src/HOL/Finite_Set.thy
author paulson
Thu, 25 Mar 2004 10:32:21 +0100
changeset 14485 ea2707645af8
parent 14449 d5c3d21df790
child 14504 7a3d80e276d4
permissions -rw-r--r--
new material from Avigad
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title:      HOL/Finite_Set.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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                Additions by Jeremy Avigad in Feb 2004
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    License:    GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {* Finite sets *}
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theory Finite_Set = Divides + Power + Inductive:
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subsection {* Types Classes @{text plus_ac0} and @{text times_ac1} *}
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axclass plus_ac0 < plus, zero
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  commute:     "x + y = y + x"
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  assoc:       "(x + y) + z = x + (y + z)"
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  zero [simp]: "0 + x = x"
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lemma plus_ac0_left_commute: "x + (y+z) = y + ((x+z)::'a::plus_ac0)"
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apply (rule plus_ac0.commute [THEN trans])
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apply (rule plus_ac0.assoc [THEN trans])
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apply (rule plus_ac0.commute [THEN arg_cong])
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done
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lemma plus_ac0_zero_right [simp]: "x + 0 = (x ::'a::plus_ac0)"
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apply (rule plus_ac0.commute [THEN trans])
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apply (rule plus_ac0.zero)
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done
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lemmas plus_ac0 = plus_ac0.assoc plus_ac0.commute plus_ac0_left_commute
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                  plus_ac0.zero plus_ac0_zero_right
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instance almost_semiring < plus_ac0
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proof qed (rule add_commute add_assoc almost_semiring.add_0)+
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axclass times_ac1 < times, one
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  commute:     "x * y = y * x"
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  assoc:       "(x * y) * z = x * (y * z)"
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  one [simp]:  "1 * x = x"
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theorem times_ac1_left_commute: "(x::'a::times_ac1) * ((y::'a) * z) =
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  y * (x * z)"
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proof -
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  have "(x::'a::times_ac1) * (y * z) = (x * y) * z"
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    by (rule times_ac1.assoc [THEN sym])
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  also have "x * y = y * x"
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    by (rule times_ac1.commute)
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  also have "(y * x) * z = y * (x * z)"
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    by (rule times_ac1.assoc)
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  finally show ?thesis .
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qed
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theorem times_ac1_one_right [simp]: "(x::'a::times_ac1) * 1 = x"
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proof -
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  have "x * 1 = 1 * x"
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    by (rule times_ac1.commute)
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  also have "... = x"
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    by (rule times_ac1.one)
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  finally show ?thesis .
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qed
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instance almost_semiring < times_ac1
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  apply intro_classes
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  apply (rule mult_commute)
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  apply (rule mult_assoc, simp)
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  done
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theorems times_ac1 = times_ac1.assoc times_ac1.commute times_ac1_left_commute
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  times_ac1.one times_ac1_one_right
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subsection {* Collection of finite sets *}
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consts Finites :: "'a set set"
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syntax
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  finite :: "'a set => bool"
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translations
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  "finite A" == "A : Finites"
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inductive Finites
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  intros
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    emptyI [simp, intro!]: "{} : Finites"
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    insertI [simp, intro!]: "A : Finites ==> insert a A : Finites"
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axclass finite \<subseteq> type
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  finite: "finite UNIV"
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lemma ex_new_if_finite: -- "does not depend on def of finite at all"
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 "[| ~finite(UNIV::'a set); finite A  |] ==> \<exists>a::'a. a \<notin> A"
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by(subgoal_tac "A \<noteq> UNIV", blast, blast)
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lemma finite_induct [case_names empty insert, induct set: Finites]:
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  "finite F ==>
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    P {} ==> (!!F x. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
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  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
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proof -
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  assume "P {}" and
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    insert: "!!F x. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
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  assume "finite F"
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  thus "P F"
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  proof induct
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    show "P {}" .
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    fix F x assume F: "finite F" and P: "P F"
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    show "P (insert x F)"
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    proof cases
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      assume "x \<in> F"
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      hence "insert x F = F" by (rule insert_absorb)
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      with P show ?thesis by (simp only:)
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    next
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      assume "x \<notin> F"
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      from F this P show ?thesis by (rule insert)
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    qed
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  qed
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qed
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lemma finite_subset_induct [consumes 2, case_names empty insert]:
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  "finite F ==> F \<subseteq> A ==>
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    P {} ==> (!!F a. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==>
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    P F"
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proof -
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  assume "P {}" and insert:
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    "!!F a. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
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  assume "finite F"
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  thus "F \<subseteq> A ==> P F"
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  proof induct
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    show "P {}" .
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    fix F x assume "finite F" and "x \<notin> F"
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      and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
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    show "P (insert x F)"
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    proof (rule insert)
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      from i show "x \<in> A" by blast
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      from i have "F \<subseteq> A" by blast
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      with P show "P F" .
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    qed
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  qed
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qed
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   137
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lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
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  -- {* The union of two finite sets is finite. *}
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  by (induct set: Finites) simp_all
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lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
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   143
  -- {* Every subset of a finite set is finite. *}
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   144
proof -
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   145
  assume "finite B"
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  thus "!!A. A \<subseteq> B ==> finite A"
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  proof induct
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    case empty
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    thus ?case by simp
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  next
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wenzelm
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   151
    case (insert F x A)
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parents:
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   152
    have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   153
    show "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   154
    proof cases
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   155
      assume x: "x \<in> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   156
      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
   157
      with r have "finite (A - {x})" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   158
      hence "finite (insert x (A - {x}))" ..
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   159
      also have "insert x (A - {x}) = A" by (rule insert_Diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   160
      finally show ?thesis .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   161
    next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   162
      show "A \<subseteq> F ==> ?thesis" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   163
      assume "x \<notin> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   164
      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
   165
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   166
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   167
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   168
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   169
lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
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parents:
diff changeset
   170
  by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
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wenzelm
parents:
diff changeset
   171
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   172
lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
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parents:
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   173
  -- {* The converse obviously fails. *}
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wenzelm
parents:
diff changeset
   174
  by (blast intro: finite_subset)
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wenzelm
parents:
diff changeset
   175
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   176
lemma finite_insert [simp]: "finite (insert a A) = finite A"
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parents:
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   177
  apply (subst insert_is_Un)
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   178
  apply (simp only: finite_Un, blast)
12396
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wenzelm
parents:
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   179
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   180
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   181
lemma finite_empty_induct:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
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   182
  "finite A ==>
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wenzelm
parents:
diff changeset
   183
  P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   184
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
diff changeset
   185
  assume "finite A"
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wenzelm
parents:
diff changeset
   186
    and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   187
  have "P (A - A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   188
  proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   189
    fix c b :: "'a set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   190
    presume c: "finite c" and b: "finite b"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   191
      and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   192
    from c show "c \<subseteq> b ==> P (b - c)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   193
    proof induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   194
      case empty
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   195
      from P1 show ?case by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   196
    next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   197
      case (insert F x)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   198
      have "P (b - F - {x})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   199
      proof (rule P2)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   200
        from _ b show "finite (b - F)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   201
        from insert show "x \<in> b - F" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   202
        from insert show "P (b - F)" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   203
      qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   204
      also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   205
      finally show ?case .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   206
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   207
  next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   208
    show "A \<subseteq> A" ..
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   209
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   210
  thus "P {}" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   211
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   212
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   213
lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   214
  by (rule Diff_subset [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   215
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   216
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
   217
  apply (subst Diff_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   218
  apply (case_tac "a : A - B")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   219
   apply (rule finite_insert [symmetric, THEN trans])
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parents: 13825
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   220
   apply (subst insert_Diff, simp_all)
12396
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wenzelm
parents:
diff changeset
   221
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   222
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   223
13825
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parents: 13737
diff changeset
   224
subsubsection {* Image and Inverse Image over Finite Sets *}
ef4c41e7956a new inverse image lemmas
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diff changeset
   225
ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   226
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   227
  -- {* The image of a finite set is finite. *}
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   228
  by (induct set: Finites) simp_all
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   229
14430
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parents: 14331
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   230
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
   231
  apply (frule finite_imageI)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   232
  apply (erule finite_subset, assumption)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   233
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   234
13825
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parents: 13737
diff changeset
   235
lemma finite_range_imageI:
ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   236
    "finite (range g) ==> finite (range (%x. f (g x)))"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   237
  apply (drule finite_imageI, simp)
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   238
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   239
12396
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wenzelm
parents:
diff changeset
   240
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
   241
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   242
  have aux: "!!A. finite (A - {}) = finite A" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   243
  fix B :: "'a set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   244
  assume "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   245
  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   246
    apply induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   247
     apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   248
    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
   249
     apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   250
     apply (simp (no_asm_use) add: inj_on_def)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   251
     apply (blast dest!: aux [THEN iffD1], atomize)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   252
    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   253
    apply (frule subsetD [OF equalityD2 insertI1], clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   254
    apply (rule_tac x = xa in bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   255
     apply (simp_all add: inj_on_image_set_diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   256
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   257
qed (rule refl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   258
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   259
13825
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paulson
parents: 13737
diff changeset
   260
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   261
  -- {* The inverse image of a singleton under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   262
         is included in a singleton. *}
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   263
  apply (auto simp add: inj_on_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   264
  apply (blast intro: the_equality [symmetric])
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   265
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   266
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   267
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   268
  -- {* The inverse image of a finite set under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   269
         is finite. *}
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   270
  apply (induct set: Finites, simp_all)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   271
  apply (subst vimage_insert)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   272
  apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   273
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   274
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   275
12396
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wenzelm
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diff changeset
   276
subsubsection {* The finite UNION of finite sets *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   277
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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diff changeset
   278
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   279
  by (induct set: Finites) simp_all
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   280
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   281
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   282
  Strengthen RHS to
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   283
  @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
12396
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wenzelm
parents:
diff changeset
   284
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   285
  We'd need to prove
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   286
  @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
12396
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wenzelm
parents:
diff changeset
   287
  by induction. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   288
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   289
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
   290
  by (blast intro: finite_UN_I finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   291
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   292
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   293
subsubsection {* Sigma of finite sets *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   294
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   295
lemma finite_SigmaI [simp]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   296
    "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
   297
  by (unfold Sigma_def) (blast intro!: finite_UN_I)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   298
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   299
lemma finite_Prod_UNIV:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   300
    "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
   301
  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   302
   apply (erule ssubst)
14208
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paulson
parents: 13825
diff changeset
   303
   apply (erule finite_SigmaI, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   304
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   305
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   306
instance unit :: finite
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   307
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   308
  have "finite {()}" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   309
  also have "{()} = UNIV" by auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   310
  finally show "finite (UNIV :: unit set)" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   311
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   312
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   313
instance * :: (finite, finite) finite
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   314
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   315
  show "finite (UNIV :: ('a \<times> 'b) set)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   316
  proof (rule finite_Prod_UNIV)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   317
    show "finite (UNIV :: 'a set)" by (rule finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   318
    show "finite (UNIV :: 'b set)" by (rule finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   319
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   320
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   321
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   322
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   323
subsubsection {* The powerset of a finite set *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   324
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   325
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   326
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   327
  assume "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   328
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   329
  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
   330
next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   331
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   332
  thus "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   333
    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   334
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   335
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   336
lemma finite_converse [iff]: "finite (r^-1) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   337
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   338
   apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   339
   apply (rule iffI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   340
    apply (erule finite_imageD [unfolded inj_on_def])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   341
    apply (simp split add: split_split)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   342
   apply (erule finite_imageI)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   343
  apply (simp add: converse_def image_def, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   344
  apply (rule bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   345
   prefer 2 apply assumption
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   346
  apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   347
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   348
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   349
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   350
subsubsection {* Finiteness of transitive closure *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   351
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   352
text {* (Thanks to Sidi Ehmety) *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   353
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   354
lemma finite_Field: "finite r ==> finite (Field r)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   355
  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   356
  apply (induct set: Finites)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   357
   apply (auto simp add: Field_def Domain_insert Range_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   358
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   359
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   360
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   361
  apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   362
  apply (erule trancl_induct)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   363
   apply (auto simp add: Field_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   364
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   365
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   366
lemma finite_trancl: "finite (r^+) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   367
  apply auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   368
   prefer 2
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   369
   apply (rule trancl_subset_Field2 [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   370
   apply (rule finite_SigmaI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   371
    prefer 3
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 13595
diff changeset
   372
    apply (blast intro: r_into_trancl' finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   373
   apply (auto simp add: finite_Field)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   374
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   375
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   376
lemma finite_cartesian_product: "[| finite A; finite B |] ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   377
    finite (A <*> B)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   378
  by (rule finite_SigmaI)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   379
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   380
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   381
subsection {* Finite cardinality *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   382
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   383
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   384
  This definition, although traditional, is ugly to work with: @{text
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   385
  "card A == LEAST n. EX f. A = {f i | i. i < n}"}.  Therefore we have
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   386
  switched to an inductive one:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   387
*}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   388
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   389
consts cardR :: "('a set \<times> nat) set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   390
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   391
inductive cardR
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   392
  intros
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   393
    EmptyI: "({}, 0) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   394
    InsertI: "(A, n) : cardR ==> a \<notin> A ==> (insert a A, Suc n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   395
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   396
constdefs
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   397
  card :: "'a set => nat"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   398
  "card A == THE n. (A, n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   399
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   400
inductive_cases cardR_emptyE: "({}, n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   401
inductive_cases cardR_insertE: "(insert a A,n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   402
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   403
lemma cardR_SucD: "(A, n) : cardR ==> a : A --> (EX m. n = Suc m)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   404
  by (induct set: cardR) simp_all
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   405
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   406
lemma cardR_determ_aux1:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   407
    "(A, m): cardR ==> (!!n a. m = Suc n ==> a:A ==> (A - {a}, n) : cardR)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   408
  apply (induct set: cardR, auto)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   409
  apply (simp add: insert_Diff_if, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   410
  apply (drule cardR_SucD)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   411
  apply (blast intro!: cardR.intros)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   412
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   413
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   414
lemma cardR_determ_aux2: "(insert a A, Suc m) : cardR ==> a \<notin> A ==> (A, m) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   415
  by (drule cardR_determ_aux1) auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   416
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   417
lemma cardR_determ: "(A, m): cardR ==> (!!n. (A, n) : cardR ==> n = m)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   418
  apply (induct set: cardR)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   419
   apply (safe elim!: cardR_emptyE cardR_insertE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   420
  apply (rename_tac B b m)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   421
  apply (case_tac "a = b")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   422
   apply (subgoal_tac "A = B")
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   423
    prefer 2 apply (blast elim: equalityE, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   424
  apply (subgoal_tac "EX C. A = insert b C & B = insert a C")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   425
   prefer 2
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   426
   apply (rule_tac x = "A Int B" in exI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   427
   apply (blast elim: equalityE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   428
  apply (frule_tac A = B in cardR_SucD)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   429
  apply (blast intro!: cardR.intros dest!: cardR_determ_aux2)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   430
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   431
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   432
lemma cardR_imp_finite: "(A, n) : cardR ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   433
  by (induct set: cardR) simp_all
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   434
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   435
lemma finite_imp_cardR: "finite A ==> EX n. (A, n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   436
  by (induct set: Finites) (auto intro!: cardR.intros)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   437
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   438
lemma card_equality: "(A,n) : cardR ==> card A = n"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   439
  by (unfold card_def) (blast intro: cardR_determ)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   440
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   441
lemma card_empty [simp]: "card {} = 0"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   442
  by (unfold card_def) (blast intro!: cardR.intros elim!: cardR_emptyE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   443
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   444
lemma card_insert_disjoint [simp]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   445
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   446
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   447
  assume x: "x \<notin> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   448
  hence aux: "!!n. ((insert x A, n) : cardR) = (EX m. (A, m) : cardR & n = Suc m)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   449
    apply (auto intro!: cardR.intros)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   450
    apply (rule_tac A1 = A in finite_imp_cardR [THEN exE])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   451
     apply (force dest: cardR_imp_finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   452
    apply (blast intro!: cardR.intros intro: cardR_determ)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   453
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   454
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   455
  thus ?thesis
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   456
    apply (simp add: card_def aux)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   457
    apply (rule the_equality)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   458
     apply (auto intro: finite_imp_cardR
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   459
       cong: conj_cong simp: card_def [symmetric] card_equality)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   460
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   461
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   462
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   463
lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   464
  apply auto
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   465
  apply (drule_tac a = x in mk_disjoint_insert, clarify)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   466
  apply (rotate_tac -1, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   467
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   468
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   469
lemma card_insert_if:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   470
    "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   471
  by (simp add: insert_absorb)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   472
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   473
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   474
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   475
apply(simp del:insert_Diff_single)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   476
done
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   477
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   478
lemma card_Diff_singleton:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   479
    "finite A ==> x: A ==> card (A - {x}) = card A - 1"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   480
  by (simp add: card_Suc_Diff1 [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   481
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   482
lemma card_Diff_singleton_if:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   483
    "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   484
  by (simp add: card_Diff_singleton)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   485
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   486
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   487
  by (simp add: card_insert_if card_Suc_Diff1)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   488
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   489
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   490
  by (simp add: card_insert_if)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   491
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   492
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   493
  apply (induct set: Finites, simp, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   494
  apply (subgoal_tac "finite A & A - {x} <= F")
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   495
   prefer 2 apply (blast intro: finite_subset, atomize)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   496
  apply (drule_tac x = "A - {x}" in spec)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   497
  apply (simp add: card_Diff_singleton_if split add: split_if_asm)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   498
  apply (case_tac "card A", auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   499
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   500
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   501
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   502
  apply (simp add: psubset_def linorder_not_le [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   503
  apply (blast dest: card_seteq)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   504
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   505
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   506
lemma card_mono: "finite B ==> A <= B ==> card A <= card B"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   507
  apply (case_tac "A = B", simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   508
  apply (simp add: linorder_not_less [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   509
  apply (blast dest: card_seteq intro: order_less_imp_le)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   510
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   511
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   512
lemma card_Un_Int: "finite A ==> finite B
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   513
    ==> card A + card B = card (A Un B) + card (A Int B)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   514
  apply (induct set: Finites, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   515
  apply (simp add: insert_absorb Int_insert_left)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   516
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   517
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   518
lemma card_Un_disjoint: "finite A ==> finite B
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   519
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   520
  by (simp add: card_Un_Int)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   521
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   522
lemma card_Diff_subset:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   523
    "finite A ==> B <= A ==> card A - card B = card (A - B)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   524
  apply (subgoal_tac "(A - B) Un B = A")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   525
   prefer 2 apply blast
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   526
  apply (rule nat_add_right_cancel [THEN iffD1])
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   527
  apply (rule card_Un_disjoint [THEN subst])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   528
     apply (erule_tac [4] ssubst)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   529
     prefer 3 apply blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   530
    apply (simp_all add: add_commute not_less_iff_le
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   531
      add_diff_inverse card_mono finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   532
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   533
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   534
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   535
  apply (rule Suc_less_SucD)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   536
  apply (simp add: card_Suc_Diff1)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   537
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   538
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   539
lemma card_Diff2_less:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   540
    "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   541
  apply (case_tac "x = y")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   542
   apply (simp add: card_Diff1_less)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   543
  apply (rule less_trans)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   544
   prefer 2 apply (auto intro!: card_Diff1_less)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   545
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   546
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   547
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   548
  apply (case_tac "x : A")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   549
   apply (simp_all add: card_Diff1_less less_imp_le)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   550
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   551
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   552
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   553
by (erule psubsetI, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   554
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   555
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   556
subsubsection {* Cardinality of image *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   557
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   558
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   559
  apply (induct set: Finites, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   560
  apply (simp add: le_SucI finite_imageI card_insert_if)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   561
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   562
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   563
lemma card_image: "finite A ==> inj_on f A ==> card (f ` A) = card A"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   564
  apply (induct set: Finites, simp_all, atomize, safe)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   565
   apply (unfold inj_on_def, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   566
  apply (subst card_insert_disjoint)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   567
    apply (erule finite_imageI, blast, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   568
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   569
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   570
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   571
  by (simp add: card_seteq card_image)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   572
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   573
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   574
subsubsection {* Cardinality of the Powerset *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   575
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   576
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   577
  apply (induct set: Finites)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   578
   apply (simp_all add: Pow_insert)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   579
  apply (subst card_Un_disjoint, blast)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   580
    apply (blast intro: finite_imageI, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   581
  apply (subgoal_tac "inj_on (insert x) (Pow F)")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   582
   apply (simp add: card_image Pow_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   583
  apply (unfold inj_on_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   584
  apply (blast elim!: equalityE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   585
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   586
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   587
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   588
  \medskip Relates to equivalence classes.  Based on a theorem of
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   589
  F. Kammüller's.  The @{prop "finite C"} premise is redundant.
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   590
*}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   591
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   592
lemma dvd_partition:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   593
  "finite C ==> finite (Union C) ==>
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   594
    ALL c : C. k dvd card c ==>
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   595
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   596
  k dvd card (Union C)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   597
  apply (induct set: Finites, simp_all, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   598
  apply (subst card_Un_disjoint)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   599
  apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   600
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   601
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   602
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   603
subsection {* A fold functional for finite sets *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   604
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   605
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   606
  For @{text n} non-negative we have @{text "fold f e {x1, ..., xn} =
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   607
  f x1 (... (f xn e))"} where @{text f} is at least left-commutative.
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   608
*}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   609
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   610
consts
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   611
  foldSet :: "('b => 'a => 'a) => 'a => ('b set \<times> 'a) set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   612
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   613
inductive "foldSet f e"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   614
  intros
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   615
    emptyI [intro]: "({}, e) : foldSet f e"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   616
    insertI [intro]: "x \<notin> A ==> (A, y) : foldSet f e ==> (insert x A, f x y) : foldSet f e"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   617
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   618
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f e"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   619
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   620
constdefs
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   621
  fold :: "('b => 'a => 'a) => 'a => 'b set => 'a"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   622
  "fold f e A == THE x. (A, x) : foldSet f e"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   623
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   624
lemma Diff1_foldSet: "(A - {x}, y) : foldSet f e ==> x: A ==> (A, f x y) : foldSet f e"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   625
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   626
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   627
lemma foldSet_imp_finite [simp]: "(A, x) : foldSet f e ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   628
  by (induct set: foldSet) auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   629
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   630
lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f e"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   631
  by (induct set: Finites) auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   632
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   633
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   634
subsubsection {* Left-commutative operations *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   635
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   636
locale LC =
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   637
  fixes f :: "'b => 'a => 'a"    (infixl "\<cdot>" 70)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   638
  assumes left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   639
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   640
lemma (in LC) foldSet_determ_aux:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   641
  "ALL A x. card A < n --> (A, x) : foldSet f e -->
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   642
    (ALL y. (A, y) : foldSet f e --> y = x)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   643
  apply (induct n)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   644
   apply (auto simp add: less_Suc_eq)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   645
  apply (erule foldSet.cases, blast)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   646
  apply (erule foldSet.cases, blast, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   647
  txt {* force simplification of @{text "card A < card (insert ...)"}. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   648
  apply (erule rev_mp)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   649
  apply (simp add: less_Suc_eq_le)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   650
  apply (rule impI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   651
  apply (rename_tac Aa xa ya Ab xb yb, case_tac "xa = xb")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   652
   apply (subgoal_tac "Aa = Ab")
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   653
    prefer 2 apply (blast elim!: equalityE, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   654
  txt {* case @{prop "xa \<notin> xb"}. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   655
  apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb : Aa & xa : Ab")
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   656
   prefer 2 apply (blast elim!: equalityE, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   657
  apply (subgoal_tac "Aa = insert xb Ab - {xa}")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   658
   prefer 2 apply blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   659
  apply (subgoal_tac "card Aa <= card Ab")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   660
   prefer 2
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   661
   apply (rule Suc_le_mono [THEN subst])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   662
   apply (simp add: card_Suc_Diff1)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   663
  apply (rule_tac A1 = "Aa - {xb}" and f1 = f in finite_imp_foldSet [THEN exE])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   664
  apply (blast intro: foldSet_imp_finite finite_Diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   665
  apply (frule (1) Diff1_foldSet)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   666
  apply (subgoal_tac "ya = f xb x")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   667
   prefer 2 apply (blast del: equalityCE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   668
  apply (subgoal_tac "(Ab - {xa}, x) : foldSet f e")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   669
   prefer 2 apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   670
  apply (subgoal_tac "yb = f xa x")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   671
   prefer 2 apply (blast del: equalityCE dest: Diff1_foldSet)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   672
  apply (simp (no_asm_simp) add: left_commute)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   673
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   674
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   675
lemma (in LC) foldSet_determ: "(A, x) : foldSet f e ==> (A, y) : foldSet f e ==> y = x"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   676
  by (blast intro: foldSet_determ_aux [rule_format])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   677
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   678
lemma (in LC) fold_equality: "(A, y) : foldSet f e ==> fold f e A = y"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   679
  by (unfold fold_def) (blast intro: foldSet_determ)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   680
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   681
lemma fold_empty [simp]: "fold f e {} = e"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   682
  by (unfold fold_def) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   683
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   684
lemma (in LC) fold_insert_aux: "x \<notin> A ==>
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   685
    ((insert x A, v) : foldSet f e) =
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   686
    (EX y. (A, y) : foldSet f e & v = f x y)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   687
  apply auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   688
  apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   689
   apply (fastsimp dest: foldSet_imp_finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   690
  apply (blast intro: foldSet_determ)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   691
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   692
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   693
lemma (in LC) fold_insert:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   694
    "finite A ==> x \<notin> A ==> fold f e (insert x A) = f x (fold f e A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   695
  apply (unfold fold_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   696
  apply (simp add: fold_insert_aux)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   697
  apply (rule the_equality)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   698
  apply (auto intro: finite_imp_foldSet
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   699
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   700
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   701
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   702
lemma (in LC) fold_commute: "finite A ==> (!!e. f x (fold f e A) = fold f (f x e) A)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   703
  apply (induct set: Finites, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   704
  apply (simp add: left_commute fold_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   705
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   706
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   707
lemma (in LC) fold_nest_Un_Int:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   708
  "finite A ==> finite B
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   709
    ==> fold f (fold f e B) A = fold f (fold f e (A Int B)) (A Un B)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   710
  apply (induct set: Finites, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   711
  apply (simp add: fold_insert fold_commute Int_insert_left insert_absorb)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   712
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   713
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   714
lemma (in LC) fold_nest_Un_disjoint:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   715
  "finite A ==> finite B ==> A Int B = {}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   716
    ==> fold f e (A Un B) = fold f (fold f e B) A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   717
  by (simp add: fold_nest_Un_Int)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   718
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   719
declare foldSet_imp_finite [simp del]
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   720
    empty_foldSetE [rule del]  foldSet.intros [rule del]
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   721
  -- {* Delete rules to do with @{text foldSet} relation. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   722
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   723
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   724
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   725
subsubsection {* Commutative monoids *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   726
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   727
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   728
  We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   729
  instead of @{text "'b => 'a => 'a"}.
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   730
*}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   731
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   732
locale ACe =
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   733
  fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   734
    and e :: 'a
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   735
  assumes ident [simp]: "x \<cdot> e = x"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   736
    and commute: "x \<cdot> y = y \<cdot> x"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   737
    and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   738
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   739
lemma (in ACe) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   740
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   741
  have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   742
  also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   743
  also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   744
  finally show ?thesis .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   745
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   746
12718
ade42a6c22ad lemmas (in ACe) AC;
wenzelm
parents: 12693
diff changeset
   747
lemmas (in ACe) AC = assoc commute left_commute
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   748
12693
827818b891c7 qualified exports from locales;
wenzelm
parents: 12396
diff changeset
   749
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   750
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   751
  have "x \<cdot> e = x" by (rule ident)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   752
  thus ?thesis by (subst commute)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   753
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   754
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   755
lemma (in ACe) fold_Un_Int:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   756
  "finite A ==> finite B ==>
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   757
    fold f e A \<cdot> fold f e B = fold f e (A Un B) \<cdot> fold f e (A Int B)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   758
  apply (induct set: Finites, simp)
13400
dbb608cd11c4 accomodate cumulative locale predicates;
wenzelm
parents: 13390
diff changeset
   759
  apply (simp add: AC insert_absorb Int_insert_left
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13400
diff changeset
   760
    LC.fold_insert [OF LC.intro])
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   761
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   762
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   763
lemma (in ACe) fold_Un_disjoint:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   764
  "finite A ==> finite B ==> A Int B = {} ==>
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   765
    fold f e (A Un B) = fold f e A \<cdot> fold f e B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   766
  by (simp add: fold_Un_Int)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   767
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   768
lemma (in ACe) fold_Un_disjoint2:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   769
  "finite A ==> finite B ==> A Int B = {} ==>
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   770
    fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   771
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   772
  assume b: "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   773
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   774
  thus "A Int B = {} ==>
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   775
    fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   776
  proof induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   777
    case empty
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   778
    thus ?case by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   779
  next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   780
    case (insert F x)
13571
d76a798281f4 less use of x-symbols
paulson
parents: 13490
diff changeset
   781
    have "fold (f o g) e (insert x F \<union> B) = fold (f o g) e (insert x (F \<union> B))"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   782
      by simp
13571
d76a798281f4 less use of x-symbols
paulson
parents: 13490
diff changeset
   783
    also have "... = (f o g) x (fold (f o g) e (F \<union> B))"
13400
dbb608cd11c4 accomodate cumulative locale predicates;
wenzelm
parents: 13390
diff changeset
   784
      by (rule LC.fold_insert [OF LC.intro])
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13400
diff changeset
   785
        (insert b insert, auto simp add: left_commute)
13571
d76a798281f4 less use of x-symbols
paulson
parents: 13490
diff changeset
   786
    also from insert have "fold (f o g) e (F \<union> B) =
d76a798281f4 less use of x-symbols
paulson
parents: 13490
diff changeset
   787
      fold (f o g) e F \<cdot> fold (f o g) e B" by blast
d76a798281f4 less use of x-symbols
paulson
parents: 13490
diff changeset
   788
    also have "(f o g) x ... = (f o g) x (fold (f o g) e F) \<cdot> fold (f o g) e B"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   789
      by (simp add: AC)
13571
d76a798281f4 less use of x-symbols
paulson
parents: 13490
diff changeset
   790
    also have "(f o g) x (fold (f o g) e F) = fold (f o g) e (insert x F)"
13400
dbb608cd11c4 accomodate cumulative locale predicates;
wenzelm
parents: 13390
diff changeset
   791
      by (rule LC.fold_insert [OF LC.intro, symmetric]) (insert b insert,
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13400
diff changeset
   792
	auto simp add: left_commute)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   793
    finally show ?case .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   794
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   795
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   796
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   797
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   798
subsection {* Generalized summation over a set *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   799
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   800
constdefs
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   801
  setsum :: "('a => 'b) => 'a set => 'b::plus_ac0"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   802
  "setsum f A == if finite A then fold (op + o f) 0 A else 0"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   803
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   804
syntax
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   805
  "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0"    ("\<Sum>_:_. _" [0, 51, 10] 10)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   806
syntax (xsymbols)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   807
  "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0"    ("\<Sum>_\<in>_. _" [0, 51, 10] 10)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   808
translations
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   809
  "\<Sum>i:A. b" == "setsum (%i. b) A"  -- {* Beware of argument permutation! *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   810
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   811
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   812
lemma setsum_empty [simp]: "setsum f {} = 0"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   813
  by (simp add: setsum_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   814
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   815
lemma setsum_insert [simp]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   816
    "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
13365
a2c4faad4d35 adapted to locale defs;
wenzelm
parents: 12937
diff changeset
   817
  by (simp add: setsum_def
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13400
diff changeset
   818
    LC.fold_insert [OF LC.intro] plus_ac0_left_commute)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   819
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   820
lemma setsum_reindex [rule_format]: "finite B ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   821
                  inj_on f B --> setsum h (f ` B) = setsum (h \<circ> f) B"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   822
apply (rule finite_induct, assumption, force)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   823
apply (rule impI, auto)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   824
apply (simp add: inj_on_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   825
apply (subgoal_tac "f x \<notin> f ` F")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   826
apply (subgoal_tac "finite (f ` F)")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   827
apply (auto simp add: setsum_insert)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   828
apply (simp add: inj_on_def)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   829
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   830
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   831
lemma setsum_reindex_id: "finite B ==> inj_on f B ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   832
    setsum f B = setsum id (f ` B)"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   833
by (auto simp add: setsum_reindex id_o)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   834
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   835
lemma setsum_reindex_cong: "finite A ==> inj_on f A ==> 
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   836
    B = f ` A ==> g = h \<circ> f ==> setsum h B = setsum g A"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   837
  by (frule setsum_reindex, assumption, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   838
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   839
lemma setsum_cong:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   840
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   841
  apply (case_tac "finite B")
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   842
   prefer 2 apply (simp add: setsum_def, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   843
  apply (subgoal_tac "ALL C. C <= B --> (ALL x:C. f x = g x) --> setsum f C = setsum g C")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   844
   apply simp
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   845
  apply (erule finite_induct, simp)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   846
  apply (simp add: subset_insert_iff, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   847
  apply (subgoal_tac "finite C")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   848
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   849
  apply (subgoal_tac "C = insert x (C - {x})")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   850
   prefer 2 apply blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   851
  apply (erule ssubst)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   852
  apply (drule spec)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   853
  apply (erule (1) notE impE)
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   854
  apply (simp add: Ball_def del:insert_Diff_single)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   855
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   856
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   857
lemma setsum_0: "setsum (%i. 0) A = 0"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   858
  apply (case_tac "finite A")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   859
   prefer 2 apply (simp add: setsum_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   860
  apply (erule finite_induct, auto)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   861
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   862
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   863
lemma setsum_0': "ALL a:F. f a = 0 ==> setsum f F = 0"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   864
  apply (subgoal_tac "setsum f F = setsum (%x. 0) F")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   865
  apply (erule ssubst, rule setsum_0)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   866
  apply (rule setsum_cong, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   867
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   868
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   869
lemma card_eq_setsum: "finite A ==> card A = setsum (%x. 1) A"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   870
  -- {* Could allow many @{text "card"} proofs to be simplified. *}
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   871
  by (induct set: Finites) auto
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   872
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   873
lemma setsum_Un_Int: "finite A ==> finite B
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   874
    ==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   875
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   876
  apply (induct set: Finites, simp)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   877
  apply (simp add: plus_ac0 Int_insert_left insert_absorb)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   878
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   879
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   880
lemma setsum_Un_disjoint: "finite A ==> finite B
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   881
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   882
  apply (subst setsum_Un_Int [symmetric], auto)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   883
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   884
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   885
lemma setsum_UN_disjoint:
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   886
    "finite I ==> (ALL i:I. finite (A i)) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   887
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   888
      setsum f (UNION I A) = setsum (%i. setsum f (A i)) I"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   889
  apply (induct set: Finites, simp, atomize)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   890
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   891
   prefer 2 apply blast
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   892
  apply (subgoal_tac "A x Int UNION F A = {}")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   893
   prefer 2 apply blast
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   894
  apply (simp add: setsum_Un_disjoint)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   895
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   896
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   897
lemma setsum_Union_disjoint:
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   898
  "finite C ==> (ALL A:C. finite A) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   899
        (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   900
      setsum f (Union C) = setsum (setsum f) C"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   901
  apply (frule setsum_UN_disjoint [of C id f])
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   902
  apply (unfold Union_def id_def, assumption+)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   903
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   904
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   905
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==> 
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   906
    (\<Sum> x:A. (\<Sum> y:B x. f x y)) = 
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   907
    (\<Sum> z:(SIGMA x:A. B x). f (fst z) (snd z))"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   908
  apply (subst Sigma_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   909
  apply (subst setsum_UN_disjoint)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   910
  apply assumption
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   911
  apply (rule ballI)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   912
  apply (drule_tac x = i in bspec, assumption)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   913
  apply (subgoal_tac "(UN y:(B i). {(i, y)}) <= (%y. (i, y)) ` (B i)") 
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   914
  apply (rule finite_surj)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   915
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   916
  apply (rule setsum_cong, rule refl)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   917
  apply (subst setsum_UN_disjoint)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   918
  apply (erule bspec, assumption)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   919
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   920
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   921
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   922
lemma setsum_cartesian_product: "finite A ==> finite B ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   923
    (\<Sum> x:A. (\<Sum> y:B. f x y)) = 
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   924
    (\<Sum> z:A <*> B. f (fst z) (snd z))"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   925
  by (erule setsum_Sigma, auto);
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   926
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   927
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   928
  apply (case_tac "finite A")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   929
   prefer 2 apply (simp add: setsum_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   930
  apply (erule finite_induct, auto)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   931
  apply (simp add: plus_ac0)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   932
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   933
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   934
subsubsection {* Properties in more restricted classes of structures *}
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   935
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   936
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   937
  apply (case_tac "finite A")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   938
   prefer 2 apply (simp add: setsum_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   939
  apply (erule rev_mp)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   940
  apply (erule finite_induct, auto)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   941
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   942
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   943
lemma setsum_eq_0_iff [simp]:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   944
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   945
  by (induct set: Finites) auto
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   946
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   947
lemma setsum_constant_nat [simp]:
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   948
    "finite A ==> (\<Sum>x: A. y) = (card A) * y"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   949
  -- {* Later generalized to any semiring. *}
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   950
  by (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   951
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   952
lemma setsum_Un: "finite A ==> finite B ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   953
    (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   954
  -- {* For the natural numbers, we have subtraction. *}
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   955
  by (subst setsum_Un_Int [symmetric], auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   956
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   957
lemma setsum_Un_ring: "finite A ==> finite B ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   958
    (setsum f (A Un B) :: 'a :: ring) =
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   959
      setsum f A + setsum f B - setsum f (A Int B)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   960
  apply (subst setsum_Un_Int [symmetric], auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   961
  apply (simp add: compare_rls)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   962
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   963
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   964
lemma setsum_diff1: "(setsum f (A - {a}) :: nat) =
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   965
    (if a:A then setsum f A - f a else setsum f A)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   966
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   967
   prefer 2 apply (simp add: setsum_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   968
  apply (erule finite_induct)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   969
   apply (auto simp add: insert_Diff_if)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   970
  apply (drule_tac a = a in mk_disjoint_insert, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   971
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   972
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   973
lemma setsum_negf: "finite A ==> setsum (%x. - (f x)::'a::ring) A =
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   974
  - setsum f A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   975
  by (induct set: Finites, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   976
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   977
lemma setsum_subtractf: "finite A ==> setsum (%x. ((f x)::'a::ring) - g x) A =
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   978
  setsum f A - setsum g A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   979
  by (simp add: diff_minus setsum_addf setsum_negf)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   980
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   981
lemma setsum_nonneg: "[| finite A;
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   982
    \<forall>x \<in> A. (0::'a::ordered_semiring) \<le> f x |] ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   983
    0 \<le>  setsum f A";
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   984
  apply (induct set: Finites, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   985
  apply (subgoal_tac "0 + 0 \<le> f x + setsum f F", simp)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   986
  apply (blast intro: add_mono)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   987
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   988
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   989
subsubsection {* Cardinality of unions and Sigma sets *}
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   990
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   991
lemma card_UN_disjoint:
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   992
    "finite I ==> (ALL i:I. finite (A i)) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   993
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   994
      card (UNION I A) = setsum (%i. card (A i)) I"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   995
  apply (subst card_eq_setsum)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   996
  apply (subst finite_UN, assumption+)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   997
  apply (subgoal_tac "setsum (%i. card (A i)) I =
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   998
      setsum (%i. (setsum (%x. 1) (A i))) I")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   999
  apply (erule ssubst)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1000
  apply (erule setsum_UN_disjoint, assumption+)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1001
  apply (rule setsum_cong)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1002
  apply simp+
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1003
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1004
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1005
lemma card_Union_disjoint:
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1006
  "finite C ==> (ALL A:C. finite A) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1007
        (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1008
      card (Union C) = setsum card C"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1009
  apply (frule card_UN_disjoint [of C id])
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1010
  apply (unfold Union_def id_def, assumption+)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1011
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1012
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1013
lemma SigmaI_insert: "y \<notin> A ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1014
  (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1015
  by auto
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1016
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1017
lemma card_cartesian_product_singleton: "finite A ==>
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1018
    card({x} <*> A) = card(A)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1019
  apply (subgoal_tac "inj_on (%y .(x,y)) A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1020
  apply (frule card_image, assumption)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1021
  apply (subgoal_tac "(Pair x ` A) = {x} <*> A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1022
  apply (auto simp add: inj_on_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1023
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1024
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1025
lemma card_SigmaI [rule_format,simp]: "finite A ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1026
  (ALL a:A. finite (B a)) -->
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1027
  card (SIGMA x: A. B x) = (\<Sum>a: A. card (B a))"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1028
  apply (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1029
  apply (subst SigmaI_insert, assumption)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1030
  apply (subst card_Un_disjoint)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1031
  apply (auto intro: finite_SigmaI simp add: card_cartesian_product_singleton)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1032
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1033
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1034
lemma card_cartesian_product: "[| finite A; finite B |] ==>
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1035
  card (A <*> B) = card(A) * card(B)"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1036
  by simp
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1037
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1038
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1039
subsection {* Generalized product over a set *}
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1040
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1041
constdefs
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1042
  setprod :: "('a => 'b) => 'a set => 'b::times_ac1"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1043
  "setprod f A == if finite A then fold (op * o f) 1 A else 1"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1044
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1045
syntax
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1046
  "_setprod" :: "idt => 'a set => 'b => 'b::plus_ac0"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1047
                ("\<Prod>_:_. _" [0, 51, 10] 10)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1048
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1049
syntax (xsymbols)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1050
  "_setprod" :: "idt => 'a set => 'b => 'b::plus_ac0"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1051
                ("\<Prod>_\<in>_. _" [0, 51, 10] 10)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1052
translations
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1053
  "\<Prod>i:A. b" == "setprod (%i. b) A"  -- {* Beware of argument permutation! *}
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1054
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1055
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1056
lemma setprod_empty [simp]: "setprod f {} = 1"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1057
  by (auto simp add: setprod_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1058
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1059
lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1060
    setprod f (insert a A) = f a * setprod f A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1061
  by (auto simp add: setprod_def LC_def LC.fold_insert
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1062
      times_ac1_left_commute)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1063
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1064
lemma setprod_reindex [rule_format]: "finite B ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1065
                  inj_on f B --> setprod h (f ` B) = setprod (h \<circ> f) B"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1066
apply (rule finite_induct, assumption, force)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1067
apply (rule impI, auto)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1068
apply (simp add: inj_on_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1069
apply (subgoal_tac "f x \<notin> f ` F")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1070
apply (subgoal_tac "finite (f ` F)")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1071
apply (auto simp add: setprod_insert)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1072
apply (simp add: inj_on_def)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1073
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1074
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1075
lemma setprod_reindex_id: "finite B ==> inj_on f B ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1076
    setprod f B = setprod id (f ` B)"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1077
by (auto simp add: setprod_reindex id_o)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1078
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1079
lemma setprod_reindex_cong: "finite A ==> inj_on f A ==> 
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1080
    B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1081
  by (frule setprod_reindex, assumption, simp)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1082
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1083
lemma setprod_cong:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1084
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1085
  apply (case_tac "finite B")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1086
   prefer 2 apply (simp add: setprod_def, simp)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1087
  apply (subgoal_tac "ALL C. C <= B --> (ALL x:C. f x = g x) --> setprod f C = setprod g C")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1088
   apply simp
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1089
  apply (erule finite_induct, simp)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1090
  apply (simp add: subset_insert_iff, clarify)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1091
  apply (subgoal_tac "finite C")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1092
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1093
  apply (subgoal_tac "C = insert x (C - {x})")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1094
   prefer 2 apply blast
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1095
  apply (erule ssubst)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1096
  apply (drule spec)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1097
  apply (erule (1) notE impE)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1098
  apply (simp add: Ball_def del:insert_Diff_single)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1099
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1100
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1101
lemma setprod_1: "setprod (%i. 1) A = 1"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1102
  apply (case_tac "finite A")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1103
  apply (erule finite_induct, auto simp add: times_ac1)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1104
  apply (simp add: setprod_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1105
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1106
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1107
lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1108
  apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1109
  apply (erule ssubst, rule setprod_1)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1110
  apply (rule setprod_cong, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1111
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1112
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1113
lemma setprod_Un_Int: "finite A ==> finite B
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1114
    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1115
  apply (induct set: Finites, simp)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1116
  apply (simp add: times_ac1 insert_absorb)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1117
  apply (simp add: times_ac1 Int_insert_left insert_absorb)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1118
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1119
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1120
lemma setprod_Un_disjoint: "finite A ==> finite B
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1121
  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1122
  apply (subst setprod_Un_Int [symmetric], auto simp add: times_ac1)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1123
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1124
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1125
lemma setprod_UN_disjoint:
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1126
    "finite I ==> (ALL i:I. finite (A i)) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1127
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1128
      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1129
  apply (induct set: Finites, simp, atomize)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1130
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1131
   prefer 2 apply blast
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1132
  apply (subgoal_tac "A x Int UNION F A = {}")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1133
   prefer 2 apply blast
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1134
  apply (simp add: setprod_Un_disjoint)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1135
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1136
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1137
lemma setprod_Union_disjoint:
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1138
  "finite C ==> (ALL A:C. finite A) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1139
        (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1140
      setprod f (Union C) = setprod (setprod f) C"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1141
  apply (frule setprod_UN_disjoint [of C id f])
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1142
  apply (unfold Union_def id_def, assumption+)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1143
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1144
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1145
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==> 
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1146
    (\<Prod> x:A. (\<Prod> y: B x. f x y)) = 
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1147
    (\<Prod> z:(SIGMA x:A. B x). f (fst z) (snd z))"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1148
  apply (subst Sigma_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1149
  apply (subst setprod_UN_disjoint)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1150
  apply assumption
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1151
  apply (rule ballI)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1152
  apply (drule_tac x = i in bspec, assumption)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1153
  apply (subgoal_tac "(UN y:(B i). {(i, y)}) <= (%y. (i, y)) ` (B i)") 
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1154
  apply (rule finite_surj)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1155
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1156
  apply (rule setprod_cong, rule refl)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1157
  apply (subst setprod_UN_disjoint)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1158
  apply (erule bspec, assumption)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1159
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1160
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1161
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1162
lemma setprod_cartesian_product: "finite A ==> finite B ==> 
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1163
    (\<Prod> x:A. (\<Prod> y: B. f x y)) = 
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1164
    (\<Prod> z:(A <*> B). f (fst z) (snd z))"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1165
  by (erule setprod_Sigma, auto)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1166
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1167
lemma setprod_timesf: "setprod (%x. f x * g x) A =
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1168
    (setprod f A * setprod g A)"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1169
  apply (case_tac "finite A")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1170
   prefer 2 apply (simp add: setprod_def times_ac1)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1171
  apply (erule finite_induct, auto)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1172
  apply (simp add: times_ac1)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1173
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1174
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1175
subsubsection {* Properties in more restricted classes of structures *}
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1176
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1177
lemma setprod_eq_1_iff [simp]:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1178
    "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1179
  by (induct set: Finites) auto
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1180
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1181
lemma setprod_constant: "finite A ==> (\<Prod>x: A. (y::'a::ringpower)) =
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1182
    y^(card A)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1183
  apply (erule finite_induct)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1184
  apply (auto simp add: power_Suc)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1185
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1186
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1187
lemma setprod_zero: "finite A ==> EX x: A. f x = (0::'a::semiring) ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1188
    setprod f A = 0"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1189
  apply (induct set: Finites, force, clarsimp)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1190
  apply (erule disjE, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1191
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1192
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1193
lemma setprod_nonneg [rule_format]: "(ALL x: A. (0::'a::ordered_ring) \<le> f x)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1194
     --> 0 \<le> setprod f A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1195
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1196
  apply (induct set: Finites, force, clarsimp)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1197
  apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1198
  apply (rule mult_mono, assumption+)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1199
  apply (auto simp add: setprod_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1200
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1201
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1202
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_ring) < f x)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1203
     --> 0 < setprod f A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1204
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1205
  apply (induct set: Finites, force, clarsimp)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1206
  apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1207
  apply (rule mult_strict_mono, assumption+)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1208
  apply (auto simp add: setprod_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1209
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1210
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1211
lemma setprod_nonzero [rule_format]:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1212
    "(ALL x y. (x::'a::semiring) * y = 0 --> x = 0 | y = 0) ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1213
      finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1214
  apply (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1215
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1216
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1217
lemma setprod_zero_eq:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1218
    "(ALL x y. (x::'a::semiring) * y = 0 --> x = 0 | y = 0) ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1219
     finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1220
  apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1221
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1222
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1223
lemma setprod_nonzero_field:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1224
    "finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1225
  apply (rule setprod_nonzero, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1226
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1227
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1228
lemma setprod_zero_eq_field:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1229
    "finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1230
  apply (rule setprod_zero_eq, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1231
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1232
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1233
lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1234
    (setprod f (A Un B) :: 'a ::{field})
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1235
      = setprod f A * setprod f B / setprod f (A Int B)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1236
  apply (subst setprod_Un_Int [symmetric], auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1237
  apply (subgoal_tac "finite (A Int B)")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1238
  apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1239
  apply (subst times_divide_eq_right [THEN sym], auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1240
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1241
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1242
lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1243
    (setprod f (A - {a}) :: 'a :: {field}) =
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1244
      (if a:A then setprod f A / f a else setprod f A)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1245
  apply (erule finite_induct)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1246
   apply (auto simp add: insert_Diff_if)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1247
  apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1248
  apply (erule ssubst)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1249
  apply (subst times_divide_eq_right [THEN sym])
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1250
  apply (auto simp add: mult_ac)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1251
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1252
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1253
lemma setprod_inversef: "finite A ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1254
    ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1255
      setprod (inverse \<circ> f) A = inverse (setprod f A)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1256
  apply (erule finite_induct)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1257
  apply (simp, simp)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1258
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1259
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1260
lemma setprod_dividef:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1261
     "[|finite A;
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1262
        \<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1263
      ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1264
  apply (subgoal_tac
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1265
         "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1266
  apply (erule ssubst)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1267
  apply (subst divide_inverse)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1268
  apply (subst setprod_timesf)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1269
  apply (subst setprod_inversef, assumption+, rule refl)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1270
  apply (rule setprod_cong, rule refl)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1271
  apply (subst divide_inverse, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1272
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1273
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1274
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1275
subsection{* Min and Max of finite linearly ordered sets *}
13490
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1276
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1277
text{* Seemed easier to define directly than via fold. *}
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1278
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1279
lemma ex_Max: fixes S :: "('a::linorder)set"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1280
  assumes fin: "finite S" shows "S \<noteq> {} ==> \<exists>m\<in>S. \<forall>s \<in> S. s \<le> m"
13490
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1281
using fin
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1282
proof (induct)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1283
  case empty thus ?case by simp
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1284
next
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1285
  case (insert S x)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1286
  show ?case
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1287
  proof (cases)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1288
    assume "S = {}" thus ?thesis by simp
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1289
  next
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1290
    assume nonempty: "S \<noteq> {}"
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1291
    then obtain m where m: "m\<in>S" "\<forall>s\<in>S. s \<le> m" using insert by blast
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1292
    show ?thesis
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1293
    proof (cases)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1294
      assume "x \<le> m" thus ?thesis using m by blast
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1295
    next
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1296
      assume "~ x \<le> m" thus ?thesis using m
13490
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1297
	by(simp add:linorder_not_le order_less_le)(blast intro: order_trans)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1298
    qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1299
  qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1300
qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1301
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1302
lemma ex_Min: fixes S :: "('a::linorder)set"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1303
  assumes fin: "finite S" shows "S \<noteq> {} ==> \<exists>m\<in>S. \<forall>s \<in> S. m \<le> s"
13490
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1304
using fin
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1305
proof (induct)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1306
  case empty thus ?case by simp
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1307
next
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1308
  case (insert S x)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1309
  show ?case
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1310
  proof (cases)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1311
    assume "S = {}" thus ?thesis by simp
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1312
  next
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1313
    assume nonempty: "S \<noteq> {}"
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1314
    then obtain m where m: "m\<in>S" "\<forall>s\<in>S. m \<le> s" using insert by blast
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1315
    show ?thesis
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1316
    proof (cases)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1317
      assume "m \<le> x" thus ?thesis using m by blast
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1318
    next
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1319
      assume "~ m \<le> x" thus ?thesis using m
13490
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1320
	by(simp add:linorder_not_le order_less_le)(blast intro: order_trans)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1321
    qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1322
  qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1323
qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1324
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1325
constdefs
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1326
 Min :: "('a::linorder)set => 'a"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1327
"Min S  ==  THE m. m \<in> S \<and> (\<forall>s \<in> S. m \<le> s)"
13490
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1328
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1329
 Max :: "('a::linorder)set => 'a"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1330
"Max S  ==  THE m. m \<in> S \<and> (\<forall>s \<in> S. s \<le> m)"
13490
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1331
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1332
lemma Min[simp]: assumes a: "finite S" "S \<noteq> {}"
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1333
                 shows "Min S \<in> S \<and> (\<forall>s \<in> S. Min S \<le> s)" (is "?P(Min S)")
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1334
proof (unfold Min_def, rule theI')
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1335
  show "\<exists>!m. ?P m"
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1336
  proof (rule ex_ex1I)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1337
    show "\<exists>m. ?P m" using ex_Min[OF a] by blast
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1338
  next
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1339
    fix m1 m2 assume "?P m1" "?P m2"
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1340
    thus "m1 = m2" by (blast dest:order_antisym)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1341
  qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1342
qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1343
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1344
lemma Max[simp]: assumes a: "finite S" "S \<noteq> {}"
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1345
                 shows "Max S \<in> S \<and> (\<forall>s \<in> S. s \<le> Max S)" (is "?P(Max S)")
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1346
proof (unfold Max_def, rule theI')
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1347
  show "\<exists>!m. ?P m"
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1348
  proof (rule ex_ex1I)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1349
    show "\<exists>m. ?P m" using ex_Max[OF a] by blast
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1350
  next
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1351
    fix m1 m2 assume "?P m1" "?P m2"
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1352
    thus "m1 = m2" by (blast dest:order_antisym)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1353
  qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1354
qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1355
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1356
subsection {* Theorems about @{text "choose"} *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1357
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1358
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1359
  \medskip Basic theorem about @{text "choose"}.  By Florian
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1360
  Kammüller, tidied by LCP.
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1361
*}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1362
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1363
lemma card_s_0_eq_empty:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1364
    "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1365
  apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1366
  apply (simp cong add: rev_conj_cong)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1367
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1368
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1369
lemma choose_deconstruct: "finite M ==> x \<notin> M
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1370
  ==> {s. s <= insert x M & card(s) = Suc k}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1371
       = {s. s <= M & card(s) = Suc k} Un
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1372
         {s. EX t. t <= M & card(t) = k & s = insert x t}"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1373
  apply safe
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1374
   apply (auto intro: finite_subset [THEN card_insert_disjoint])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1375
  apply (drule_tac x = "xa - {x}" in spec)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1376
  apply (subgoal_tac "x \<notin> xa", auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1377
  apply (erule rev_mp, subst card_Diff_singleton)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1378
  apply (auto intro: finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1379
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1380
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1381
lemma card_inj_on_le:
13595
7e6cdcd113a2 Proof tidying
paulson
parents: 13571
diff changeset
  1382
    "[|inj_on f A; f ` A \<subseteq> B; finite A; finite B |] ==> card A <= card B"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1383
  by (auto intro: card_mono simp add: card_image [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1384
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1385
lemma card_bij_eq:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1386
    "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
13595
7e6cdcd113a2 Proof tidying
paulson
parents: 13571
diff changeset
  1387
       finite A; finite B |] ==> card A = card B"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1388
  by (auto intro: le_anti_sym card_inj_on_le)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1389