src/HOL/Finite_Set.thy
author huffman
Tue, 09 Dec 2008 15:31:38 -0800
changeset 29025 8c8859c0d734
parent 28853 69eb69659bf3
child 29223 e09c53289830
permissions -rw-r--r--
move lemmas from Numeral_Type.thy to other theories
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
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
     4
                with contributions by Jeremy Avigad
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
27981
feb0c01cf0fb tuned import order
haftmann
parents: 27611
diff changeset
    10
imports Datatype Divides Transitive_Closure
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
23736
bf8d4a46452d Renamed inductive2 to inductive.
berghofe
parents: 23706
diff changeset
    15
inductive finite :: "'a set => bool"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
    16
  where
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
    17
    emptyI [simp, intro!]: "finite {}"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
    18
  | insertI [simp, intro!]: "finite A ==> finite (insert a A)"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    19
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13735
diff changeset
    20
lemma ex_new_if_finite: -- "does not depend on def of finite at all"
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
    21
  assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
    22
  shows "\<exists>a::'a. a \<notin> A"
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
    23
proof -
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 27981
diff changeset
    24
  from assms have "A \<noteq> UNIV" by blast
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
    25
  thus ?thesis by blast
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
    26
qed
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    27
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
    28
lemma finite_induct [case_names empty insert, induct set: finite]:
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    29
  "finite F ==>
15327
0230a10582d3 changed the order of !!-quantifiers in finite set induction.
nipkow
parents: 15318
diff changeset
    30
    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
    31
  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    32
proof -
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13400
diff changeset
    33
  assume "P {}" and
15327
0230a10582d3 changed the order of !!-quantifiers in finite set induction.
nipkow
parents: 15318
diff changeset
    34
    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
    35
  assume "finite F"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    36
  thus "P F"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    37
  proof induct
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    38
    show "P {}" by fact
15327
0230a10582d3 changed the order of !!-quantifiers in finite set induction.
nipkow
parents: 15318
diff changeset
    39
    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
    40
    show "P (insert x F)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    41
    proof cases
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    42
      assume "x \<in> F"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    43
      hence "insert x F = F" by (rule insert_absorb)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    44
      with P show ?thesis by (simp only:)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    45
    next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    46
      assume "x \<notin> F"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    47
      from F this P show ?thesis by (rule insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    48
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    49
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    50
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    51
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    52
lemma finite_ne_induct[case_names singleton insert, consumes 2]:
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    53
assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    54
 \<lbrakk> \<And>x. P{x};
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    55
   \<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
    56
 \<Longrightarrow> P F"
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    57
using fin
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    58
proof induct
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    59
  case empty thus ?case by simp
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    60
next
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    61
  case (insert x F)
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    62
  show ?case
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    63
  proof cases
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    64
    assume "F = {}"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    65
    thus ?thesis using `P {x}` by simp
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    66
  next
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    67
    assume "F \<noteq> {}"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    68
    thus ?thesis using insert by blast
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    69
  qed
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    70
qed
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
    71
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    72
lemma finite_subset_induct [consumes 2, case_names empty insert]:
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    73
  assumes "finite F" and "F \<subseteq> A"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    74
    and empty: "P {}"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    75
    and insert: "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    76
  shows "P F"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    77
proof -
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    78
  from `finite F` and `F \<subseteq> A`
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    79
  show ?thesis
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    80
  proof induct
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    81
    show "P {}" by fact
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    82
  next
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    83
    fix x F
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    84
    assume "finite F" and "x \<notin> F" and
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    85
      P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    86
    show "P (insert x F)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    87
    proof (rule insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    88
      from i show "x \<in> A" by blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    89
      from i have "F \<subseteq> A" by blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    90
      with P show "P F" .
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    91
      show "finite F" by fact
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
    92
      show "x \<notin> F" by fact
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    93
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    94
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    95
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
    96
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23736
diff changeset
    97
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
    98
text{* Finite sets are the images of initial segments of natural numbers: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
    99
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   100
lemma finite_imp_nat_seg_image_inj_on:
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   101
  assumes fin: "finite A" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   102
  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
   103
using fin
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   104
proof induct
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   105
  case empty
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   106
  show ?case  
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   107
  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
   108
  qed
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   109
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   110
  case (insert a A)
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   111
  have notinA: "a \<notin> A" by fact
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   112
  from insert.hyps obtain n f
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   113
    where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   114
  hence "insert a A = f(n:=a) ` {i. i < Suc n}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   115
        "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   116
    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
   117
  thus ?case by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   118
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   119
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   120
lemma nat_seg_image_imp_finite:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   121
  "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   122
proof (induct n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   123
  case 0 thus ?case by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   124
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   125
  case (Suc n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   126
  let ?B = "f ` {i. i < n}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   127
  have finB: "finite ?B" by(rule Suc.hyps[OF refl])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   128
  show ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   129
  proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   130
    assume "\<exists>k<n. f n = f k"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   131
    hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   132
    thus ?thesis using finB by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   133
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   134
    assume "\<not>(\<exists> k<n. f n = f k)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   135
    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
   136
    thus ?thesis using finB by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   137
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   138
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   139
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   140
lemma finite_conv_nat_seg_image:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   141
  "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   142
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
   143
26441
7914697ff104 no "attach UNIV" any more
haftmann
parents: 26146
diff changeset
   144
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   145
subsubsection{* Finiteness and set theoretic constructions *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   146
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   147
lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   148
  -- {* The union of two finite sets is finite. *}
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   149
  by (induct set: finite) simp_all
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   150
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   151
lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   152
  -- {* Every subset of a finite set is finite. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   153
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   154
  assume "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   155
  thus "!!A. A \<subseteq> B ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   156
  proof induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   157
    case empty
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   158
    thus ?case by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   159
  next
15327
0230a10582d3 changed the order of !!-quantifiers in finite set induction.
nipkow
parents: 15318
diff changeset
   160
    case (insert x F A)
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   161
    have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" by fact+
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   162
    show "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   163
    proof cases
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   164
      assume x: "x \<in> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   165
      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
   166
      with r have "finite (A - {x})" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   167
      hence "finite (insert x (A - {x}))" ..
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   168
      also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   169
      finally show ?thesis .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   170
    next
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   171
      show "A \<subseteq> F ==> ?thesis" by fact
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   172
      assume "x \<notin> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   173
      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
   174
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   175
  qed
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
18423
d7859164447f new lemmas
nipkow
parents: 17782
diff changeset
   178
lemma finite_Collect_subset[simp]: "finite A \<Longrightarrow> finite{x \<in> A. P x}"
17761
2c42d0a94f58 new lemmas
nipkow
parents: 17589
diff changeset
   179
using finite_subset[of "{x \<in> A. P x}" "A"] by blast
2c42d0a94f58 new lemmas
nipkow
parents: 17589
diff changeset
   180
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   181
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
   182
  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
   183
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   184
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
   185
  -- {* The converse obviously fails. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   186
  by (blast intro: finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   187
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   188
lemma finite_insert [simp]: "finite (insert a A) = finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   189
  apply (subst insert_is_Un)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   190
  apply (simp only: finite_Un, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   191
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   192
15281
bd4611956c7b More lemmas
nipkow
parents: 15234
diff changeset
   193
lemma finite_Union[simp, intro]:
bd4611956c7b More lemmas
nipkow
parents: 15234
diff changeset
   194
 "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
bd4611956c7b More lemmas
nipkow
parents: 15234
diff changeset
   195
by (induct rule:finite_induct) simp_all
bd4611956c7b More lemmas
nipkow
parents: 15234
diff changeset
   196
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   197
lemma finite_empty_induct:
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   198
  assumes "finite A"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   199
    and "P A"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   200
    and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   201
  shows "P {}"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   202
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   203
  have "P (A - A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   204
  proof -
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   205
    {
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   206
      fix c b :: "'a set"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   207
      assume c: "finite c" and b: "finite b"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   208
	and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   209
      have "c \<subseteq> b ==> P (b - c)"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   210
	using c
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   211
      proof induct
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   212
	case empty
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   213
	from P1 show ?case by simp
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   214
      next
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   215
	case (insert x F)
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   216
	have "P (b - F - {x})"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   217
	proof (rule P2)
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   218
          from _ b show "finite (b - F)" by (rule finite_subset) blast
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   219
          from insert show "x \<in> b - F" by simp
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   220
          from insert show "P (b - F)" by simp
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   221
	qed
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   222
	also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   223
	finally show ?case .
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   224
      qed
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   225
    }
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   226
    then show ?thesis by this (simp_all add: assms)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   227
  qed
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   228
  then show ?thesis by simp
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   229
qed
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 [simp]: "finite B ==> finite (B - Ba)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   232
  by (rule Diff_subset [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   233
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   234
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
   235
  apply (subst Diff_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   236
  apply (case_tac "a : A - B")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   237
   apply (rule finite_insert [symmetric, THEN trans])
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   238
   apply (subst insert_Diff, simp_all)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   239
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   240
19870
ef037d1b32d1 new results
paulson
parents: 19868
diff changeset
   241
lemma finite_Diff_singleton [simp]: "finite (A - {a}) = finite A"
ef037d1b32d1 new results
paulson
parents: 19868
diff changeset
   242
  by simp
ef037d1b32d1 new results
paulson
parents: 19868
diff changeset
   243
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   244
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   245
text {* Image and Inverse Image over Finite Sets *}
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   246
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   247
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   248
  -- {* The image of a finite set is finite. *}
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   249
  by (induct set: finite) simp_all
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   250
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   251
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
   252
  apply (frule finite_imageI)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   253
  apply (erule finite_subset, assumption)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   254
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   255
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   256
lemma finite_range_imageI:
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   257
    "finite (range g) ==> finite (range (%x. f (g x)))"
27418
564117b58d73 remove simp attribute from range_composition
huffman
parents: 27165
diff changeset
   258
  apply (drule finite_imageI, simp add: range_composition)
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   259
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   260
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   261
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
   262
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   263
  have aux: "!!A. finite (A - {}) = finite A" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   264
  fix B :: "'a set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   265
  assume "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   266
  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   267
    apply induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   268
     apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   269
    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
   270
     apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   271
     apply (simp (no_asm_use) add: inj_on_def)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   272
     apply (blast dest!: aux [THEN iffD1], atomize)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   273
    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   274
    apply (frule subsetD [OF equalityD2 insertI1], clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   275
    apply (rule_tac x = xa in bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   276
     apply (simp_all add: inj_on_image_set_diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   277
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   278
qed (rule refl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   279
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   280
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   281
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   282
  -- {* The inverse image of a singleton under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   283
         is included in a singleton. *}
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   284
  apply (auto simp add: inj_on_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   285
  apply (blast intro: the_equality [symmetric])
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   286
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   287
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   288
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   289
  -- {* The inverse image of a finite set under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   290
         is finite. *}
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   291
  apply (induct set: finite)
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
   292
   apply simp_all
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   293
  apply (subst vimage_insert)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   294
  apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   295
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   296
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   297
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   298
text {* The finite UNION of finite sets *}
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
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   301
  by (induct set: finite) simp_all
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   302
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   303
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   304
  Strengthen RHS to
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   305
  @{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
   306
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   307
  We'd need to prove
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   308
  @{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
   309
  by induction. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   310
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   311
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
   312
  by (blast intro: finite_UN_I finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   313
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   314
17022
b257300c3a9c added Brian Hufmann's finite instances
nipkow
parents: 16775
diff changeset
   315
lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"
b257300c3a9c added Brian Hufmann's finite instances
nipkow
parents: 16775
diff changeset
   316
by (simp add: Plus_def)
b257300c3a9c added Brian Hufmann's finite instances
nipkow
parents: 16775
diff changeset
   317
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   318
text {* Sigma of finite sets *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   319
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   320
lemma finite_SigmaI [simp]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   321
    "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
   322
  by (unfold Sigma_def) (blast intro!: finite_UN_I)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   323
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   324
lemma finite_cartesian_product: "[| finite A; finite B |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   325
    finite (A <*> B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   326
  by (rule finite_SigmaI)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   327
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   328
lemma finite_Prod_UNIV:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   329
    "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
   330
  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   331
   apply (erule ssubst)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   332
   apply (erule finite_SigmaI, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   333
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   334
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   335
lemma finite_cartesian_productD1:
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   336
     "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   337
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
   338
apply (drule_tac x=n in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   339
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
   340
apply (auto simp add: o_def) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   341
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   342
apply (drule equalityD1) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   343
apply (rename_tac y x)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   344
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
   345
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   346
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   347
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
   348
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   349
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   350
lemma finite_cartesian_productD2:
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   351
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   352
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
   353
apply (drule_tac x=n in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   354
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
   355
apply (auto simp add: o_def) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   356
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   357
apply (drule equalityD1)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   358
apply (rename_tac x y)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   359
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
   360
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   361
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   362
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
   363
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   364
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   365
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   366
text {* The powerset of a finite set *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   367
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   368
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   369
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   370
  assume "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   371
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   372
  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
   373
next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   374
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   375
  thus "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   376
    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   377
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   378
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   379
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   380
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   381
by(blast intro: finite_subset[OF subset_Pow_Union])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   382
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   383
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   384
lemma finite_converse [iff]: "finite (r^-1) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   385
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   386
   apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   387
   apply (rule iffI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   388
    apply (erule finite_imageD [unfolded inj_on_def])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   389
    apply (simp split add: split_split)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   390
   apply (erule finite_imageI)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   391
  apply (simp add: converse_def image_def, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   392
  apply (rule bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   393
   prefer 2 apply assumption
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   394
  apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   395
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   396
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   397
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   398
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   399
Ehmety) *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   400
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   401
lemma finite_Field: "finite r ==> finite (Field r)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   402
  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   403
  apply (induct set: finite)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   404
   apply (auto simp add: Field_def Domain_insert Range_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   405
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   406
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   407
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   408
  apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   409
  apply (erule trancl_induct)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   410
   apply (auto simp add: Field_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   411
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   412
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   413
lemma finite_trancl: "finite (r^+) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   414
  apply auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   415
   prefer 2
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   416
   apply (rule trancl_subset_Field2 [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   417
   apply (rule finite_SigmaI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   418
    prefer 3
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 13595
diff changeset
   419
    apply (blast intro: r_into_trancl' finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   420
   apply (auto simp add: finite_Field)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   421
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   422
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   423
26441
7914697ff104 no "attach UNIV" any more
haftmann
parents: 26146
diff changeset
   424
subsection {* Class @{text finite}  *}
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   425
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   426
setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*}
26441
7914697ff104 no "attach UNIV" any more
haftmann
parents: 26146
diff changeset
   427
class finite = itself +
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   428
  assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   429
setup {* Sign.parent_path *}
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   430
hide const finite
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   431
27430
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   432
context finite
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   433
begin
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   434
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   435
lemma finite [simp]: "finite (A \<Colon> 'a set)"
26441
7914697ff104 no "attach UNIV" any more
haftmann
parents: 26146
diff changeset
   436
  by (rule subset_UNIV finite_UNIV finite_subset)+
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   437
27430
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   438
end
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   439
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   440
lemma UNIV_unit [noatp]:
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   441
  "UNIV = {()}" by auto
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   442
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   443
instance unit :: finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   444
  by default (simp add: UNIV_unit)
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   445
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   446
lemma UNIV_bool [noatp]:
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   447
  "UNIV = {False, True}" by auto
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   448
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   449
instance bool :: finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   450
  by default (simp add: UNIV_bool)
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   451
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   452
instance * :: (finite, finite) finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   453
  by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   454
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   455
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   456
  by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   457
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   458
instance "fun" :: (finite, finite) finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   459
proof
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   460
  show "finite (UNIV :: ('a => 'b) set)"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   461
  proof (rule finite_imageD)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   462
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
26792
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   463
    have "range ?graph \<subseteq> Pow UNIV" by simp
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   464
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   465
      by (simp only: finite_Pow_iff finite)
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   466
    ultimately show "finite (range ?graph)"
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   467
      by (rule finite_subset)
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   468
    show "inj ?graph" by (rule inj_graph)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   469
  qed
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   470
qed
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   471
27981
feb0c01cf0fb tuned import order
haftmann
parents: 27611
diff changeset
   472
instance "+" :: (finite, finite) finite
feb0c01cf0fb tuned import order
haftmann
parents: 27611
diff changeset
   473
  by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
feb0c01cf0fb tuned import order
haftmann
parents: 27611
diff changeset
   474
feb0c01cf0fb tuned import order
haftmann
parents: 27611
diff changeset
   475
instance option :: (finite) finite
feb0c01cf0fb tuned import order
haftmann
parents: 27611
diff changeset
   476
  by default (simp add: insert_None_conv_UNIV [symmetric])
feb0c01cf0fb tuned import order
haftmann
parents: 27611
diff changeset
   477
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   478
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   479
subsection {* A fold functional for finite sets *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   480
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   481
text {* The intended behaviour is
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   482
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   483
if @{text f} is ``left-commutative'':
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   484
*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   485
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   486
locale fun_left_comm =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   487
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   488
  assumes fun_left_comm: "f x (f y z) = f y (f x z)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   489
begin
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   490
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   491
text{* On a functional level it looks much nicer: *}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   492
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   493
lemma fun_comp_comm:  "f x \<circ> f y = f y \<circ> f x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   494
by (simp add: fun_left_comm expand_fun_eq)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   495
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   496
end
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   497
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   498
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   499
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   500
  emptyI [intro]: "fold_graph f z {} z" |
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   501
  insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   502
      \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   503
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   504
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   505
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   506
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   507
[code del]: "fold f z A = (THE y. fold_graph f z A y)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   508
15498
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   509
text{*A tempting alternative for the definiens is
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   510
@{term "if finite A then THE y. fold_graph f z A y else e"}.
15498
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   511
It allows the removal of finiteness assumptions from the theorems
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   512
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   513
The proofs become ugly. It is not worth the effort. (???) *}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   514
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   515
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   516
lemma Diff1_fold_graph:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   517
  "fold_graph f z (A - {x}) y \<Longrightarrow> x \<in> A \<Longrightarrow> fold_graph f z A (f x y)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   518
by (erule insert_Diff [THEN subst], rule fold_graph.intros, auto)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   519
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   520
lemma fold_graph_imp_finite: "fold_graph f z A x \<Longrightarrow> finite A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   521
by (induct set: fold_graph) auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   522
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   523
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   524
by (induct set: finite) auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   525
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   526
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   527
subsubsection{*From @{const fold_graph} to @{term fold}*}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   528
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   529
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
19868
wenzelm
parents: 19793
diff changeset
   530
  by (auto simp add: less_Suc_eq) 
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   531
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   532
lemma insert_image_inj_on_eq:
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   533
     "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   534
        inj_on h {i. i < Suc m}|] 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   535
      ==> A = h ` {i. i < m}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   536
apply (auto simp add: image_less_Suc inj_on_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   537
apply (blast intro: less_trans) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   538
done
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   539
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   540
lemma insert_inj_onE:
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   541
  assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   542
      and inj_on: "inj_on h {i::nat. i<n}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   543
  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
   544
proof (cases n)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   545
  case 0 thus ?thesis using aA by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   546
next
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   547
  case (Suc m)
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   548
  have nSuc: "n = Suc m" by fact
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   549
  have mlessn: "m<n" by (simp add: nSuc)
15532
9712d41db5b8 simplified a proof
paulson
parents: 15526
diff changeset
   550
  from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
27165
e1c49eb8cee6 Hid swap
nipkow
parents: 26792
diff changeset
   551
  let ?hm = "Fun.swap k m h"
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   552
  have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   553
    by (simp add: inj_on_swap_iff inj_on)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   554
  show ?thesis
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   555
  proof (intro exI conjI)
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   556
    show "inj_on ?hm {i. i < m}" using inj_hm
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   557
      by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   558
    show "m<n" by (rule mlessn)
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   559
    show "A = ?hm ` {i. i < m}" 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   560
    proof (rule insert_image_inj_on_eq)
27165
e1c49eb8cee6 Hid swap
nipkow
parents: 26792
diff changeset
   561
      show "inj_on (Fun.swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   562
      show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   563
      show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   564
	using aA hkeq nSuc klessn
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   565
	by (auto simp add: swap_def image_less_Suc fun_upd_image 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   566
			   less_Suc_eq inj_on_image_set_diff [OF inj_on])
15479
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   567
    qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   568
  qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   569
qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   570
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   571
context fun_left_comm
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   572
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   573
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   574
lemma fold_graph_determ_aux:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   575
  "A = h`{i::nat. i<n} \<Longrightarrow> inj_on h {i. i<n}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   576
   \<Longrightarrow> fold_graph f z A x \<Longrightarrow> fold_graph f z A x'
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   577
   \<Longrightarrow> x' = x"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   578
proof (induct n arbitrary: A x x' h rule: less_induct)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   579
  case (less n)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   580
  have IH: "\<And>m h A x x'. m < n \<Longrightarrow> A = h ` {i. i<m}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   581
      \<Longrightarrow> inj_on h {i. i<m} \<Longrightarrow> fold_graph f z A x
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   582
      \<Longrightarrow> fold_graph f z A x' \<Longrightarrow> x' = x" by fact
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   583
  have Afoldx: "fold_graph f z A x" and Afoldx': "fold_graph f z A x'"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   584
    and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" by fact+
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   585
  show ?case
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   586
  proof (rule fold_graph.cases [OF Afoldx])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   587
    assume "A = {}" and "x = z"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   588
    with Afoldx' show "x' = x" by auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   589
  next
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   590
    fix B b u
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   591
    assume AbB: "A = insert b B" and x: "x = f b u"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   592
      and notinB: "b \<notin> B" and Bu: "fold_graph f z B u"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   593
    show "x'=x" 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   594
    proof (rule fold_graph.cases [OF Afoldx'])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   595
      assume "A = {}" and "x' = z"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   596
      with AbB show "x' = x" by blast
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   597
    next
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   598
      fix C c v
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   599
      assume AcC: "A = insert c C" and x': "x' = f c v"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   600
        and notinC: "c \<notin> C" and Cv: "fold_graph f z C v"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   601
      from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   602
      from insert_inj_onE [OF Beq notinB injh]
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   603
      obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   604
        and Beq: "B = hB ` {i. i < mB}" and lessB: "mB < n" by auto 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   605
      from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   606
      from insert_inj_onE [OF Ceq notinC injh]
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   607
      obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   608
        and Ceq: "C = hC ` {i. i < mC}" and lessC: "mC < n" by auto 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   609
      show "x'=x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   610
      proof cases
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   611
        assume "b=c"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   612
	then moreover have "B = C" using AbB AcC notinB notinC by auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   613
	ultimately show ?thesis  using Bu Cv x x' IH [OF lessC Ceq inj_onC]
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   614
          by auto
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   615
      next
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   616
	assume diff: "b \<noteq> c"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   617
	let ?D = "B - {c}"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   618
	have B: "B = insert c ?D" and C: "C = insert b ?D"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   619
	  using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   620
	have "finite A" by(rule fold_graph_imp_finite [OF Afoldx])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   621
	with AbB have "finite ?D" by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   622
	then obtain d where Dfoldd: "fold_graph f z ?D d"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   623
	  using finite_imp_fold_graph by iprover
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   624
	moreover have cinB: "c \<in> B" using B by auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   625
	ultimately have "fold_graph f z B (f c d)" by(rule Diff1_fold_graph)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   626
	hence "f c d = u" by (rule IH [OF lessB Beq inj_onB Bu]) 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   627
        moreover have "f b d = v"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   628
	proof (rule IH[OF lessC Ceq inj_onC Cv])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   629
	  show "fold_graph f z C (f b d)" using C notinB Dfoldd by fastsimp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   630
	qed
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   631
	ultimately show ?thesis
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   632
          using fun_left_comm [of c b] x x' by (auto simp add: o_def)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   633
      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   634
    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   635
  qed
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   636
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   637
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   638
lemma fold_graph_determ:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   639
  "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   640
apply (frule fold_graph_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   641
apply (blast intro: fold_graph_determ_aux [rule_format])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   642
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   643
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   644
lemma fold_equality:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   645
  "fold_graph f z A y \<Longrightarrow> fold f z A = y"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   646
by (unfold fold_def) (blast intro: fold_graph_determ)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   647
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   648
text{* The base case for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   649
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   650
lemma (in -) fold_empty [simp]: "fold f z {} = z"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   651
by (unfold fold_def) blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   652
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   653
text{* The various recursion equations for @{const fold}: *}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   654
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   655
lemma fold_insert_aux: "x \<notin> A
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   656
  \<Longrightarrow> fold_graph f z (insert x A) v \<longleftrightarrow>
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   657
      (\<exists>y. fold_graph f z A y \<and> v = f x y)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   658
apply auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   659
apply (rule_tac A1 = A and f1 = f in finite_imp_fold_graph [THEN exE])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   660
 apply (fastsimp dest: fold_graph_imp_finite)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   661
apply (blast intro: fold_graph_determ)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   662
done
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   663
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   664
lemma fold_insert [simp]:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   665
  "finite A ==> x \<notin> A ==> fold f z (insert x A) = f x (fold f z A)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   666
apply (simp add: fold_def fold_insert_aux)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   667
apply (rule the_equality)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   668
 apply (auto intro: finite_imp_fold_graph
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   669
        cong add: conj_cong simp add: fold_def[symmetric] fold_equality)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   670
done
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   671
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   672
lemma fold_fun_comm:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   673
  "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   674
proof (induct rule: finite_induct)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   675
  case empty then show ?case by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   676
next
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   677
  case (insert y A) then show ?case
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   678
    by (simp add: fun_left_comm[of x])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   679
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   680
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   681
lemma fold_insert2:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   682
  "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   683
by (simp add: fold_insert fold_fun_comm)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   684
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   685
lemma fold_rec:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   686
assumes "finite A" and "x \<in> A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   687
shows "fold f z A = f x (fold f z (A - {x}))"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   688
proof -
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   689
  have A: "A = insert x (A - {x})" using `x \<in> A` by blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   690
  then have "fold f z A = fold f z (insert x (A - {x}))" by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   691
  also have "\<dots> = f x (fold f z (A - {x}))"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   692
    by (rule fold_insert) (simp add: `finite A`)+
15535
nipkow
parents: 15532
diff changeset
   693
  finally show ?thesis .
nipkow
parents: 15532
diff changeset
   694
qed
nipkow
parents: 15532
diff changeset
   695
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   696
lemma fold_insert_remove:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   697
  assumes "finite A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   698
  shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   699
proof -
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   700
  from `finite A` have "finite (insert x A)" by auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   701
  moreover have "x \<in> insert x A" by auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   702
  ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   703
    by (rule fold_rec)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   704
  then show ?thesis by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   705
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   706
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   707
end
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   708
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   709
text{* A simplified version for idempotent functions: *}
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   710
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   711
locale fun_left_comm_idem = fun_left_comm +
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   712
  assumes fun_left_idem: "f x (f x z) = f x z"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   713
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   714
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   715
text{* The nice version: *}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   716
lemma fun_comp_idem : "f x o f x = f x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   717
by (simp add: fun_left_idem expand_fun_eq)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   718
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   719
lemma fold_insert_idem:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   720
  assumes fin: "finite A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   721
  shows "fold f z (insert x A) = f x (fold f z A)"
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   722
proof cases
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   723
  assume "x \<in> A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   724
  then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   725
  then show ?thesis using assms by (simp add:fun_left_idem)
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   726
next
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   727
  assume "x \<notin> A" then show ?thesis using assms by simp
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   728
qed
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   729
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   730
declare fold_insert[simp del] fold_insert_idem[simp]
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   731
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   732
lemma fold_insert_idem2:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   733
  "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   734
by(simp add:fold_fun_comm)
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   735
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   736
end
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   737
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   738
subsubsection{* The derived combinator @{text fold_image} *}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   739
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   740
definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   741
where "fold_image f g = fold (%x y. f (g x) y)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   742
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   743
lemma fold_image_empty[simp]: "fold_image f g z {} = z"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   744
by(simp add:fold_image_def)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   745
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   746
context ab_semigroup_mult
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   747
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   748
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   749
lemma fold_image_insert[simp]:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   750
assumes "finite A" and "a \<notin> A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   751
shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   752
proof -
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   753
  interpret I: fun_left_comm ["%x y. (g x) * y"]
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   754
    by unfold_locales (simp add: mult_ac)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   755
  show ?thesis using assms by(simp add:fold_image_def I.fold_insert)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   756
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   757
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   758
(*
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   759
lemma fold_commute:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   760
  "finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   761
  apply (induct set: finite)
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
   762
   apply simp
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   763
  apply (simp add: mult_left_commute [of x])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   764
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   765
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   766
lemma fold_nest_Un_Int:
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   767
  "finite A ==> finite B
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   768
    ==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   769
  apply (induct set: finite)
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
   770
   apply simp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   771
  apply (simp add: fold_commute Int_insert_left insert_absorb)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   772
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   773
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   774
lemma fold_nest_Un_disjoint:
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   775
  "finite A ==> finite B ==> A Int B = {}
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   776
    ==> fold times g z (A Un B) = fold times g (fold times g z B) A"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   777
  by (simp add: fold_nest_Un_Int)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   778
*)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   779
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   780
lemma fold_image_reindex:
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   781
assumes fin: "finite A"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   782
shows "inj_on h A \<Longrightarrow> fold_image times g z (h`A) = fold_image times (g\<circ>h) z A"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   783
using fin apply induct
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   784
 apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   785
apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   786
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   787
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   788
(*
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   789
text{*
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   790
  Fusion theorem, as described in Graham Hutton's paper,
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   791
  A Tutorial on the Universality and Expressiveness of Fold,
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   792
  JFP 9:4 (355-372), 1999.
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   793
*}
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   794
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   795
lemma fold_fusion:
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 27430
diff changeset
   796
  assumes "ab_semigroup_mult g"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   797
  assumes fin: "finite A"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   798
    and hyp: "\<And>x y. h (g x y) = times x (h y)"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   799
  shows "h (fold g j w A) = fold times j (h w) A"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 27430
diff changeset
   800
proof -
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 27430
diff changeset
   801
  interpret ab_semigroup_mult [g] by fact
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 27430
diff changeset
   802
  show ?thesis using fin hyp by (induct set: finite) simp_all
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 27430
diff changeset
   803
qed
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   804
*)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   805
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   806
lemma fold_image_cong:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   807
  "finite A \<Longrightarrow>
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   808
  (!!x. x:A ==> g x = h x) ==> fold_image times g z A = fold_image times h z A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   809
apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   810
 apply simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   811
apply (erule finite_induct, simp)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   812
apply (simp add: subset_insert_iff, clarify)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   813
apply (subgoal_tac "finite C")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   814
 prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   815
apply (subgoal_tac "C = insert x (C - {x})")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   816
 prefer 2 apply blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   817
apply (erule ssubst)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   818
apply (drule spec)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   819
apply (erule (1) notE impE)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   820
apply (simp add: Ball_def del: insert_Diff_single)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   821
done
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   822
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   823
end
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   824
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   825
context comm_monoid_mult
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   826
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   827
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   828
lemma fold_image_Un_Int:
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   829
  "finite A ==> finite B ==>
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   830
    fold_image times g 1 A * fold_image times g 1 B =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   831
    fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   832
by (induct set: finite) 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   833
   (auto simp add: mult_ac insert_absorb Int_insert_left)
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   834
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   835
corollary fold_Un_disjoint:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   836
  "finite A ==> finite B ==> A Int B = {} ==>
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   837
   fold_image times g 1 (A Un B) =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   838
   fold_image times g 1 A * fold_image times g 1 B"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   839
by (simp add: fold_image_Un_Int)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   840
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   841
lemma fold_image_UN_disjoint:
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   842
  "\<lbrakk> finite I; ALL i:I. finite (A i);
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   843
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   844
   \<Longrightarrow> fold_image times g 1 (UNION I A) =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   845
       fold_image times (%i. fold_image times g 1 (A i)) 1 I"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   846
apply (induct set: finite, simp, atomize)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   847
apply (subgoal_tac "ALL i:F. x \<noteq> i")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   848
 prefer 2 apply blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   849
apply (subgoal_tac "A x Int UNION F A = {}")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   850
 prefer 2 apply blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   851
apply (simp add: fold_Un_disjoint)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   852
done
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   854
lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   855
  fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   856
  fold_image times (split g) 1 (SIGMA x:A. B x)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   857
apply (subst Sigma_def)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   858
apply (subst fold_image_UN_disjoint, assumption, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   859
 apply blast
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   860
apply (erule fold_image_cong)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   861
apply (subst fold_image_UN_disjoint, simp, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   862
 apply blast
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   863
apply simp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   864
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   865
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   866
lemma fold_image_distrib: "finite A \<Longrightarrow>
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   867
   fold_image times (%x. g x * h x) 1 A =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   868
   fold_image times g 1 A *  fold_image times h 1 A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   869
by (erule finite_induct) (simp_all add: mult_ac)
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   870
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   871
end
22917
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
   872
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
   873
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   874
subsection {* Generalized summation over a set *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   875
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   876
interpretation comm_monoid_add: comm_monoid_mult ["0::'a::comm_monoid_add" "op +"]
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 27981
diff changeset
   877
  proof qed (auto intro: add_assoc add_commute)
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   878
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   879
definition setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   880
where "setsum f A == if finite A then fold_image (op +) f 0 A else 0"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   881
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   882
abbreviation
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21249
diff changeset
   883
  Setsum  ("\<Sum>_" [1000] 999) where
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   884
  "\<Sum>A == setsum (%x. x) A"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   885
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   886
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   887
written @{text"\<Sum>x\<in>A. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   888
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   889
syntax
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   890
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   891
syntax (xsymbols)
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   892
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   893
syntax (HTML output)
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   894
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   895
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   896
translations -- {* Beware of argument permutation! *}
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   897
  "SUM i:A. b" == "CONST setsum (%i. b) A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   898
  "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   899
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   900
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   901
 @{text"\<Sum>x|P. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   902
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   903
syntax
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   904
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   905
syntax (xsymbols)
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   906
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   907
syntax (HTML output)
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   908
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   909
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   910
translations
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   911
  "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   912
  "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   913
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   914
print_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   915
let
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   916
  fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   917
    if x<>y then raise Match
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   918
    else let val x' = Syntax.mark_bound x
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   919
             val t' = subst_bound(x',t)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   920
             val P' = subst_bound(x',P)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   921
         in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   922
in [("setsum", setsum_tr')] end
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   923
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   924
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   925
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   926
lemma setsum_empty [simp]: "setsum f {} = 0"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   927
by (simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   928
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   929
lemma setsum_insert [simp]:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   930
  "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   931
by (simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   932
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   933
lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   934
by (simp add: setsum_def)
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   935
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   936
lemma setsum_reindex:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   937
     "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   938
by(auto simp add: setsum_def comm_monoid_add.fold_image_reindex dest!:finite_imageD)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   939
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   940
lemma setsum_reindex_id:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   941
     "inj_on f B ==> setsum f B = setsum id (f ` B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   942
by (auto simp add: setsum_reindex)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   943
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   944
lemma setsum_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   945
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   946
by(fastsimp simp: setsum_def intro: comm_monoid_add.fold_image_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   947
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16632
diff changeset
   948
lemma strong_setsum_cong[cong]:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16632
diff changeset
   949
  "A = B ==> (!!x. x:B =simp=> f x = g x)
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16632
diff changeset
   950
   ==> setsum (%x. f x) A = setsum (%x. g x) B"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   951
by(fastsimp simp: simp_implies_def setsum_def intro: comm_monoid_add.fold_image_cong)
16632
ad2895beef79 Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents: 16550
diff changeset
   952
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
   953
lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A";
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   954
by (rule setsum_cong[OF refl], auto);
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
   955
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   956
lemma setsum_reindex_cong:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   957
   "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   958
    ==> setsum h B = setsum g A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   959
by (simp add: setsum_reindex cong: setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   960
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   961
lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   962
apply (clarsimp simp: setsum_def)
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   963
apply (erule finite_induct, auto)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   964
done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   965
15543
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   966
lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   967
by(simp add:setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   968
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   969
lemma setsum_Un_Int: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   970
  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
   971
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   972
by(simp add: setsum_def comm_monoid_add.fold_image_Un_Int [symmetric])
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   973
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   974
lemma setsum_Un_disjoint: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   975
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   976
by (subst setsum_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   977
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   978
(*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
   979
  the lhs need not be, since UNION I A could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   980
lemma setsum_UN_disjoint:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   981
    "finite I ==> (ALL i:I. finite (A i)) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   982
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   983
      setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   984
by(simp add: setsum_def comm_monoid_add.fold_image_UN_disjoint cong: setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   985
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   986
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
   987
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
   988
lemma setsum_Union_disjoint:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   989
  "[| (ALL A:C. finite A);
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   990
      (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
   991
   ==> setsum f (Union C) = setsum (setsum f) C"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   992
apply (cases "finite C") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   993
 prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   994
  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
   995
 apply (unfold Union_def id_def, assumption+)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   996
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   997
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   998
(*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
   999
  the rhs need not be, since SIGMA A B could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1000
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
  1001
    (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1002
by(simp add:setsum_def comm_monoid_add.fold_image_Sigma split_def cong:setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1003
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1004
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
  1005
lemma setsum_cartesian_product: 
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
  1006
   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1007
apply (cases "finite A") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1008
 apply (cases "finite B") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1009
  apply (simp add: setsum_Sigma)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1010
 apply (cases "A={}", simp)
15543
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
  1011
 apply (simp) 
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1012
apply (auto simp add: setsum_def
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1013
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1014
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1015
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1016
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1017
by(simp add:setsum_def comm_monoid_add.fold_image_distrib)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1018
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1019
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1020
subsubsection {* Properties in more restricted classes of structures *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1021
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1022
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1023
apply (case_tac "finite A")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1024
 prefer 2 apply (simp add: setsum_def)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1025
apply (erule rev_mp)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1026
apply (erule finite_induct, auto)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1027
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1028
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1029
lemma setsum_eq_0_iff [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1030
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1031
by (induct set: finite) auto
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1032
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1033
lemma setsum_Un_nat: "finite A ==> finite B ==>
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1034
  (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1035
  -- {* For the natural numbers, we have subtraction. *}
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1036
by (subst setsum_Un_Int [symmetric], auto simp add: ring_simps)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1037
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1038
lemma setsum_Un: "finite A ==> finite B ==>
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1039
  (setsum f (A Un B) :: 'a :: ab_group_add) =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1040
   setsum f A + setsum f B - setsum f (A Int B)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1041
by (subst setsum_Un_Int [symmetric], auto simp add: ring_simps)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1042
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1043
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1044
  (if a:A then setsum f A - f a else setsum f A)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1045
apply (case_tac "finite A")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1046
 prefer 2 apply (simp add: setsum_def)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1047
apply (erule finite_induct)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1048
 apply (auto simp add: insert_Diff_if)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1049
apply (drule_tac a = a in mk_disjoint_insert, auto)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1050
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1051
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1052
lemma setsum_diff1: "finite A \<Longrightarrow>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1053
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1054
  (if a:A then setsum f A - f a else setsum f A)"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1055
by (erule finite_induct) (auto simp add: insert_Diff_if)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1056
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1057
lemma setsum_diff1'[rule_format]:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1058
  "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1059
apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1060
apply (auto simp add: insert_Diff_if add_ac)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1061
done
15552
8ab8e425410b added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents: 15543
diff changeset
  1062
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1063
(* By Jeremy Siek: *)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1064
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1065
lemma setsum_diff_nat: 
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1066
assumes "finite B" and "B \<subseteq> A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1067
shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1068
using assms
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1069
proof induct
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1070
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1071
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1072
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1073
    and xFinA: "insert x F \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1074
    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
  1075
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1076
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1077
    by (simp add: setsum_diff1_nat)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1078
  from xFinA have "F \<subseteq> A" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1079
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1080
  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
  1081
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1082
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1083
  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
  1084
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1085
  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
  1086
  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
  1087
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1088
  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
  1089
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1090
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1091
lemma setsum_diff:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1092
  assumes le: "finite A" "B \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1093
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1094
proof -
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1095
  from le have finiteB: "finite B" using finite_subset by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1096
  show ?thesis using finiteB le
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1097
  proof induct
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1098
    case empty
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1099
    thus ?case by auto
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1100
  next
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1101
    case (insert x F)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1102
    thus ?case using le finiteB 
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1103
      by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1104
  qed
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1105
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1106
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1107
lemma setsum_mono:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1108
  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
  1109
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1110
proof (cases "finite K")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1111
  case True
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1112
  thus ?thesis using le
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1113
  proof induct
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1114
    case empty
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1115
    thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1116
  next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1117
    case insert
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1118
    thus ?case using add_mono by fastsimp
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1119
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1120
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1121
  case False
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1122
  thus ?thesis
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1123
    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
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1126
lemma setsum_strict_mono:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1127
  fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1128
  assumes "finite A"  "A \<noteq> {}"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1129
    and "!!x. x:A \<Longrightarrow> f x < g x"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1130
  shows "setsum f A < setsum g A"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1131
  using prems
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1132
proof (induct rule: finite_ne_induct)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1133
  case singleton thus ?case by simp
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1134
next
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1135
  case insert thus ?case by (auto simp: add_strict_mono)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1136
qed
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1137
15535
nipkow
parents: 15532
diff changeset
  1138
lemma setsum_negf:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1139
  "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
15535
nipkow
parents: 15532
diff changeset
  1140
proof (cases "finite A")
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1141
  case True thus ?thesis by (induct set: finite) auto
15535
nipkow
parents: 15532
diff changeset
  1142
next
nipkow
parents: 15532
diff changeset
  1143
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1144
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1145
15535
nipkow
parents: 15532
diff changeset
  1146
lemma setsum_subtractf:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1147
  "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1148
    setsum f A - setsum g A"
15535
nipkow
parents: 15532
diff changeset
  1149
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1150
  case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
nipkow
parents: 15532
diff changeset
  1151
next
nipkow
parents: 15532
diff changeset
  1152
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1153
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1154
15535
nipkow
parents: 15532
diff changeset
  1155
lemma setsum_nonneg:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1156
  assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1157
  shows "0 \<le> setsum f A"
15535
nipkow
parents: 15532
diff changeset
  1158
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1159
  case True thus ?thesis using nn
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1160
  proof induct
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1161
    case empty then show ?case by simp
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1162
  next
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1163
    case (insert x F)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1164
    then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1165
    with insert show ?case by simp
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1166
  qed
15535
nipkow
parents: 15532
diff changeset
  1167
next
nipkow
parents: 15532
diff changeset
  1168
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1169
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1170
15535
nipkow
parents: 15532
diff changeset
  1171
lemma setsum_nonpos:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1172
  assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1173
  shows "setsum f A \<le> 0"
15535
nipkow
parents: 15532
diff changeset
  1174
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1175
  case True thus ?thesis using np
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1176
  proof induct
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1177
    case empty then show ?case by simp
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1178
  next
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1179
    case (insert x F)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1180
    then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1181
    with insert show ?case by simp
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1182
  qed
15535
nipkow
parents: 15532
diff changeset
  1183
next
nipkow
parents: 15532
diff changeset
  1184
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1185
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1186
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1187
lemma setsum_mono2:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1188
fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1189
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1190
shows "setsum f A \<le> setsum f B"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1191
proof -
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1192
  have "setsum f A \<le> setsum f A + setsum f (B-A)"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1193
    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1194
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1195
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1196
  also have "A \<union> (B-A) = B" using sub by blast
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1197
  finally show ?thesis .
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1198
qed
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1199
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1200
lemma setsum_mono3: "finite B ==> A <= B ==> 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1201
    ALL x: B - A. 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1202
      0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==>
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1203
        setsum f A <= setsum f B"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1204
  apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1205
  apply (erule ssubst)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1206
  apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1207
  apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1208
  apply (rule add_left_mono)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1209
  apply (erule setsum_nonneg)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1210
  apply (subst setsum_Un_disjoint [THEN sym])
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1211
  apply (erule finite_subset, assumption)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1212
  apply (rule finite_subset)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1213
  prefer 2
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1214
  apply assumption
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1215
  apply auto
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1216
  apply (rule setsum_cong)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1217
  apply auto
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1218
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1219
19279
48b527d0331b Renamed setsum_mult to setsum_right_distrib.
ballarin
parents: 18493
diff changeset
  1220
lemma setsum_right_distrib: 
22934
64ecb3d6790a generalize setsum lemmas from semiring_0_cancel to semiring_0
huffman
parents: 22917
diff changeset
  1221
  fixes f :: "'a => ('b::semiring_0)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1222
  shows "r * setsum f A = setsum (%n. r * f n) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1223
proof (cases "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1224
  case True
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1225
  thus ?thesis
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1226
  proof induct
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1227
    case empty thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1228
  next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1229
    case (insert x A) thus ?case by (simp add: right_distrib)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1230
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1231
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1232
  case False thus ?thesis by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1233
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1234
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1235
lemma setsum_left_distrib:
22934
64ecb3d6790a generalize setsum lemmas from semiring_0_cancel to semiring_0
huffman
parents: 22917
diff changeset
  1236
  "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1237
proof (cases "finite A")
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1238
  case True
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1239
  then show ?thesis
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1240
  proof induct
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1241
    case empty thus ?case by simp
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1242
  next
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1243
    case (insert x A) thus ?case by (simp add: left_distrib)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1244
  qed
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1245
next
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1246
  case False thus ?thesis by (simp add: setsum_def)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1247
qed
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1248
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1249
lemma setsum_divide_distrib:
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1250
  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1251
proof (cases "finite A")
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1252
  case True
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1253
  then show ?thesis
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1254
  proof induct
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1255
    case empty thus ?case by simp
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1256
  next
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1257
    case (insert x A) thus ?case by (simp add: add_divide_distrib)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1258
  qed
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1259
next
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1260
  case False thus ?thesis by (simp add: setsum_def)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1261
qed
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1262
15535
nipkow
parents: 15532
diff changeset
  1263
lemma setsum_abs[iff]: 
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25205
diff changeset
  1264
  fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1265
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
15535
nipkow
parents: 15532
diff changeset
  1266
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1267
  case True
nipkow
parents: 15532
diff changeset
  1268
  thus ?thesis
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1269
  proof induct
15535
nipkow
parents: 15532
diff changeset
  1270
    case empty thus ?case by simp
nipkow
parents: 15532
diff changeset
  1271
  next
nipkow
parents: 15532
diff changeset
  1272
    case (insert x A)
nipkow
parents: 15532
diff changeset
  1273
    thus ?case by (auto intro: abs_triangle_ineq order_trans)
nipkow
parents: 15532
diff changeset
  1274
  qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1275
next
15535
nipkow
parents: 15532
diff changeset
  1276
  case False thus ?thesis by (simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1277
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1278
15535
nipkow
parents: 15532
diff changeset
  1279
lemma setsum_abs_ge_zero[iff]: 
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25205
diff changeset
  1280
  fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1281
  shows "0 \<le> setsum (%i. abs(f i)) A"
15535
nipkow
parents: 15532
diff changeset
  1282
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1283
  case True
nipkow
parents: 15532
diff changeset
  1284
  thus ?thesis
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1285
  proof induct
15535
nipkow
parents: 15532
diff changeset
  1286
    case empty thus ?case by simp
nipkow
parents: 15532
diff changeset
  1287
  next
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21626
diff changeset
  1288
    case (insert x A) thus ?case by (auto simp: add_nonneg_nonneg)
15535
nipkow
parents: 15532
diff changeset
  1289
  qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1290
next
15535
nipkow
parents: 15532
diff changeset
  1291
  case False thus ?thesis by (simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1292
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1293
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1294
lemma abs_setsum_abs[simp]: 
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25205
diff changeset
  1295
  fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1296
  shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1297
proof (cases "finite A")
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1298
  case True
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1299
  thus ?thesis
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1300
  proof induct
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1301
    case empty thus ?case by simp
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1302
  next
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1303
    case (insert a A)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1304
    hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1305
    also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1306
    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1307
      by (simp del: abs_of_nonneg)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1308
    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1309
    finally show ?case .