src/HOL/Finite_Set.thy
author paulson
Thu, 15 Jul 2004 15:32:32 +0200
changeset 15047 fa62de5862b9
parent 15042 fa7d27ef7e59
child 15074 277b3a4da341
permissions -rw-r--r--
redefining sumr to be a translation to setsum
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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*)
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header {* Finite sets *}
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theory Finite_Set = Divides + Power + Inductive:
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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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  assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
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  shows "\<exists>a::'a. a \<notin> A"
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proof -
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  from prems have "A \<noteq> UNIV" by blast
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  thus ?thesis by blast
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qed
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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
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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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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    69
    show "P {}" .
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    70
    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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    72
    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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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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  -- {* Every subset of a finite set is finite. *}
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    87
proof -
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    88
  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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    case (insert F x A)
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    95
    have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" .
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    show "finite A"
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    proof cases
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    98
      assume x: "x \<in> A"
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    99
      with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
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   100
      with r have "finite (A - {x})" .
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   101
      hence "finite (insert x (A - {x}))" ..
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   102
      also have "insert x (A - {x}) = A" by (rule insert_Diff)
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   103
      finally show ?thesis .
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   104
    next
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   105
      show "A \<subseteq> F ==> ?thesis" .
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   106
      assume "x \<notin> A"
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   107
      with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
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   108
    qed
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  qed
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qed
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   111
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   112
lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
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   113
  by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
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   114
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lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
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   116
  -- {* The converse obviously fails. *}
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  by (blast intro: finite_subset)
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lemma finite_insert [simp]: "finite (insert a A) = finite A"
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  apply (subst insert_is_Un)
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   121
  apply (simp only: finite_Un, blast)
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  done
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   123
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lemma finite_empty_induct:
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   125
  "finite A ==>
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   126
  P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
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   127
proof -
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   128
  assume "finite A"
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   129
    and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
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parents:
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   130
  have "P (A - A)"
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parents:
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   131
  proof -
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   132
    fix c b :: "'a set"
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   133
    presume c: "finite c" and b: "finite b"
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   134
      and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
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wenzelm
parents:
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   135
    from c show "c \<subseteq> b ==> P (b - c)"
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parents:
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   136
    proof induct
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parents:
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   137
      case empty
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wenzelm
parents:
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   138
      from P1 show ?case by simp
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   139
    next
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wenzelm
parents:
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   140
      case (insert F x)
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wenzelm
parents:
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   141
      have "P (b - F - {x})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   142
      proof (rule P2)
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wenzelm
parents:
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   143
        from _ b show "finite (b - F)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   144
        from insert show "x \<in> b - F" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   145
        from insert show "P (b - F)" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   146
      qed
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wenzelm
parents:
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   147
      also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
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wenzelm
parents:
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   148
      finally show ?case .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   149
    qed
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wenzelm
parents:
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   150
  next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   151
    show "A \<subseteq> A" ..
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   152
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   153
  thus "P {}" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   154
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   155
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
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  by (rule Diff_subset [THEN finite_subset])
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lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
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  apply (subst Diff_insert)
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  apply (case_tac "a : A - B")
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   apply (rule finite_insert [symmetric, THEN trans])
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   apply (subst insert_Diff, simp_all)
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  done
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subsubsection {* Image and Inverse Image over Finite Sets *}
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lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
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  -- {* The image of a finite set is finite. *}
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  by (induct set: Finites) simp_all
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lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
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  apply (frule finite_imageI)
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  apply (erule finite_subset, assumption)
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  done
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lemma finite_range_imageI:
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    "finite (range g) ==> finite (range (%x. f (g x)))"
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  apply (drule finite_imageI, simp)
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  done
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lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
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proof -
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  have aux: "!!A. finite (A - {}) = finite A" by simp
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  fix B :: "'a set"
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  assume "finite B"
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  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
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    apply induct
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     apply simp
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    apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
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     apply clarify
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     apply (simp (no_asm_use) add: inj_on_def)
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     apply (blast dest!: aux [THEN iffD1], atomize)
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    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
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    apply (frule subsetD [OF equalityD2 insertI1], clarify)
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    apply (rule_tac x = xa in bexI)
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     apply (simp_all add: inj_on_image_set_diff)
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    done
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qed (rule refl)
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lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
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  -- {* The inverse image of a singleton under an injective function
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         is included in a singleton. *}
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  apply (auto simp add: inj_on_def)
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  apply (blast intro: the_equality [symmetric])
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  done
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lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
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  -- {* The inverse image of a finite set under an injective function
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         is finite. *}
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  apply (induct set: Finites, simp_all)
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  apply (subst vimage_insert)
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  apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
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  done
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   217
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subsubsection {* The finite UNION of finite sets *}
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lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
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  by (induct set: Finites) simp_all
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text {*
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  Strengthen RHS to
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  @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
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  We'd need to prove
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  @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
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  by induction. *}
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lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
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  by (blast intro: finite_UN_I finite_subset)
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subsubsection {* Sigma of finite sets *}
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lemma finite_SigmaI [simp]:
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    "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
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  by (unfold Sigma_def) (blast intro!: finite_UN_I)
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   241
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lemma finite_Prod_UNIV:
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    "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
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  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
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   apply (erule ssubst)
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   apply (erule finite_SigmaI, auto)
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  done
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instance unit :: finite
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proof
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  have "finite {()}" by simp
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  also have "{()} = UNIV" by auto
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  finally show "finite (UNIV :: unit set)" .
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qed
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   256
instance * :: (finite, finite) finite
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proof
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   258
  show "finite (UNIV :: ('a \<times> 'b) set)"
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  proof (rule finite_Prod_UNIV)
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    show "finite (UNIV :: 'a set)" by (rule finite)
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    show "finite (UNIV :: 'b set)" by (rule finite)
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  qed
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qed
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subsubsection {* The powerset of a finite set *}
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lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
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proof
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  assume "finite (Pow A)"
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  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
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  thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
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next
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  assume "finite A"
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  thus "finite (Pow A)"
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    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
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qed
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lemma finite_converse [iff]: "finite (r^-1) = finite r"
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  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
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   281
   apply simp
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   282
   apply (rule iffI)
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   283
    apply (erule finite_imageD [unfolded inj_on_def])
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   284
    apply (simp split add: split_split)
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   285
   apply (erule finite_imageI)
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  apply (simp add: converse_def image_def, auto)
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   287
  apply (rule bexI)
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   288
   prefer 2 apply assumption
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  apply simp
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  done
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   291
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subsubsection {* Finiteness of transitive closure *}
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   295
text {* (Thanks to Sidi Ehmety) *}
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   297
lemma finite_Field: "finite r ==> finite (Field r)"
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  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
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  apply (induct set: Finites)
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   apply (auto simp add: Field_def Domain_insert Range_insert)
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   301
  done
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   302
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   303
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
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  apply clarify
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   305
  apply (erule trancl_induct)
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   306
   apply (auto simp add: Field_def)
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   307
  done
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   308
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   309
lemma finite_trancl: "finite (r^+) = finite r"
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   310
  apply auto
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   311
   prefer 2
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   312
   apply (rule trancl_subset_Field2 [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   313
   apply (rule finite_SigmaI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   314
    prefer 3
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 13595
diff changeset
   315
    apply (blast intro: r_into_trancl' finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   316
   apply (auto simp add: finite_Field)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   317
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   318
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   319
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
   320
    finite (A <*> B)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   321
  by (rule finite_SigmaI)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   322
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   323
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   324
subsection {* Finite cardinality *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   325
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   326
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   327
  This definition, although traditional, is ugly to work with: @{text
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   328
  "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
   329
  switched to an inductive one:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   330
*}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   331
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   332
consts cardR :: "('a set \<times> nat) set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   333
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   334
inductive cardR
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   335
  intros
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   336
    EmptyI: "({}, 0) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   337
    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
   338
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   339
constdefs
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   340
  card :: "'a set => nat"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   341
  "card A == THE n. (A, n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   342
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   343
inductive_cases cardR_emptyE: "({}, n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   344
inductive_cases cardR_insertE: "(insert a A,n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   345
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   346
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
   347
  by (induct set: cardR) simp_all
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   348
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   349
lemma cardR_determ_aux1:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   350
    "(A, m): cardR ==> (!!n a. m = Suc n ==> a:A ==> (A - {a}, n) : cardR)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   351
  apply (induct set: cardR, auto)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   352
  apply (simp add: insert_Diff_if, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   353
  apply (drule cardR_SucD)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   354
  apply (blast intro!: cardR.intros)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   355
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   356
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   357
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
   358
  by (drule cardR_determ_aux1) auto
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 cardR_determ: "(A, m): cardR ==> (!!n. (A, n) : cardR ==> n = m)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   361
  apply (induct set: cardR)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   362
   apply (safe elim!: cardR_emptyE cardR_insertE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   363
  apply (rename_tac B b m)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   364
  apply (case_tac "a = b")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   365
   apply (subgoal_tac "A = B")
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   366
    prefer 2 apply (blast elim: equalityE, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   367
  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
   368
   prefer 2
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   369
   apply (rule_tac x = "A Int B" in exI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   370
   apply (blast elim: equalityE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   371
  apply (frule_tac A = B in cardR_SucD)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   372
  apply (blast intro!: cardR.intros dest!: cardR_determ_aux2)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   373
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   374
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   375
lemma cardR_imp_finite: "(A, n) : cardR ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   376
  by (induct set: cardR) simp_all
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   377
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   378
lemma finite_imp_cardR: "finite A ==> EX n. (A, n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   379
  by (induct set: Finites) (auto intro!: cardR.intros)
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
lemma card_equality: "(A,n) : cardR ==> card A = n"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   382
  by (unfold card_def) (blast intro: cardR_determ)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   383
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   384
lemma card_empty [simp]: "card {} = 0"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   385
  by (unfold card_def) (blast intro!: cardR.intros elim!: cardR_emptyE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   386
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   387
lemma card_insert_disjoint [simp]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   388
  "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
   389
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   390
  assume x: "x \<notin> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   391
  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
   392
    apply (auto intro!: cardR.intros)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   393
    apply (rule_tac A1 = A in finite_imp_cardR [THEN exE])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   394
     apply (force dest: cardR_imp_finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   395
    apply (blast intro!: cardR.intros intro: cardR_determ)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   396
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   397
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   398
  thus ?thesis
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   399
    apply (simp add: card_def aux)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   400
    apply (rule the_equality)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   401
     apply (auto intro: finite_imp_cardR
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   402
       cong: conj_cong simp: card_def [symmetric] card_equality)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   403
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   404
qed
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 card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   407
  apply auto
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   408
  apply (drule_tac a = x in mk_disjoint_insert, clarify)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   409
  apply (rotate_tac -1, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   410
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   411
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   412
lemma card_insert_if:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   413
    "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
   414
  by (simp add: insert_absorb)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   415
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   416
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   417
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   418
apply(simp del:insert_Diff_single)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   419
done
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   420
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   421
lemma card_Diff_singleton:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   422
    "finite A ==> x: A ==> card (A - {x}) = card A - 1"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   423
  by (simp add: card_Suc_Diff1 [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   424
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   425
lemma card_Diff_singleton_if:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   426
    "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
   427
  by (simp add: card_Diff_singleton)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   428
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   429
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
   430
  by (simp add: card_insert_if card_Suc_Diff1)
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 card_insert_le: "finite A ==> card A <= card (insert x A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   433
  by (simp add: card_insert_if)
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 card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   436
  apply (induct set: Finites, simp, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   437
  apply (subgoal_tac "finite A & A - {x} <= F")
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   438
   prefer 2 apply (blast intro: finite_subset, atomize)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   439
  apply (drule_tac x = "A - {x}" in spec)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   440
  apply (simp add: card_Diff_singleton_if split add: split_if_asm)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   441
  apply (case_tac "card A", auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   442
  done
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 psubset_card_mono: "finite B ==> A < B ==> card A < card B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   445
  apply (simp add: psubset_def linorder_not_le [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   446
  apply (blast dest: card_seteq)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   447
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   448
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   449
lemma card_mono: "finite B ==> A <= B ==> card A <= card B"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   450
  apply (case_tac "A = B", simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   451
  apply (simp add: linorder_not_less [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   452
  apply (blast dest: card_seteq intro: order_less_imp_le)
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
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   455
lemma card_Un_Int: "finite A ==> finite B
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   456
    ==> card A + card B = card (A Un B) + card (A Int B)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   457
  apply (induct set: Finites, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   458
  apply (simp add: insert_absorb Int_insert_left)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   459
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   460
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   461
lemma card_Un_disjoint: "finite A ==> finite B
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   462
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   463
  by (simp add: card_Un_Int)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   464
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   465
lemma card_Diff_subset:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   466
    "finite A ==> B <= A ==> card A - card B = card (A - B)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   467
  apply (subgoal_tac "(A - B) Un B = A")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   468
   prefer 2 apply blast
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   469
  apply (rule nat_add_right_cancel [THEN iffD1])
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   470
  apply (rule card_Un_disjoint [THEN subst])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   471
     apply (erule_tac [4] ssubst)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   472
     prefer 3 apply blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   473
    apply (simp_all add: add_commute not_less_iff_le
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   474
      add_diff_inverse card_mono finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   475
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   476
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   477
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
   478
  apply (rule Suc_less_SucD)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   479
  apply (simp add: card_Suc_Diff1)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   480
  done
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_Diff2_less:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   483
    "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
   484
  apply (case_tac "x = y")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   485
   apply (simp add: card_Diff1_less)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   486
  apply (rule less_trans)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   487
   prefer 2 apply (auto intro!: card_Diff1_less)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   488
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   489
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   490
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   491
  apply (case_tac "x : A")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   492
   apply (simp_all add: card_Diff1_less less_imp_le)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   493
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   494
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   495
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   496
by (erule psubsetI, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   497
14889
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   498
lemma insert_partition:
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   499
     "[| x \<notin> F; \<forall>c1\<in>insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 --> c1 \<inter> c2 = {}|] 
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   500
      ==> x \<inter> \<Union> F = {}"
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   501
by auto
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   502
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   503
(* main cardinality theorem *)
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   504
lemma card_partition [rule_format]:
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   505
     "finite C ==>  
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   506
        finite (\<Union> C) -->  
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   507
        (\<forall>c\<in>C. card c = k) -->   
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   508
        (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->  
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   509
        k * card(C) = card (\<Union> C)"
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   510
apply (erule finite_induct, simp)
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   511
apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition 
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   512
       finite_subset [of _ "\<Union> (insert x F)"])
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   513
done
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   514
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   515
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   516
subsubsection {* Cardinality of image *}
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_image_le: "finite A ==> card (f ` A) <= card A"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   519
  apply (induct set: Finites, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   520
  apply (simp add: le_SucI finite_imageI card_insert_if)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   521
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   522
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   523
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
   524
  apply (induct set: Finites, simp_all, atomize, safe)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   525
   apply (unfold inj_on_def, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   526
  apply (subst card_insert_disjoint)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   527
    apply (erule finite_imageI, blast, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   528
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   529
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   530
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
   531
  by (simp add: card_seteq card_image)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   532
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
subsubsection {* Cardinality of the Powerset *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   535
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   536
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
   537
  apply (induct set: Finites)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   538
   apply (simp_all add: Pow_insert)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   539
  apply (subst card_Un_disjoint, blast)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   540
    apply (blast intro: finite_imageI, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   541
  apply (subgoal_tac "inj_on (insert x) (Pow F)")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   542
   apply (simp add: card_image Pow_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   543
  apply (unfold inj_on_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   544
  apply (blast elim!: equalityE)
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
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   548
  \medskip Relates to equivalence classes.  Based on a theorem of
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   549
  F. Kammüller's.  The @{prop "finite C"} premise is redundant.
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   550
*}
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 dvd_partition:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   553
  "finite C ==> finite (Union C) ==>
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   554
    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
   555
    (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
   556
  k dvd card (Union C)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   557
  apply (induct set: Finites, simp_all, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   558
  apply (subst card_Un_disjoint)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   559
  apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   560
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   561
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
subsection {* A fold functional for finite sets *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   564
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   565
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   566
  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
   567
  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
   568
*}
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
consts
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   571
  foldSet :: "('b => 'a => 'a) => 'a => ('b set \<times> 'a) set"
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
inductive "foldSet f e"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   574
  intros
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   575
    emptyI [intro]: "({}, e) : foldSet f e"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   576
    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
   577
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   578
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f e"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   579
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   580
constdefs
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   581
  fold :: "('b => 'a => 'a) => 'a => 'b set => 'a"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   582
  "fold f e A == THE x. (A, x) : foldSet f e"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   583
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   584
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
   585
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
12396
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
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
   588
  by (induct set: foldSet) auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   589
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   590
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
   591
  by (induct set: Finites) auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   592
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   593
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   594
subsubsection {* Left-commutative operations *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   595
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   596
locale LC =
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   597
  fixes f :: "'b => 'a => 'a"    (infixl "\<cdot>" 70)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   598
  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
   599
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   600
lemma (in LC) foldSet_determ_aux:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   601
  "ALL A x. card A < n --> (A, x) : foldSet f e -->
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   602
    (ALL y. (A, y) : foldSet f e --> y = x)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   603
  apply (induct n)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   604
   apply (auto simp add: less_Suc_eq)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   605
  apply (erule foldSet.cases, blast)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   606
  apply (erule foldSet.cases, blast, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   607
  txt {* force simplification of @{text "card A < card (insert ...)"}. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   608
  apply (erule rev_mp)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   609
  apply (simp add: less_Suc_eq_le)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   610
  apply (rule impI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   611
  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
   612
   apply (subgoal_tac "Aa = Ab")
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   613
    prefer 2 apply (blast elim!: equalityE, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   614
  txt {* case @{prop "xa \<notin> xb"}. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   615
  apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb : Aa & xa : Ab")
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   616
   prefer 2 apply (blast elim!: equalityE, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   617
  apply (subgoal_tac "Aa = insert xb Ab - {xa}")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   618
   prefer 2 apply blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   619
  apply (subgoal_tac "card Aa <= card Ab")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   620
   prefer 2
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   621
   apply (rule Suc_le_mono [THEN subst])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   622
   apply (simp add: card_Suc_Diff1)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   623
  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
   624
  apply (blast intro: foldSet_imp_finite finite_Diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   625
  apply (frule (1) Diff1_foldSet)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   626
  apply (subgoal_tac "ya = f xb x")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   627
   prefer 2 apply (blast del: equalityCE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   628
  apply (subgoal_tac "(Ab - {xa}, x) : foldSet f e")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   629
   prefer 2 apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   630
  apply (subgoal_tac "yb = f xa x")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   631
   prefer 2 apply (blast del: equalityCE dest: Diff1_foldSet)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   632
  apply (simp (no_asm_simp) add: left_commute)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   633
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   634
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   635
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
   636
  by (blast intro: foldSet_determ_aux [rule_format])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   637
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   638
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
   639
  by (unfold fold_def) (blast intro: foldSet_determ)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   640
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   641
lemma fold_empty [simp]: "fold f e {} = e"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   642
  by (unfold fold_def) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   643
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   644
lemma (in LC) fold_insert_aux: "x \<notin> A ==>
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   645
    ((insert x A, v) : foldSet f e) =
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   646
    (EX y. (A, y) : foldSet f e & v = f x y)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   647
  apply auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   648
  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
   649
   apply (fastsimp dest: foldSet_imp_finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   650
  apply (blast intro: foldSet_determ)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   651
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   652
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   653
lemma (in LC) fold_insert:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   654
    "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
   655
  apply (unfold fold_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   656
  apply (simp add: fold_insert_aux)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   657
  apply (rule the_equality)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   658
  apply (auto intro: finite_imp_foldSet
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   659
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   660
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   661
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   662
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
   663
  apply (induct set: Finites, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   664
  apply (simp add: left_commute fold_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   665
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   666
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   667
lemma (in LC) fold_nest_Un_Int:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   668
  "finite A ==> finite B
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   669
    ==> 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
   670
  apply (induct set: Finites, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   671
  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
   672
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   673
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   674
lemma (in LC) fold_nest_Un_disjoint:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   675
  "finite A ==> finite B ==> A Int B = {}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   676
    ==> 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
   677
  by (simp add: fold_nest_Un_Int)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   678
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   679
declare foldSet_imp_finite [simp del]
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   680
    empty_foldSetE [rule del]  foldSet.intros [rule del]
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   681
  -- {* Delete rules to do with @{text foldSet} relation. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   682
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
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   685
subsubsection {* Commutative monoids *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   686
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   687
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   688
  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
   689
  instead of @{text "'b => 'a => 'a"}.
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   690
*}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   691
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   692
locale ACe =
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   693
  fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   694
    and e :: 'a
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   695
  assumes ident [simp]: "x \<cdot> e = x"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   696
    and commute: "x \<cdot> y = y \<cdot> x"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   697
    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
   698
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   699
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
   700
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   701
  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
   702
  also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   703
  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
   704
  finally show ?thesis .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   705
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   706
12718
ade42a6c22ad lemmas (in ACe) AC;
wenzelm
parents: 12693
diff changeset
   707
lemmas (in ACe) AC = assoc commute left_commute
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   708
12693
827818b891c7 qualified exports from locales;
wenzelm
parents: 12396
diff changeset
   709
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   710
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   711
  have "x \<cdot> e = x" by (rule ident)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   712
  thus ?thesis by (subst commute)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   713
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   714
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   715
lemma (in ACe) fold_Un_Int:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   716
  "finite A ==> finite B ==>
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   717
    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
   718
  apply (induct set: Finites, simp)
13400
dbb608cd11c4 accomodate cumulative locale predicates;
wenzelm
parents: 13390
diff changeset
   719
  apply (simp add: AC insert_absorb Int_insert_left
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13400
diff changeset
   720
    LC.fold_insert [OF LC.intro])
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   721
  done
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
lemma (in ACe) fold_Un_disjoint:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   724
  "finite A ==> finite B ==> A Int B = {} ==>
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   725
    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
   726
  by (simp add: fold_Un_Int)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   727
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   728
lemma (in ACe) fold_Un_disjoint2:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   729
  "finite A ==> finite B ==> A Int B = {} ==>
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   730
    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
   731
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   732
  assume b: "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   733
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   734
  thus "A Int B = {} ==>
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   735
    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
   736
  proof induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   737
    case empty
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   738
    thus ?case by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   739
  next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   740
    case (insert F x)
13571
d76a798281f4 less use of x-symbols
paulson
parents: 13490
diff changeset
   741
    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
   742
      by simp
13571
d76a798281f4 less use of x-symbols
paulson
parents: 13490
diff changeset
   743
    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
   744
      by (rule LC.fold_insert [OF LC.intro])
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13400
diff changeset
   745
        (insert b insert, auto simp add: left_commute)
13571
d76a798281f4 less use of x-symbols
paulson
parents: 13490
diff changeset
   746
    also from insert have "fold (f o g) e (F \<union> B) =
d76a798281f4 less use of x-symbols
paulson
parents: 13490
diff changeset
   747
      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
   748
    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
   749
      by (simp add: AC)
13571
d76a798281f4 less use of x-symbols
paulson
parents: 13490
diff changeset
   750
    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
   751
      by (rule LC.fold_insert [OF LC.intro, symmetric]) (insert b insert,
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
   752
        auto simp add: left_commute)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   753
    finally show ?case .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   754
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   755
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   756
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   757
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   758
subsection {* Generalized summation over a set *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   759
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   760
constdefs
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
   761
  setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   762
  "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
   763
15042
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   764
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   765
written @{text"\<Sum>x\<in>A. e"}. *}
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   766
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   767
syntax
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
   768
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_:_. _)" [0, 51, 10] 10)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   769
syntax (xsymbols)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
   770
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
14565
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14504
diff changeset
   771
syntax (HTML output)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
   772
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   773
translations
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   774
  "\<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
   775
15042
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   776
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   777
 @{text"\<Sum>x|P. e"}. *}
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   778
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   779
syntax
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   780
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ | / _./ _)" [0,0,10] 10)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   781
syntax (xsymbols)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   782
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   783
syntax (HTML output)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   784
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   785
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   786
translations "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   787
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   788
print_translation {*
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   789
let
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   790
  fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   791
    (if x<>y then raise Match
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   792
     else let val x' = Syntax.mark_bound x
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   793
              val t' = subst_bound(x',t)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   794
              val P' = subst_bound(x',P)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   795
          in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   796
in
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   797
[("setsum", setsum_tr')]
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   798
end
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   799
*}
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
   800
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   801
text{* As Jeremy Avigad notes, setprod needs the same treatment \dots *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   802
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   803
lemma setsum_empty [simp]: "setsum f {} = 0"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   804
  by (simp add: setsum_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   805
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   806
lemma setsum_insert [simp]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   807
    "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   808
  by (simp add: setsum_def LC.fold_insert [OF LC.intro] add_left_commute)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   809
14944
efbaecbdc05c strengthened some theorems
paulson
parents: 14889
diff changeset
   810
lemma setsum_reindex [rule_format]:
efbaecbdc05c strengthened some theorems
paulson
parents: 14889
diff changeset
   811
     "finite B ==> inj_on f B --> setsum h (f ` B) = setsum (h \<circ> f) B"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   812
apply (rule finite_induct, assumption, force)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   813
apply (rule impI, auto)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   814
apply (simp add: inj_on_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   815
apply (subgoal_tac "f x \<notin> f ` F")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   816
apply (subgoal_tac "finite (f ` F)")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   817
apply (auto simp add: setsum_insert)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   818
apply (simp add: inj_on_def)
14944
efbaecbdc05c strengthened some theorems
paulson
parents: 14889
diff changeset
   819
done
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   820
14944
efbaecbdc05c strengthened some theorems
paulson
parents: 14889
diff changeset
   821
lemma setsum_reindex_id:
efbaecbdc05c strengthened some theorems
paulson
parents: 14889
diff changeset
   822
     "finite B ==> inj_on f B ==> setsum f B = setsum id (f ` B)"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   823
by (auto simp add: setsum_reindex id_o)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   824
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   825
lemma setsum_cong:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   826
  "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
   827
  apply (case_tac "finite B")
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   828
   prefer 2 apply (simp add: setsum_def, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   829
  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
   830
   apply simp
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   831
  apply (erule finite_induct, simp)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   832
  apply (simp add: subset_insert_iff, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   833
  apply (subgoal_tac "finite C")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   834
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   835
  apply (subgoal_tac "C = insert x (C - {x})")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   836
   prefer 2 apply blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   837
  apply (erule ssubst)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   838
  apply (drule spec)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   839
  apply (erule (1) notE impE)
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   840
  apply (simp add: Ball_def del:insert_Diff_single)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   841
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   842
14944
efbaecbdc05c strengthened some theorems
paulson
parents: 14889
diff changeset
   843
lemma setsum_reindex_cong:
efbaecbdc05c strengthened some theorems
paulson
parents: 14889
diff changeset
   844
     "[|finite A; inj_on f A; B = f ` A; !!a. g a = h (f a)|] 
efbaecbdc05c strengthened some theorems
paulson
parents: 14889
diff changeset
   845
      ==> setsum h B = setsum g A"
efbaecbdc05c strengthened some theorems
paulson
parents: 14889
diff changeset
   846
  by (simp add: setsum_reindex cong: setsum_cong) 
efbaecbdc05c strengthened some theorems
paulson
parents: 14889
diff changeset
   847
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   848
lemma setsum_0: "setsum (%i. 0) A = 0"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   849
  apply (case_tac "finite A")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   850
   prefer 2 apply (simp add: setsum_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   851
  apply (erule finite_induct, auto)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   852
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   853
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   854
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
   855
  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
   856
  apply (erule ssubst, rule setsum_0)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   857
  apply (rule setsum_cong, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   858
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   859
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   860
lemma card_eq_setsum: "finite A ==> card A = setsum (%x. 1) A"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   861
  -- {* Could allow many @{text "card"} proofs to be simplified. *}
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   862
  by (induct set: Finites) auto
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   863
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   864
lemma setsum_Un_Int: "finite A ==> finite B
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   865
    ==> 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
   866
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   867
  apply (induct set: Finites, simp)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
   868
  apply (simp add: add_ac Int_insert_left insert_absorb)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   869
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   870
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   871
lemma setsum_Un_disjoint: "finite A ==> finite B
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   872
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   873
  apply (subst setsum_Un_Int [symmetric], auto)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   874
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   875
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   876
lemma setsum_UN_disjoint:
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   877
    "finite I ==> (ALL i:I. finite (A i)) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   878
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   879
      setsum f (UNION I A) = setsum (%i. setsum f (A i)) I"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   880
  apply (induct set: Finites, simp, atomize)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   881
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   882
   prefer 2 apply blast
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   883
  apply (subgoal_tac "A x Int UNION F A = {}")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   884
   prefer 2 apply blast
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   885
  apply (simp add: setsum_Un_disjoint)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   886
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   887
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   888
lemma setsum_Union_disjoint:
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   889
  "finite C ==> (ALL A:C. finite A) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   890
        (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   891
      setsum f (Union C) = setsum (setsum f) C"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   892
  apply (frule setsum_UN_disjoint [of C id f])
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   893
  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
   894
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   895
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
   896
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
   897
    (\<Sum>x:A. (\<Sum>y:B x. f x y)) =
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
   898
    (\<Sum>z:(SIGMA x:A. B x). f (fst z) (snd z))"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   899
  apply (subst Sigma_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   900
  apply (subst setsum_UN_disjoint)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   901
  apply assumption
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   902
  apply (rule ballI)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   903
  apply (drule_tac x = i in bspec, assumption)
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
   904
  apply (subgoal_tac "(UN y:(B i). {(i, y)}) <= (%y. (i, y)) ` (B i)")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   905
  apply (rule finite_surj)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   906
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   907
  apply (rule setsum_cong, rule refl)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   908
  apply (subst setsum_UN_disjoint)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   909
  apply (erule bspec, assumption)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   910
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   911
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   912
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   913
lemma setsum_cartesian_product: "finite A ==> finite B ==>
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
   914
    (\<Sum>x:A. (\<Sum>y:B. f x y)) =
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
   915
    (\<Sum>z:A <*> B. f (fst z) (snd z))"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   916
  by (erule setsum_Sigma, auto);
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   917
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   918
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
   919
  apply (case_tac "finite A")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   920
   prefer 2 apply (simp add: setsum_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   921
  apply (erule finite_induct, auto)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
   922
  apply (simp add: add_ac)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   923
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   924
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   925
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
   926
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   927
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f 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 rev_mp)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   931
  apply (erule finite_induct, auto)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   932
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
   933
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   934
lemma setsum_eq_0_iff [simp]:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   935
    "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
   936
  by (induct set: Finites) auto
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   937
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   938
lemma setsum_constant_nat:
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   939
    "finite A ==> (\<Sum>x: A. y) = (card A) * y"
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   940
  -- {* Generalized to any @{text comm_semiring_1_cancel} in
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   941
        @{text IntDef} as @{text setsum_constant}. *}
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   942
  by (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   943
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   944
lemma setsum_Un: "finite A ==> finite B ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   945
    (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
   946
  -- {* For the natural numbers, we have subtraction. *}
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
   947
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   948
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   949
lemma setsum_Un_ring: "finite A ==> finite B ==>
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
   950
    (setsum f (A Un B) :: 'a :: comm_ring_1) =
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   951
      setsum f A + setsum f B - setsum f (A Int B)"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
   952
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   953
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   954
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
   955
    (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
   956
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   957
   prefer 2 apply (simp add: setsum_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   958
  apply (erule finite_induct)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   959
   apply (auto simp add: insert_Diff_if)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   960
  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
   961
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   962
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
   963
lemma setsum_negf: "finite A ==> setsum (%x. - (f x)::'a::comm_ring_1) A =
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   964
  - setsum f A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   965
  by (induct set: Finites, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   966
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
   967
lemma setsum_subtractf: "finite A ==> setsum (%x. ((f x)::'a::comm_ring_1) - g x) A =
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   968
  setsum f A - setsum g A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   969
  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
   970
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   971
lemma setsum_nonneg: "[| finite A;
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
   972
    \<forall>x \<in> A. (0::'a::ordered_semidom) \<le> f x |] ==>
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   973
    0 \<le>  setsum f A";
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   974
  apply (induct set: Finites, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   975
  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
   976
  apply (blast intro: add_mono)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   977
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   978
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   979
lemma setsum_mult: 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   980
  fixes f :: "'a => ('b::semiring_0_cancel)"
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   981
  assumes fin: "finite A" 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   982
  shows "r * setsum f A = setsum (%n. r * f n) A"
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   983
using fin 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   984
proof (induct) 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   985
  case empty thus ?case by simp
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   986
next
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   987
  case (insert A x)
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   988
  thus ?case by (simp add: right_distrib) 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   989
qed
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   990
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   991
lemma setsum_abs: 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   992
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   993
  assumes fin: "finite A" 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   994
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   995
using fin 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   996
proof (induct) 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   997
  case empty thus ?case by simp
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   998
next
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
   999
  case (insert A x)
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1000
  thus ?case by (auto intro: abs_triangle_ineq order_trans)
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1001
qed
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1002
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1003
lemma setsum_abs_ge_zero: 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1004
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1005
  assumes fin: "finite A" 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1006
  shows "0 \<le> setsum (%i. abs(f i)) A"
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1007
using fin 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1008
proof (induct) 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1009
  case empty thus ?case by simp
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1010
next
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1011
  case (insert A x) thus ?case by (auto intro: order_trans)
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1012
qed
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1013
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1014
subsubsection {* Cardinality of unions and Sigma sets *}
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1015
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1016
lemma card_UN_disjoint:
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1017
    "finite I ==> (ALL i:I. finite (A i)) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1018
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1019
      card (UNION I A) = setsum (%i. card (A i)) I"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1020
  apply (subst card_eq_setsum)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1021
  apply (subst finite_UN, assumption+)
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1022
  apply (subgoal_tac
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1023
           "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1024
  apply (simp add: setsum_UN_disjoint) 
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1025
  apply (simp add: setsum_constant_nat cong: setsum_cong) 
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1026
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1027
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1028
lemma card_Union_disjoint:
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1029
  "finite C ==> (ALL A:C. finite A) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1030
        (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1031
      card (Union C) = setsum card C"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1032
  apply (frule card_UN_disjoint [of C id])
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1033
  apply (unfold Union_def id_def, assumption+)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1034
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1035
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1036
lemma SigmaI_insert: "y \<notin> A ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1037
  (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
  1038
  by auto
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1039
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1040
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
  1041
    card({x} <*> A) = card(A)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1042
  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
  1043
  apply (frule card_image, assumption)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1044
  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
  1045
  apply (auto simp add: inj_on_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1046
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1047
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1048
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
  1049
  (ALL a:A. finite (B a)) -->
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1050
  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
  1051
  apply (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1052
  apply (subst SigmaI_insert, assumption)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1053
  apply (subst card_Un_disjoint)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1054
  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
  1055
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1056
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1057
lemma card_cartesian_product:
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1058
     "[| finite A; finite B |] ==> card (A <*> B) = card(A) * card(B)"
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1059
  by (simp add: setsum_constant_nat)
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15042
diff changeset
  1060
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1061
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1062
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1063
subsection {* Generalized product over a set *}
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1064
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1065
constdefs
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1066
  setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1067
  "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
  1068
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1069
syntax
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1070
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_:_. _)" [0, 51, 10] 10)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1071
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1072
syntax (xsymbols)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1073
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
14565
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14504
diff changeset
  1074
syntax (HTML output)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1075
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1076
translations
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1077
  "\<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
  1078
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1079
lemma setprod_empty [simp]: "setprod f {} = 1"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1080
  by (auto simp add: setprod_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1081
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1082
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
  1083
    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
  1084
  by (auto simp add: setprod_def LC_def LC.fold_insert
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1085
      mult_left_commute)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1086
14748
001323d6d75b removed a premise of card_inj_on_le
paulson
parents: 14740
diff changeset
  1087
lemma setprod_reindex [rule_format]:
001323d6d75b removed a premise of card_inj_on_le
paulson
parents: 14740
diff changeset
  1088
     "finite B ==> inj_on f B --> setprod h (f ` B) = setprod (h \<circ> f) B"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1089
apply (rule finite_induct, assumption, force)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1090
apply (rule impI, auto)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1091
apply (simp add: inj_on_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1092
apply (subgoal_tac "f x \<notin> f ` F")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1093
apply (subgoal_tac "finite (f ` F)")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1094
apply (auto simp add: setprod_insert)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1095
apply (simp add: inj_on_def)
14748
001323d6d75b removed a premise of card_inj_on_le
paulson
parents: 14740
diff changeset
  1096
done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1097
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1098
lemma setprod_reindex_id: "finite B ==> inj_on f B ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1099
    setprod f B = setprod id (f ` B)"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1100
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
  1101
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1102
lemma setprod_reindex_cong: "finite A ==> inj_on f A ==>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1103
    B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1104
  by (frule setprod_reindex, assumption, simp)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1105
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1106
lemma setprod_cong:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1107
  "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
  1108
  apply (case_tac "finite B")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1109
   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
  1110
  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
  1111
   apply simp
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1112
  apply (erule finite_induct, simp)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1113
  apply (simp add: subset_insert_iff, clarify)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1114
  apply (subgoal_tac "finite C")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1115
   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
  1116
  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
  1117
   prefer 2 apply blast
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1118
  apply (erule ssubst)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1119
  apply (drule spec)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1120
  apply (erule (1) notE impE)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1121
  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
  1122
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1123
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1124
lemma setprod_1: "setprod (%i. 1) A = 1"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1125
  apply (case_tac "finite A")
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1126
  apply (erule finite_induct, auto simp add: mult_ac)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1127
  apply (simp add: setprod_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1128
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1129
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1130
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
  1131
  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
  1132
  apply (erule ssubst, rule setprod_1)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1133
  apply (rule setprod_cong, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1134
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1135
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1136
lemma setprod_Un_Int: "finite A ==> finite B
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1137
    ==> 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
  1138
  apply (induct set: Finites, simp)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1139
  apply (simp add: mult_ac insert_absorb)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1140
  apply (simp add: mult_ac Int_insert_left insert_absorb)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1141
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1142
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1143
lemma setprod_Un_disjoint: "finite A ==> finite B
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1144
  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1145
  apply (subst setprod_Un_Int [symmetric], auto simp add: mult_ac)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1146
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1147
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1148
lemma setprod_UN_disjoint:
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1149
    "finite I ==> (ALL i:I. finite (A i)) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1150
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1151
      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1152
  apply (induct set: Finites, simp, atomize)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1153
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1154
   prefer 2 apply blast
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1155
  apply (subgoal_tac "A x Int UNION F A = {}")
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1156
   prefer 2 apply blast
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1157
  apply (simp add: setprod_Un_disjoint)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1158
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1159
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1160
lemma setprod_Union_disjoint:
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1161
  "finite C ==> (ALL A:C. finite A) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1162
        (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1163
      setprod f (Union C) = setprod (setprod f) C"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1164
  apply (frule setprod_UN_disjoint [of C id f])
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1165
  apply (unfold Union_def id_def, assumption+)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1166
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1167
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1168
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1169
    (\<Prod>x:A. (\<Prod>y: B x. f x y)) =
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1170
    (\<Prod>z:(SIGMA x:A. B x). f (fst z) (snd z))"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1171
  apply (subst Sigma_def)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1172
  apply (subst setprod_UN_disjoint)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1173
  apply assumption
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1174
  apply (rule ballI)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1175
  apply (drule_tac x = i in bspec, assumption)
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1176
  apply (subgoal_tac "(UN y:(B i). {(i, y)}) <= (%y. (i, y)) ` (B i)")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1177
  apply (rule finite_surj)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1178
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1179
  apply (rule setprod_cong, rule refl)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1180
  apply (subst setprod_UN_disjoint)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1181
  apply (erule bspec, assumption)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1182
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1183
  done
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1184
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1185
lemma setprod_cartesian_product: "finite A ==> finite B ==>
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1186
    (\<Prod>x:A. (\<Prod>y: B. f x y)) =
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1187
    (\<Prod>z:(A <*> B). f (fst z) (snd z))"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1188
  by (erule setprod_Sigma, auto)
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1189
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1190
lemma setprod_timesf: "setprod (%x. f x * g x) A =
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1191
    (setprod f A * setprod g A)"
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1192
  apply (case_tac "finite A")
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1193
   prefer 2 apply (simp add: setprod_def mult_ac)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1194
  apply (erule finite_induct, auto)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1195
  apply (simp add: mult_ac)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14449
diff changeset
  1196
  done
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1197
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1198
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
  1199
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1200
lemma setprod_eq_1_iff [simp]:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1201
    "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
  1202
  by (induct set: Finites) auto
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1203
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14944
diff changeset
  1204
lemma setprod_constant: "finite A ==> (\<Prod>x: A. (y::'a::recpower)) = y^(card A)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1205
  apply (erule finite_induct)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1206
  apply (auto simp add: power_Suc)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1207
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1208
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14944
diff changeset
  1209
lemma setprod_zero:
44ac09ba7213 ringpower to recpower
paulson
parents: 14944
diff changeset
  1210
     "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1211
  apply (induct set: Finites, force, clarsimp)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1212
  apply (erule disjE, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1213
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1214
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14944
diff changeset
  1215
lemma setprod_nonneg [rule_format]:
44ac09ba7213 ringpower to recpower
paulson
parents: 14944
diff changeset
  1216
     "(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1217
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1218
  apply (induct set: Finites, force, clarsimp)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1219
  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
  1220
  apply (rule mult_mono, assumption+)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1221
  apply (auto simp add: setprod_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1222
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1223
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1224
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1225
     --> 0 < setprod f A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1226
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1227
  apply (induct set: Finites, force, clarsimp)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1228
  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
  1229
  apply (rule mult_strict_mono, assumption+)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1230
  apply (auto simp add: setprod_def)
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_nonzero [rule_format]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1234
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1235
      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
  1236
  apply (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1237
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1238
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1239
lemma setprod_zero_eq:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14661
diff changeset
  1240
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1241
     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
  1242
  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
  1243
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1244
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1245
lemma setprod_nonzero_field:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1246
    "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
  1247
  apply (rule setprod_nonzero, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1248
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1249
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1250
lemma setprod_zero_eq_field:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1251
    "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
  1252
  apply (rule setprod_zero_eq, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1253
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1254
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1255
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
  1256
    (setprod f (A Un B) :: 'a ::{field})
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1257
      = 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
  1258
  apply (subst setprod_Un_Int [symmetric], auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1259
  apply (subgoal_tac "finite (A Int B)")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1260
  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
  1261
  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
  1262
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1263
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1264
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
  1265
    (setprod f (A - {a}) :: 'a :: {field}) =
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1266
      (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
  1267
  apply (erule finite_induct)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1268
   apply (auto simp add: insert_Diff_if)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1269
  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
  1270
  apply (erule ssubst)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1271
  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
  1272
  apply (auto simp add: mult_ac)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1273
  done
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
lemma setprod_inversef: "finite A ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1276
    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
  1277
      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
  1278
  apply (erule finite_induct)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1279
  apply (simp, simp)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1280
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1281
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1282
lemma setprod_dividef:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1283
     "[|finite A;
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1284
        \<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
  1285
      ==> 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
  1286
  apply (subgoal_tac
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1287
         "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
  1288
  apply (erule ssubst)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1289
  apply (subst divide_inverse)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1290
  apply (subst setprod_timesf)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1291
  apply (subst setprod_inversef, assumption+, rule refl)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1292
  apply (rule setprod_cong, rule refl)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1293
  apply (subst divide_inverse, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1294
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1295
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1296
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1297
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
  1298
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1299
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
  1300
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1301
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
  1302
  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
  1303
using fin
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1304
proof (induct)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1305
  case empty thus ?case by simp
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1306
next
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1307
  case (insert S x)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1308
  show ?case
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1309
  proof (cases)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1310
    assume "S = {}" thus ?thesis by simp
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1311
  next
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1312
    assume nonempty: "S \<noteq> {}"
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1313
    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
  1314
    show ?thesis
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1315
    proof (cases)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1316
      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
  1317
    next
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1318
      assume "~ x \<le> m" thus ?thesis using m
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1319
        by(simp add:linorder_not_le order_less_le)(blast intro: order_trans)
13490
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1320
    qed
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
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1324
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
  1325
  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
  1326
using fin
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1327
proof (induct)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1328
  case empty thus ?case by simp
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1329
next
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1330
  case (insert S x)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1331
  show ?case
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1332
  proof (cases)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1333
    assume "S = {}" thus ?thesis by simp
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1334
  next
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1335
    assume nonempty: "S \<noteq> {}"
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1336
    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
  1337
    show ?thesis
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1338
    proof (cases)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1339
      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
  1340
    next
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1341
      assume "~ m \<le> x" thus ?thesis using m
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1342
        by(simp add:linorder_not_le order_less_le)(blast intro: order_trans)
13490
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1343
    qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1344
  qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1345
qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1346
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1347
constdefs
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1348
  Min :: "('a::linorder)set => 'a"
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1349
  "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
  1350
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1351
  Max :: "('a::linorder)set => 'a"
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1352
  "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
  1353
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1354
lemma Min [simp]:
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1355
  assumes a: "finite S"  "S \<noteq> {}"
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1356
  shows "Min S \<in> S \<and> (\<forall>s \<in> S. Min S \<le> s)" (is "?P(Min S)")
13490
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1357
proof (unfold Min_def, rule theI')
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1358
  show "\<exists>!m. ?P m"
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1359
  proof (rule ex_ex1I)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1360
    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
  1361
  next
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1362
    fix m1 m2 assume "?P m1" and "?P m2"
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1363
    thus "m1 = m2" by (blast dest: order_antisym)
13490
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1364
  qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1365
qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1366
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1367
lemma Max [simp]:
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1368
  assumes a: "finite S"  "S \<noteq> {}"
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1369
  shows "Max S \<in> S \<and> (\<forall>s \<in> S. s \<le> Max S)" (is "?P(Max S)")
13490
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1370
proof (unfold Max_def, rule theI')
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1371
  show "\<exists>!m. ?P m"
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1372
  proof (rule ex_ex1I)
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1373
    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
  1374
  next
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1375
    fix m1 m2 assume "?P m1" "?P m2"
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1376
    thus "m1 = m2" by (blast dest: order_antisym)
13490
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1377
  qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1378
qed
44bdc150211b Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents: 13421
diff changeset
  1379
14661
9ead82084de8 tuned notation;
wenzelm
parents: 14565
diff changeset
  1380
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1381
subsection {* Theorems about @{text "choose"} *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1382
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset