsum f B - setsum f (A Int B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   996
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   997
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   998
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   999
    (if a:A then setsum f A - f a else setsum f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1000
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1001
   prefer 2 apply (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1002
  apply (erule finite_induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1003
   apply (auto simp add: insert_Diff_if)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1004
  apply (drule_tac a = a in mk_disjoint_insert, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1005
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1006
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1007
lemma setsum_diff1: "finite A \<Longrightarrow>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1008
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1009
  (if a:A then setsum f A - f a else setsum f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1010
  by (erule finite_induct) (auto simp add: insert_Diff_if)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1011
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1012
(* By Jeremy Siek: *)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1013
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1014
lemma setsum_diff_nat: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1015
  assumes finB: "finite B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1016
  shows "B \<subseteq> A \<Longrightarrow> (setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1017
using finB
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1018
proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1019
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1020
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1021
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1022
    and xFinA: "insert x F \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1023
    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1024
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1025
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1026
    by (simp add: setsum_diff1_nat)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1027
  from xFinA have "F \<subseteq> A" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1028
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1029
  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1030
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1031
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1032
  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1033
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1034
  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1035
  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1036
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1037
  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1038
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1039
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1040
lemma setsum_diff:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1041
  assumes le: "finite A" "B \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1042
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1043
proof -
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1044
  from le have finiteB: "finite B" using finite_subset by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1045
  show ?thesis using finiteB le
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1046
    proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1047
      case empty
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1048
      thus ?case by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1049
    next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1050
      case (insert x F)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1051
      thus ?case using le finiteB 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1052
	by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1053
    qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1054
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1055
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1056
lemma setsum_mono:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1057
  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1058
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1059
proof (cases "finite K")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1060
  case True
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1061
  thus ?thesis using le
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1062
  proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1063
    case empty
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1064
    thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1065
  next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1066
    case insert
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1067
    thus ?case using add_mono 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1068
      by force
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1069
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1070
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1071
  case False
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1072
  thus ?thesis
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1073
    by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1074
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1075
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1076
lemma setsum_mono2_nat:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1077
  assumes fin: "finite B" and sub: "A \<subseteq> B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1078
shows "setsum f A \<le> (setsum f B :: nat)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1079
proof -
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1080
  have "setsum f A \<le> setsum f A + setsum f (B-A)" by arith
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1081
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1082
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1083
  also have "A \<union> (B-A) = B" using sub by blast
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1084
  finally show ?thesis .
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1085
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1086
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1087
lemma setsum_negf: "finite A ==> setsum (%x. - (f x)::'a::ab_group_add) A =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1088
  - setsum f A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1089
  by (induct set: Finites, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1090
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1091
lemma setsum_subtractf: "finite A ==> setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1092
  setsum f A - setsum g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1093
  by (simp add: diff_minus setsum_addf setsum_negf)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1094
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1095
lemma setsum_nonneg: "[| finite A;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1096
    \<forall>x \<in> A. (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) \<le> f x |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1097
    0 \<le> setsum f A";
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1098
  apply (induct set: Finites, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1099
  apply (subgoal_tac "0 + 0 \<le> f x + setsum f F", simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1100
  apply (blast intro: add_mono)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1101
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1102
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1103
lemma setsum_nonpos: "[| finite A;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1104
    \<forall>x \<in> A. f x \<le> (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1105
    setsum f A \<le> 0";
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1106
  apply (induct set: Finites, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1107
  apply (subgoal_tac "f x + setsum f F \<le> 0 + 0", simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1108
  apply (blast intro: add_mono)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1109
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1110
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1111
lemma setsum_mult: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1112
  fixes f :: "'a => ('b::semiring_0_cancel)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1113
  shows "r * setsum f A = setsum (%n. r * f n) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1114
proof (cases "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1115
  case True
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1116
  thus ?thesis
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1117
  proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1118
    case empty thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1119
  next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1120
    case (insert x A) thus ?case by (simp add: right_distrib)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1121
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1122
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1123
  case False thus ?thesis by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1124
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1125
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1126
lemma setsum_abs: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1127
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1128
  assumes fin: "finite A" 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1129
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1130
using fin 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1131
proof (induct) 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1132
  case empty thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1133
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1134
  case (insert x A)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1135
  thus ?case by (auto intro: abs_triangle_ineq order_trans)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1136
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1137
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1138
lemma setsum_abs_ge_zero: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1139
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1140
  assumes fin: "finite A" 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1141
  shows "0 \<le> setsum (%i. abs(f i)) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1142
using fin 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1143
proof (induct) 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1144
  case empty thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1145
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1146
  case (insert x A) thus ?case by (auto intro: order_trans)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1147
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1148
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1149
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1150
subsection {* Generalized product over a set *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1151
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1152
constdefs
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1153
  setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1154
  "setprod f A == if finite A then fold (op *) f 1 A else 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1155
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1156
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1157
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_:_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1158
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1159
syntax (xsymbols)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1160
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1161
syntax (HTML output)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1162
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1163
translations
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1164
  "\<Prod>i:A. b" == "setprod (%i. b) A"  -- {* Beware of argument permutation! *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1165
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1166
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1167
  "_Setprod" :: "'a set => 'a::comm_monoid_mult"  ("\<Prod>_" [1000] 999)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1168
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1169
parse_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1170
  let
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1171
    fun Setprod_tr [A] = Syntax.const "setprod" $ Abs ("", dummyT, Bound 0) $ A
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1172
  in [("_Setprod", Setprod_tr)] end;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1173
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1174
print_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1175
let fun setprod_tr' [Abs(x,Tx,t), A] =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1176
    if t = Bound 0 then Syntax.const "_Setprod" $ A else raise Match
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1177
in
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1178
[("setprod", setprod_tr')]
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1179
end
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1180
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1181
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1182
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1183
lemma setprod_empty [simp]: "setprod f {} = 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1184
  by (auto simp add: setprod_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1185
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1186
lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1187
    setprod f (insert a A) = f a * setprod f A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1188
by (simp add: setprod_def ACf.fold_insert [OF ACf_mult])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1189
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1190
lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1191
  by (simp add: setprod_def)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1192
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1193
lemma setprod_reindex:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1194
     "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1195
by(auto simp: setprod_def ACf.fold_reindex[OF ACf_mult] dest!:finite_imageD)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1196
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1197
lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1198
by (auto simp add: setprod_reindex)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1199
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1200
lemma setprod_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1201
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1202
by(fastsimp simp: setprod_def intro: ACf.fold_cong[OF ACf_mult])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1203
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1204
lemma setprod_reindex_cong: "inj_on f A ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1205
    B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1206
  by (frule setprod_reindex, simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1207
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1208
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1209
lemma setprod_1: "setprod (%i. 1) A = 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1210
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1211
  apply (erule finite_induct, auto simp add: mult_ac)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1212
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1213
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1214
lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1215
  apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1216
  apply (erule ssubst, rule setprod_1)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1217
  apply (rule setprod_cong, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1218
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1219
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1220
lemma setprod_Un_Int: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1221
    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1222
by(simp add: setprod_def ACe.fold_Un_Int[OF ACe_mult,symmetric])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1223
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1224
lemma setprod_Un_disjoint: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1225
  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1226
by (subst setprod_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1227
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1228
lemma setprod_UN_disjoint:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1229
    "finite I ==> (ALL i:I. finite (A i)) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1230
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1231
      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1232
by(simp add: setprod_def ACe.fold_UN_disjoint[OF ACe_mult] cong: setprod_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1233
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1234
lemma setprod_Union_disjoint:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1235
  "[| (ALL A:C. finite A);
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1236
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |] 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1237
   ==> setprod f (Union C) = setprod (setprod f) C"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1238
apply (cases "finite C") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1239
 prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1240
  apply (frule setprod_UN_disjoint [of C id f])
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1241
 apply (unfold Union_def id_def, assumption+)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1242
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1243
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1244
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1245
    (\<Prod>x:A. (\<Prod>y: B x. f x y)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1246
    (\<Prod>z:(SIGMA x:A. B x). f (fst z) (snd z))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1247
by(simp add:setprod_def ACe.fold_Sigma[OF ACe_mult] split_def cong:setprod_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1248
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1249
text{*Here we can eliminate the finiteness assumptions, by cases.*}
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1250
lemma setprod_cartesian_product: 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1251
     "(\<Prod>x:A. (\<Prod>y: B. f x y)) = (\<Prod>z:(A <*> B). f (fst z) (snd z))"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1252
apply (cases "finite A") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1253
 apply (cases "finite B") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1254
  apply (simp add: setprod_Sigma)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1255
 apply (cases "A={}", simp)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1256
 apply (simp add: setprod_1) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1257
apply (auto simp add: setprod_def
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1258
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1259
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1260
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1261
lemma setprod_timesf:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1262
     "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1263
by(simp add:setprod_def ACe.fold_distrib[OF ACe_mult])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1264
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1265
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1266
subsubsection {* Properties in more restricted classes of structures *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1267
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1268
lemma setprod_eq_1_iff [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1269
    "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1270
  by (induct set: Finites) auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1271
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1272
lemma setprod_zero:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1273
     "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1274
  apply (induct set: Finites, force, clarsimp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1275
  apply (erule disjE, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1276
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1277
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1278
lemma setprod_nonneg [rule_format]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1279
     "(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1280
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1281
  apply (induct set: Finites, force, clarsimp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1282
  apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1283
  apply (rule mult_mono, assumption+)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1284
  apply (auto simp add: setprod_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1285
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1286
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1287
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1288
     --> 0 < setprod f A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1289
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1290
  apply (induct set: Finites, force, clarsimp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1291
  apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1292
  apply (rule mult_strict_mono, assumption+)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1293
  apply (auto simp add: setprod_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1294
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1295
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1296
lemma setprod_nonzero [rule_format]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1297
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1298
      finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1299
  apply (erule finite_induct, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1300
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1301
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1302
lemma setprod_zero_eq:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1303
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1304
     finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1305
  apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1306
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1307
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1308
lemma setprod_nonzero_field:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1309
    "finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1310
  apply (rule setprod_nonzero, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1311
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1312
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1313
lemma setprod_zero_eq_field:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1314
    "finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1315
  apply (rule setprod_zero_eq, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1316
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1317
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1318
lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1319
    (setprod f (A Un B) :: 'a ::{field})
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1320
      = setprod f A * setprod f B / setprod f (A Int B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1321
  apply (subst setprod_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1322
  apply (subgoal_tac "finite (A Int B)")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1323
  apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1324
  apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1325
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1326
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1327
lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1328
    (setprod f (A - {a}) :: 'a :: {field}) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1329
      (if a:A then setprod f A / f a else setprod f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1330
  apply (erule finite_induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1331
   apply (auto simp add: insert_Diff_if)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1332
  apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1333
  apply (erule ssubst)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1334
  apply (subst times_divide_eq_right [THEN sym])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1335
  apply (auto simp add: mult_ac times_divide_eq_right divide_self)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1336
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1337
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1338
lemma setprod_inversef: "finite A ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1339
    ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1340
      setprod (inverse \<circ> f) A = inverse (setprod f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1341
  apply (erule finite_induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1342
  apply (simp, simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1343
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1344
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1345
lemma setprod_dividef:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1346
     "[|finite A;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1347
        \<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1348
      ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1349
  apply (subgoal_tac
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1350
         "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1351
  apply (erule ssubst)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1352
  apply (subst divide_inverse)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1353
  apply (subst setprod_timesf)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1354
  apply (subst setprod_inversef, assumption+, rule refl)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1355
  apply (rule setprod_cong, rule refl)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1356
  apply (subst divide_inverse, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1357
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1358
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1359
subsection {* Finite cardinality *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1360
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1361
text {* This definition, although traditional, is ugly to work with:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1362
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1363
But now that we have @{text setsum} things are easy:
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1364
*}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1365
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1366
constdefs
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1367
  card :: "'a set => nat"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1368
  "card A == setsum (%x. 1::nat) A"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1369
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1370
lemma card_empty [simp]: "card {} = 0"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1371
  by (simp add: card_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1372
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1373
lemma card_infinite [simp]: "~ finite A ==> card A = 0"