src/HOL/Finite_Set.thy
author paulson
Fri Sep 26 10:34:57 2003 +0200 (2003-09-26)
changeset 14208 144f45277d5a
parent 13825 ef4c41e7956a
child 14302 6c24235e8d5d
permissions -rw-r--r--
misc tidying
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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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    License:    GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {* Finite sets *}
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theory Finite_Set = Divides + Power + Inductive + SetInterval:
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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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 "\<lbrakk> ~finite(UNIV::'a set); finite A \<rbrakk> \<Longrightarrow> \<exists>a::'a. a ~: A"
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by(subgoal_tac "A ~= UNIV", blast, blast)
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lemma finite_induct [case_names empty insert, induct set: Finites]:
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  "finite F ==>
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    P {} ==> (!!F x. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
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  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
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proof -
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  assume "P {}" and
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    insert: "!!F x. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
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  assume "finite F"
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  thus "P F"
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  proof induct
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    show "P {}" .
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    fix F x assume F: "finite F" and P: "P F"
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    show "P (insert x F)"
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    proof cases
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      assume "x \<in> F"
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      hence "insert x F = F" by (rule insert_absorb)
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      with P show ?thesis by (simp only:)
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    next
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      assume "x \<notin> F"
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      from F this P show ?thesis by (rule insert)
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    qed
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  qed
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qed
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lemma finite_subset_induct [consumes 2, case_names empty insert]:
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  "finite F ==> F \<subseteq> A ==>
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    P {} ==> (!!F a. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==>
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    P F"
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proof -
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  assume "P {}" and insert:
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    "!!F a. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
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  assume "finite F"
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  thus "F \<subseteq> A ==> P F"
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  proof induct
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    show "P {}" .
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    fix F x assume "finite F" and "x \<notin> F"
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      and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
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    show "P (insert x F)"
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    proof (rule insert)
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      from i show "x \<in> A" by blast
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      from i have "F \<subseteq> A" by blast
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      with P show "P F" .
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    qed
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  qed
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qed
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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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proof -
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  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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    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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      assume x: "x \<in> A"
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      with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
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      with r have "finite (A - {x})" .
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      hence "finite (insert x (A - {x}))" ..
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      also have "insert x (A - {x}) = A" by (rule insert_Diff)
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      finally show ?thesis .
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    next
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      show "A \<subseteq> F ==> ?thesis" .
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      assume "x \<notin> A"
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      with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
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    qed
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  qed
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qed
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lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
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  by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
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lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
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  -- {* 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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  apply (simp only: finite_Un, blast)
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  done
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lemma finite_empty_induct:
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  "finite A ==>
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  P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
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proof -
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  assume "finite A"
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    and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
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  have "P (A - A)"
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  proof -
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    fix c b :: "'a set"
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    presume c: "finite c" and b: "finite b"
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      and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
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    from c show "c \<subseteq> b ==> P (b - c)"
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    proof induct
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      case empty
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      from P1 show ?case by simp
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    next
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      case (insert F x)
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      have "P (b - F - {x})"
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      proof (rule P2)
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        from _ b show "finite (b - F)" by (rule finite_subset) blast
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        from insert show "x \<in> b - F" by simp
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        from insert show "P (b - F)" by simp
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      qed
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      also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
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      finally show ?case .
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    qed
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  next
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    show "A \<subseteq> A" ..
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  qed
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  thus "P {}" by simp
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qed
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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_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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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 ~= {}})"}?
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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 ~= {}}"}
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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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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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instance * :: (finite, finite) finite
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proof
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  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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   apply simp
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   apply (rule iffI)
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    apply (erule finite_imageD [unfolded inj_on_def])
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    apply (simp split add: split_split)
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   apply (erule finite_imageI)
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  apply (simp add: converse_def image_def, auto)
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  apply (rule bexI)
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   prefer 2 apply assumption
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  apply simp
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  done
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lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..k(}"
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  by (induct k) (simp_all add: lessThan_Suc)
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lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
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  by (induct k) (simp_all add: atMost_Suc)
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lemma finite_greaterThanLessThan [iff]:
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  fixes l :: nat shows "finite {)l..u(}"
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by (simp add: greaterThanLessThan_def)
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lemma finite_atLeastLessThan [iff]:
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  fixes l :: nat shows "finite {l..u(}"
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by (simp add: atLeastLessThan_def)
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lemma finite_greaterThanAtMost [iff]:
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  fixes l :: nat shows "finite {)l..u}"
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by (simp add: greaterThanAtMost_def)
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lemma finite_atLeastAtMost [iff]:
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  fixes l :: nat shows "finite {l..u}"
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by (simp add: atLeastAtMost_def)
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lemma bounded_nat_set_is_finite:
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    "(ALL i:N. i < (n::nat)) ==> finite N"
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  -- {* A bounded set of natural numbers is finite. *}
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  apply (rule finite_subset)
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   apply (rule_tac [2] finite_lessThan, auto)
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  done
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subsubsection {* Finiteness of transitive closure *}
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text {* (Thanks to Sidi Ehmety) *}
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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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  done
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lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
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  apply clarify
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  apply (erule trancl_induct)
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   apply (auto simp add: Field_def)
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  done
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lemma finite_trancl: "finite (r^+) = finite r"
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  apply auto
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   prefer 2
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   apply (rule trancl_subset_Field2 [THEN finite_subset])
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   apply (rule finite_SigmaI)
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    prefer 3
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    apply (blast intro: r_into_trancl' finite_subset)
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   337
   apply (auto simp add: finite_Field)
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   338
  done
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   339
wenzelm@12396
   340
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   341
subsection {* Finite cardinality *}
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   342
wenzelm@12396
   343
text {*
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   344
  This definition, although traditional, is ugly to work with: @{text
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   345
  "card A == LEAST n. EX f. A = {f i | i. i < n}"}.  Therefore we have
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   346
  switched to an inductive one:
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   347
*}
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   348
wenzelm@12396
   349
consts cardR :: "('a set \<times> nat) set"
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   350
wenzelm@12396
   351
inductive cardR
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   352
  intros
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   353
    EmptyI: "({}, 0) : cardR"
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   354
    InsertI: "(A, n) : cardR ==> a \<notin> A ==> (insert a A, Suc n) : cardR"
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   355
wenzelm@12396
   356
constdefs
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   357
  card :: "'a set => nat"
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   358
  "card A == THE n. (A, n) : cardR"
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   359
wenzelm@12396
   360
inductive_cases cardR_emptyE: "({}, n) : cardR"
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   361
inductive_cases cardR_insertE: "(insert a A,n) : cardR"
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   362
wenzelm@12396
   363
lemma cardR_SucD: "(A, n) : cardR ==> a : A --> (EX m. n = Suc m)"
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   364
  by (induct set: cardR) simp_all
wenzelm@12396
   365
wenzelm@12396
   366
lemma cardR_determ_aux1:
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   367
    "(A, m): cardR ==> (!!n a. m = Suc n ==> a:A ==> (A - {a}, n) : cardR)"
paulson@14208
   368
  apply (induct set: cardR, auto)
paulson@14208
   369
  apply (simp add: insert_Diff_if, auto)
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   370
  apply (drule cardR_SucD)
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   371
  apply (blast intro!: cardR.intros)
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   372
  done
wenzelm@12396
   373
wenzelm@12396
   374
lemma cardR_determ_aux2: "(insert a A, Suc m) : cardR ==> a \<notin> A ==> (A, m) : cardR"
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   375
  by (drule cardR_determ_aux1) auto
wenzelm@12396
   376
wenzelm@12396
   377
lemma cardR_determ: "(A, m): cardR ==> (!!n. (A, n) : cardR ==> n = m)"
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   378
  apply (induct set: cardR)
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   379
   apply (safe elim!: cardR_emptyE cardR_insertE)
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   380
  apply (rename_tac B b m)
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   381
  apply (case_tac "a = b")
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   382
   apply (subgoal_tac "A = B")
paulson@14208
   383
    prefer 2 apply (blast elim: equalityE, blast)
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   384
  apply (subgoal_tac "EX C. A = insert b C & B = insert a C")
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   385
   prefer 2
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   386
   apply (rule_tac x = "A Int B" in exI)
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   387
   apply (blast elim: equalityE)
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   388
  apply (frule_tac A = B in cardR_SucD)
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   389
  apply (blast intro!: cardR.intros dest!: cardR_determ_aux2)
wenzelm@12396
   390
  done
wenzelm@12396
   391
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   392
lemma cardR_imp_finite: "(A, n) : cardR ==> finite A"
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   393
  by (induct set: cardR) simp_all
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   394
wenzelm@12396
   395
lemma finite_imp_cardR: "finite A ==> EX n. (A, n) : cardR"
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   396
  by (induct set: Finites) (auto intro!: cardR.intros)
wenzelm@12396
   397
wenzelm@12396
   398
lemma card_equality: "(A,n) : cardR ==> card A = n"
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   399
  by (unfold card_def) (blast intro: cardR_determ)
wenzelm@12396
   400
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   401
lemma card_empty [simp]: "card {} = 0"
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   402
  by (unfold card_def) (blast intro!: cardR.intros elim!: cardR_emptyE)
wenzelm@12396
   403
wenzelm@12396
   404
lemma card_insert_disjoint [simp]:
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   405
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
wenzelm@12396
   406
proof -
wenzelm@12396
   407
  assume x: "x \<notin> A"
wenzelm@12396
   408
  hence aux: "!!n. ((insert x A, n) : cardR) = (EX m. (A, m) : cardR & n = Suc m)"
wenzelm@12396
   409
    apply (auto intro!: cardR.intros)
wenzelm@12396
   410
    apply (rule_tac A1 = A in finite_imp_cardR [THEN exE])
wenzelm@12396
   411
     apply (force dest: cardR_imp_finite)
wenzelm@12396
   412
    apply (blast intro!: cardR.intros intro: cardR_determ)
wenzelm@12396
   413
    done
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   414
  assume "finite A"
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   415
  thus ?thesis
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   416
    apply (simp add: card_def aux)
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   417
    apply (rule the_equality)
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   418
     apply (auto intro: finite_imp_cardR
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   419
       cong: conj_cong simp: card_def [symmetric] card_equality)
wenzelm@12396
   420
    done
wenzelm@12396
   421
qed
wenzelm@12396
   422
wenzelm@12396
   423
lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
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   424
  apply auto
paulson@14208
   425
  apply (drule_tac a = x in mk_disjoint_insert, clarify)
paulson@14208
   426
  apply (rotate_tac -1, auto)
wenzelm@12396
   427
  done
wenzelm@12396
   428
wenzelm@12396
   429
lemma card_insert_if:
wenzelm@12396
   430
    "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
wenzelm@12396
   431
  by (simp add: insert_absorb)
wenzelm@12396
   432
wenzelm@12396
   433
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
paulson@14208
   434
by (rule_tac t = A in insert_Diff [THEN subst], assumption, simp)
wenzelm@12396
   435
wenzelm@12396
   436
lemma card_Diff_singleton:
wenzelm@12396
   437
    "finite A ==> x: A ==> card (A - {x}) = card A - 1"
wenzelm@12396
   438
  by (simp add: card_Suc_Diff1 [symmetric])
wenzelm@12396
   439
wenzelm@12396
   440
lemma card_Diff_singleton_if:
wenzelm@12396
   441
    "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
wenzelm@12396
   442
  by (simp add: card_Diff_singleton)
wenzelm@12396
   443
wenzelm@12396
   444
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
wenzelm@12396
   445
  by (simp add: card_insert_if card_Suc_Diff1)
wenzelm@12396
   446
wenzelm@12396
   447
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
wenzelm@12396
   448
  by (simp add: card_insert_if)
wenzelm@12396
   449
wenzelm@12396
   450
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
paulson@14208
   451
  apply (induct set: Finites, simp, clarify)
wenzelm@12396
   452
  apply (subgoal_tac "finite A & A - {x} <= F")
paulson@14208
   453
   prefer 2 apply (blast intro: finite_subset, atomize)
wenzelm@12396
   454
  apply (drule_tac x = "A - {x}" in spec)
wenzelm@12396
   455
  apply (simp add: card_Diff_singleton_if split add: split_if_asm)
paulson@14208
   456
  apply (case_tac "card A", auto)
wenzelm@12396
   457
  done
wenzelm@12396
   458
wenzelm@12396
   459
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
wenzelm@12396
   460
  apply (simp add: psubset_def linorder_not_le [symmetric])
wenzelm@12396
   461
  apply (blast dest: card_seteq)
wenzelm@12396
   462
  done
wenzelm@12396
   463
wenzelm@12396
   464
lemma card_mono: "finite B ==> A <= B ==> card A <= card B"
paulson@14208
   465
  apply (case_tac "A = B", simp)
wenzelm@12396
   466
  apply (simp add: linorder_not_less [symmetric])
wenzelm@12396
   467
  apply (blast dest: card_seteq intro: order_less_imp_le)
wenzelm@12396
   468
  done
wenzelm@12396
   469
wenzelm@12396
   470
lemma card_Un_Int: "finite A ==> finite B
wenzelm@12396
   471
    ==> card A + card B = card (A Un B) + card (A Int B)"
paulson@14208
   472
  apply (induct set: Finites, simp)
wenzelm@12396
   473
  apply (simp add: insert_absorb Int_insert_left)
wenzelm@12396
   474
  done
wenzelm@12396
   475
wenzelm@12396
   476
lemma card_Un_disjoint: "finite A ==> finite B
wenzelm@12396
   477
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
wenzelm@12396
   478
  by (simp add: card_Un_Int)
wenzelm@12396
   479
wenzelm@12396
   480
lemma card_Diff_subset:
wenzelm@12396
   481
    "finite A ==> B <= A ==> card A - card B = card (A - B)"
wenzelm@12396
   482
  apply (subgoal_tac "(A - B) Un B = A")
wenzelm@12396
   483
   prefer 2 apply blast
wenzelm@12396
   484
  apply (rule add_right_cancel [THEN iffD1])
wenzelm@12396
   485
  apply (rule card_Un_disjoint [THEN subst])
wenzelm@12396
   486
     apply (erule_tac [4] ssubst)
wenzelm@12396
   487
     prefer 3 apply blast
wenzelm@12396
   488
    apply (simp_all add: add_commute not_less_iff_le
wenzelm@12396
   489
      add_diff_inverse card_mono finite_subset)
wenzelm@12396
   490
  done
wenzelm@12396
   491
wenzelm@12396
   492
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
wenzelm@12396
   493
  apply (rule Suc_less_SucD)
wenzelm@12396
   494
  apply (simp add: card_Suc_Diff1)
wenzelm@12396
   495
  done
wenzelm@12396
   496
wenzelm@12396
   497
lemma card_Diff2_less:
wenzelm@12396
   498
    "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
wenzelm@12396
   499
  apply (case_tac "x = y")
wenzelm@12396
   500
   apply (simp add: card_Diff1_less)
wenzelm@12396
   501
  apply (rule less_trans)
wenzelm@12396
   502
   prefer 2 apply (auto intro!: card_Diff1_less)
wenzelm@12396
   503
  done
wenzelm@12396
   504
wenzelm@12396
   505
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
wenzelm@12396
   506
  apply (case_tac "x : A")
wenzelm@12396
   507
   apply (simp_all add: card_Diff1_less less_imp_le)
wenzelm@12396
   508
  done
wenzelm@12396
   509
wenzelm@12396
   510
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
paulson@14208
   511
by (erule psubsetI, blast)
wenzelm@12396
   512
wenzelm@12396
   513
wenzelm@12396
   514
subsubsection {* Cardinality of image *}
wenzelm@12396
   515
wenzelm@12396
   516
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
paulson@14208
   517
  apply (induct set: Finites, simp)
wenzelm@12396
   518
  apply (simp add: le_SucI finite_imageI card_insert_if)
wenzelm@12396
   519
  done
wenzelm@12396
   520
wenzelm@12396
   521
lemma card_image: "finite A ==> inj_on f A ==> card (f ` A) = card A"
paulson@14208
   522
  apply (induct set: Finites, simp_all, atomize)
wenzelm@12396
   523
  apply safe
paulson@14208
   524
   apply (unfold inj_on_def, blast)
wenzelm@12396
   525
  apply (subst card_insert_disjoint)
paulson@14208
   526
    apply (erule finite_imageI, blast, blast)
wenzelm@12396
   527
  done
wenzelm@12396
   528
wenzelm@12396
   529
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
wenzelm@12396
   530
  by (simp add: card_seteq card_image)
wenzelm@12396
   531
wenzelm@12396
   532
wenzelm@12396
   533
subsubsection {* Cardinality of the Powerset *}
wenzelm@12396
   534
wenzelm@12396
   535
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
wenzelm@12396
   536
  apply (induct set: Finites)
wenzelm@12396
   537
   apply (simp_all add: Pow_insert)
paulson@14208
   538
  apply (subst card_Un_disjoint, blast)
paulson@14208
   539
    apply (blast intro: finite_imageI, blast)
wenzelm@12396
   540
  apply (subgoal_tac "inj_on (insert x) (Pow F)")
wenzelm@12396
   541
   apply (simp add: card_image Pow_insert)
wenzelm@12396
   542
  apply (unfold inj_on_def)
wenzelm@12396
   543
  apply (blast elim!: equalityE)
wenzelm@12396
   544
  done
wenzelm@12396
   545
wenzelm@12396
   546
text {*
wenzelm@12396
   547
  \medskip Relates to equivalence classes.  Based on a theorem of
wenzelm@12396
   548
  F. Kammüller's.  The @{prop "finite C"} premise is redundant.
wenzelm@12396
   549
*}
wenzelm@12396
   550
wenzelm@12396
   551
lemma dvd_partition:
wenzelm@12396
   552
  "finite C ==> finite (Union C) ==>
wenzelm@12396
   553
    ALL c : C. k dvd card c ==>
wenzelm@12396
   554
    (ALL c1: C. ALL c2: C. c1 ~= c2 --> c1 Int c2 = {}) ==>
wenzelm@12396
   555
  k dvd card (Union C)"
paulson@14208
   556
  apply (induct set: Finites, simp_all, clarify)
wenzelm@12396
   557
  apply (subst card_Un_disjoint)
wenzelm@12396
   558
  apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
wenzelm@12396
   559
  done
wenzelm@12396
   560
wenzelm@12396
   561
wenzelm@12396
   562
subsection {* A fold functional for finite sets *}
wenzelm@12396
   563
wenzelm@12396
   564
text {*
wenzelm@12396
   565
  For @{text n} non-negative we have @{text "fold f e {x1, ..., xn} =
wenzelm@12396
   566
  f x1 (... (f xn e))"} where @{text f} is at least left-commutative.
wenzelm@12396
   567
*}
wenzelm@12396
   568
wenzelm@12396
   569
consts
wenzelm@12396
   570
  foldSet :: "('b => 'a => 'a) => 'a => ('b set \<times> 'a) set"
wenzelm@12396
   571
wenzelm@12396
   572
inductive "foldSet f e"
wenzelm@12396
   573
  intros
wenzelm@12396
   574
    emptyI [intro]: "({}, e) : foldSet f e"
wenzelm@12396
   575
    insertI [intro]: "x \<notin> A ==> (A, y) : foldSet f e ==> (insert x A, f x y) : foldSet f e"
wenzelm@12396
   576
wenzelm@12396
   577
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f e"
wenzelm@12396
   578
wenzelm@12396
   579
constdefs
wenzelm@12396
   580
  fold :: "('b => 'a => 'a) => 'a => 'b set => 'a"
wenzelm@12396
   581
  "fold f e A == THE x. (A, x) : foldSet f e"
wenzelm@12396
   582
wenzelm@12396
   583
lemma Diff1_foldSet: "(A - {x}, y) : foldSet f e ==> x: A ==> (A, f x y) : foldSet f e"
paulson@14208
   584
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
wenzelm@12396
   585
wenzelm@12396
   586
lemma foldSet_imp_finite [simp]: "(A, x) : foldSet f e ==> finite A"
wenzelm@12396
   587
  by (induct set: foldSet) auto
wenzelm@12396
   588
wenzelm@12396
   589
lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f e"
wenzelm@12396
   590
  by (induct set: Finites) auto
wenzelm@12396
   591
wenzelm@12396
   592
wenzelm@12396
   593
subsubsection {* Left-commutative operations *}
wenzelm@12396
   594
wenzelm@12396
   595
locale LC =
wenzelm@12396
   596
  fixes f :: "'b => 'a => 'a"    (infixl "\<cdot>" 70)
wenzelm@12396
   597
  assumes left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
wenzelm@12396
   598
wenzelm@12396
   599
lemma (in LC) foldSet_determ_aux:
wenzelm@12396
   600
  "ALL A x. card A < n --> (A, x) : foldSet f e -->
wenzelm@12396
   601
    (ALL y. (A, y) : foldSet f e --> y = x)"
wenzelm@12396
   602
  apply (induct n)
wenzelm@12396
   603
   apply (auto simp add: less_Suc_eq)
paulson@14208
   604
  apply (erule foldSet.cases, blast)
paulson@14208
   605
  apply (erule foldSet.cases, blast, clarify)
wenzelm@12396
   606
  txt {* force simplification of @{text "card A < card (insert ...)"}. *}
wenzelm@12396
   607
  apply (erule rev_mp)
wenzelm@12396
   608
  apply (simp add: less_Suc_eq_le)
wenzelm@12396
   609
  apply (rule impI)
wenzelm@12396
   610
  apply (rename_tac Aa xa ya Ab xb yb, case_tac "xa = xb")
wenzelm@12396
   611
   apply (subgoal_tac "Aa = Ab")
paulson@14208
   612
    prefer 2 apply (blast elim!: equalityE, blast)
wenzelm@12396
   613
  txt {* case @{prop "xa \<notin> xb"}. *}
wenzelm@12396
   614
  apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb : Aa & xa : Ab")
paulson@14208
   615
   prefer 2 apply (blast elim!: equalityE, clarify)
wenzelm@12396
   616
  apply (subgoal_tac "Aa = insert xb Ab - {xa}")
wenzelm@12396
   617
   prefer 2 apply blast
wenzelm@12396
   618
  apply (subgoal_tac "card Aa <= card Ab")
wenzelm@12396
   619
   prefer 2
wenzelm@12396
   620
   apply (rule Suc_le_mono [THEN subst])
wenzelm@12396
   621
   apply (simp add: card_Suc_Diff1)
wenzelm@12396
   622
  apply (rule_tac A1 = "Aa - {xb}" and f1 = f in finite_imp_foldSet [THEN exE])
wenzelm@12396
   623
  apply (blast intro: foldSet_imp_finite finite_Diff)
wenzelm@12396
   624
  apply (frule (1) Diff1_foldSet)
wenzelm@12396
   625
  apply (subgoal_tac "ya = f xb x")
wenzelm@12396
   626
   prefer 2 apply (blast del: equalityCE)
wenzelm@12396
   627
  apply (subgoal_tac "(Ab - {xa}, x) : foldSet f e")
wenzelm@12396
   628
   prefer 2 apply simp
wenzelm@12396
   629
  apply (subgoal_tac "yb = f xa x")
wenzelm@12396
   630
   prefer 2 apply (blast del: equalityCE dest: Diff1_foldSet)
wenzelm@12396
   631
  apply (simp (no_asm_simp) add: left_commute)
wenzelm@12396
   632
  done
wenzelm@12396
   633
wenzelm@12396
   634
lemma (in LC) foldSet_determ: "(A, x) : foldSet f e ==> (A, y) : foldSet f e ==> y = x"
wenzelm@12396
   635
  by (blast intro: foldSet_determ_aux [rule_format])
wenzelm@12396
   636
wenzelm@12396
   637
lemma (in LC) fold_equality: "(A, y) : foldSet f e ==> fold f e A = y"
wenzelm@12396
   638
  by (unfold fold_def) (blast intro: foldSet_determ)
wenzelm@12396
   639
wenzelm@12396
   640
lemma fold_empty [simp]: "fold f e {} = e"
wenzelm@12396
   641
  by (unfold fold_def) blast
wenzelm@12396
   642
wenzelm@12396
   643
lemma (in LC) fold_insert_aux: "x \<notin> A ==>
wenzelm@12396
   644
    ((insert x A, v) : foldSet f e) =
wenzelm@12396
   645
    (EX y. (A, y) : foldSet f e & v = f x y)"
wenzelm@12396
   646
  apply auto
wenzelm@12396
   647
  apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
wenzelm@12396
   648
   apply (fastsimp dest: foldSet_imp_finite)
wenzelm@12396
   649
  apply (blast intro: foldSet_determ)
wenzelm@12396
   650
  done
wenzelm@12396
   651
wenzelm@12396
   652
lemma (in LC) fold_insert:
wenzelm@12396
   653
    "finite A ==> x \<notin> A ==> fold f e (insert x A) = f x (fold f e A)"
wenzelm@12396
   654
  apply (unfold fold_def)
wenzelm@12396
   655
  apply (simp add: fold_insert_aux)
wenzelm@12396
   656
  apply (rule the_equality)
wenzelm@12396
   657
  apply (auto intro: finite_imp_foldSet
wenzelm@12396
   658
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
wenzelm@12396
   659
  done
wenzelm@12396
   660
wenzelm@12396
   661
lemma (in LC) fold_commute: "finite A ==> (!!e. f x (fold f e A) = fold f (f x e) A)"
paulson@14208
   662
  apply (induct set: Finites, simp)
wenzelm@12396
   663
  apply (simp add: left_commute fold_insert)
wenzelm@12396
   664
  done
wenzelm@12396
   665
wenzelm@12396
   666
lemma (in LC) fold_nest_Un_Int:
wenzelm@12396
   667
  "finite A ==> finite B
wenzelm@12396
   668
    ==> fold f (fold f e B) A = fold f (fold f e (A Int B)) (A Un B)"
paulson@14208
   669
  apply (induct set: Finites, simp)
wenzelm@12396
   670
  apply (simp add: fold_insert fold_commute Int_insert_left insert_absorb)
wenzelm@12396
   671
  done
wenzelm@12396
   672
wenzelm@12396
   673
lemma (in LC) fold_nest_Un_disjoint:
wenzelm@12396
   674
  "finite A ==> finite B ==> A Int B = {}
wenzelm@12396
   675
    ==> fold f e (A Un B) = fold f (fold f e B) A"
wenzelm@12396
   676
  by (simp add: fold_nest_Un_Int)
wenzelm@12396
   677
wenzelm@12396
   678
declare foldSet_imp_finite [simp del]
wenzelm@12396
   679
    empty_foldSetE [rule del]  foldSet.intros [rule del]
wenzelm@12396
   680
  -- {* Delete rules to do with @{text foldSet} relation. *}
wenzelm@12396
   681
wenzelm@12396
   682
wenzelm@12396
   683
wenzelm@12396
   684
subsubsection {* Commutative monoids *}
wenzelm@12396
   685
wenzelm@12396
   686
text {*
wenzelm@12396
   687
  We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"}
wenzelm@12396
   688
  instead of @{text "'b => 'a => 'a"}.
wenzelm@12396
   689
*}
wenzelm@12396
   690
wenzelm@12396
   691
locale ACe =
wenzelm@12396
   692
  fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
wenzelm@12396
   693
    and e :: 'a
wenzelm@12396
   694
  assumes ident [simp]: "x \<cdot> e = x"
wenzelm@12396
   695
    and commute: "x \<cdot> y = y \<cdot> x"
wenzelm@12396
   696
    and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
wenzelm@12396
   697
wenzelm@12396
   698
lemma (in ACe) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
wenzelm@12396
   699
proof -
wenzelm@12396
   700
  have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
wenzelm@12396
   701
  also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
wenzelm@12396
   702
  also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
wenzelm@12396
   703
  finally show ?thesis .
wenzelm@12396
   704
qed
wenzelm@12396
   705
wenzelm@12718
   706
lemmas (in ACe) AC = assoc commute left_commute
wenzelm@12396
   707
wenzelm@12693
   708
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
wenzelm@12396
   709
proof -
wenzelm@12396
   710
  have "x \<cdot> e = x" by (rule ident)
wenzelm@12396
   711
  thus ?thesis by (subst commute)
wenzelm@12396
   712
qed
wenzelm@12396
   713
wenzelm@12396
   714
lemma (in ACe) fold_Un_Int:
wenzelm@12396
   715
  "finite A ==> finite B ==>
wenzelm@12396
   716
    fold f e A \<cdot> fold f e B = fold f e (A Un B) \<cdot> fold f e (A Int B)"
paulson@14208
   717
  apply (induct set: Finites, simp)
wenzelm@13400
   718
  apply (simp add: AC insert_absorb Int_insert_left
wenzelm@13421
   719
    LC.fold_insert [OF LC.intro])
wenzelm@12396
   720
  done
wenzelm@12396
   721
wenzelm@12396
   722
lemma (in ACe) fold_Un_disjoint:
wenzelm@12396
   723
  "finite A ==> finite B ==> A Int B = {} ==>
wenzelm@12396
   724
    fold f e (A Un B) = fold f e A \<cdot> fold f e B"
wenzelm@12396
   725
  by (simp add: fold_Un_Int)
wenzelm@12396
   726
wenzelm@12396
   727
lemma (in ACe) fold_Un_disjoint2:
wenzelm@12396
   728
  "finite A ==> finite B ==> A Int B = {} ==>
wenzelm@12396
   729
    fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B"
wenzelm@12396
   730
proof -
wenzelm@12396
   731
  assume b: "finite B"
wenzelm@12396
   732
  assume "finite A"
wenzelm@12396
   733
  thus "A Int B = {} ==>
wenzelm@12396
   734
    fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B"
wenzelm@12396
   735
  proof induct
wenzelm@12396
   736
    case empty
wenzelm@12396
   737
    thus ?case by simp
wenzelm@12396
   738
  next
wenzelm@12396
   739
    case (insert F x)
paulson@13571
   740
    have "fold (f o g) e (insert x F \<union> B) = fold (f o g) e (insert x (F \<union> B))"
wenzelm@12396
   741
      by simp
paulson@13571
   742
    also have "... = (f o g) x (fold (f o g) e (F \<union> B))"
wenzelm@13400
   743
      by (rule LC.fold_insert [OF LC.intro])
wenzelm@13421
   744
        (insert b insert, auto simp add: left_commute)
paulson@13571
   745
    also from insert have "fold (f o g) e (F \<union> B) =
paulson@13571
   746
      fold (f o g) e F \<cdot> fold (f o g) e B" by blast
paulson@13571
   747
    also have "(f o g) x ... = (f o g) x (fold (f o g) e F) \<cdot> fold (f o g) e B"
wenzelm@12396
   748
      by (simp add: AC)
paulson@13571
   749
    also have "(f o g) x (fold (f o g) e F) = fold (f o g) e (insert x F)"
wenzelm@13400
   750
      by (rule LC.fold_insert [OF LC.intro, symmetric]) (insert b insert,
wenzelm@13421
   751
	auto simp add: left_commute)
wenzelm@12396
   752
    finally show ?case .
wenzelm@12396
   753
  qed
wenzelm@12396
   754
qed
wenzelm@12396
   755
wenzelm@12396
   756
wenzelm@12396
   757
subsection {* Generalized summation over a set *}
wenzelm@12396
   758
wenzelm@12396
   759
constdefs
wenzelm@12396
   760
  setsum :: "('a => 'b) => 'a set => 'b::plus_ac0"
wenzelm@12396
   761
  "setsum f A == if finite A then fold (op + o f) 0 A else 0"
wenzelm@12396
   762
wenzelm@12396
   763
syntax
wenzelm@12396
   764
  "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0"    ("\<Sum>_:_. _" [0, 51, 10] 10)
wenzelm@12396
   765
syntax (xsymbols)
wenzelm@12396
   766
  "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0"    ("\<Sum>_\<in>_. _" [0, 51, 10] 10)
wenzelm@12396
   767
translations
wenzelm@12396
   768
  "\<Sum>i:A. b" == "setsum (%i. b) A"  -- {* Beware of argument permutation! *}
wenzelm@12396
   769
wenzelm@12396
   770
wenzelm@12396
   771
lemma setsum_empty [simp]: "setsum f {} = 0"
wenzelm@12396
   772
  by (simp add: setsum_def)
wenzelm@12396
   773
wenzelm@12396
   774
lemma setsum_insert [simp]:
wenzelm@12396
   775
    "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
wenzelm@13365
   776
  by (simp add: setsum_def
wenzelm@13421
   777
    LC.fold_insert [OF LC.intro] plus_ac0_left_commute)
wenzelm@12396
   778
wenzelm@12396
   779
lemma setsum_0: "setsum (\<lambda>i. 0) A = 0"
wenzelm@12396
   780
  apply (case_tac "finite A")
wenzelm@12396
   781
   prefer 2 apply (simp add: setsum_def)
paulson@14208
   782
  apply (erule finite_induct, auto)
wenzelm@12396
   783
  done
wenzelm@12396
   784
wenzelm@12396
   785
lemma setsum_eq_0_iff [simp]:
wenzelm@12396
   786
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
wenzelm@12396
   787
  by (induct set: Finites) auto
wenzelm@12396
   788
wenzelm@12396
   789
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
wenzelm@12396
   790
  apply (case_tac "finite A")
wenzelm@12396
   791
   prefer 2 apply (simp add: setsum_def)
wenzelm@12396
   792
  apply (erule rev_mp)
paulson@14208
   793
  apply (erule finite_induct, auto)
wenzelm@12396
   794
  done
wenzelm@12396
   795
wenzelm@12396
   796
lemma card_eq_setsum: "finite A ==> card A = setsum (\<lambda>x. 1) A"
wenzelm@12396
   797
  -- {* Could allow many @{text "card"} proofs to be simplified. *}
wenzelm@12396
   798
  by (induct set: Finites) auto
wenzelm@12396
   799
wenzelm@12396
   800
lemma setsum_Un_Int: "finite A ==> finite B
wenzelm@12396
   801
    ==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
wenzelm@12396
   802
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
paulson@14208
   803
  apply (induct set: Finites, simp)
wenzelm@12396
   804
  apply (simp add: plus_ac0 Int_insert_left insert_absorb)
wenzelm@12396
   805
  done
wenzelm@12396
   806
wenzelm@12396
   807
lemma setsum_Un_disjoint: "finite A ==> finite B
wenzelm@12396
   808
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
paulson@14208
   809
  apply (subst setsum_Un_Int [symmetric], auto)
wenzelm@12396
   810
  done
wenzelm@12396
   811
wenzelm@12937
   812
lemma setsum_UN_disjoint:
wenzelm@12937
   813
  fixes f :: "'a => 'b::plus_ac0"
wenzelm@12937
   814
  shows
wenzelm@12937
   815
    "finite I ==> (ALL i:I. finite (A i)) ==>
wenzelm@12937
   816
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
wenzelm@12937
   817
      setsum f (UNION I A) = setsum (\<lambda>i. setsum f (A i)) I"
paulson@14208
   818
  apply (induct set: Finites, simp, atomize)
wenzelm@12396
   819
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
wenzelm@12396
   820
   prefer 2 apply blast
wenzelm@12396
   821
  apply (subgoal_tac "A x Int UNION F A = {}")
wenzelm@12396
   822
   prefer 2 apply blast
wenzelm@12396
   823
  apply (simp add: setsum_Un_disjoint)
wenzelm@12396
   824
  done
wenzelm@12396
   825
wenzelm@12396
   826
lemma setsum_addf: "setsum (\<lambda>x. f x + g x) A = (setsum f A + setsum g A)"
wenzelm@12396
   827
  apply (case_tac "finite A")
wenzelm@12396
   828
   prefer 2 apply (simp add: setsum_def)
paulson@14208
   829
  apply (erule finite_induct, auto)
wenzelm@12396
   830
  apply (simp add: plus_ac0)
wenzelm@12396
   831
  done
wenzelm@12396
   832
wenzelm@12396
   833
lemma setsum_Un: "finite A ==> finite B ==>
wenzelm@12396
   834
    (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
wenzelm@12396
   835
  -- {* For the natural numbers, we have subtraction. *}
paulson@14208
   836
  apply (subst setsum_Un_Int [symmetric], auto)
wenzelm@12396
   837
  done
wenzelm@12396
   838
wenzelm@12396
   839
lemma setsum_diff1: "(setsum f (A - {a}) :: nat) =
wenzelm@12396
   840
    (if a:A then setsum f A - f a else setsum f A)"
wenzelm@12396
   841
  apply (case_tac "finite A")
wenzelm@12396
   842
   prefer 2 apply (simp add: setsum_def)
wenzelm@12396
   843
  apply (erule finite_induct)
wenzelm@12396
   844
   apply (auto simp add: insert_Diff_if)
paulson@14208
   845
  apply (drule_tac a = a in mk_disjoint_insert, auto)
wenzelm@12396
   846
  done
wenzelm@12396
   847
wenzelm@12396
   848
lemma setsum_cong:
wenzelm@12396
   849
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
wenzelm@12396
   850
  apply (case_tac "finite B")
paulson@14208
   851
   prefer 2 apply (simp add: setsum_def, simp)
wenzelm@12396
   852
  apply (subgoal_tac "ALL C. C <= B --> (ALL x:C. f x = g x) --> setsum f C = setsum g C")
wenzelm@12396
   853
   apply simp
paulson@14208
   854
  apply (erule finite_induct, simp)
paulson@14208
   855
  apply (simp add: subset_insert_iff, clarify)
wenzelm@12396
   856
  apply (subgoal_tac "finite C")
wenzelm@12396
   857
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
wenzelm@12396
   858
  apply (subgoal_tac "C = insert x (C - {x})")
wenzelm@12396
   859
   prefer 2 apply blast
wenzelm@12396
   860
  apply (erule ssubst)
wenzelm@12396
   861
  apply (drule spec)
wenzelm@12396
   862
  apply (erule (1) notE impE)
wenzelm@12396
   863
  apply (simp add: Ball_def)
wenzelm@12396
   864
  done
wenzelm@12396
   865
nipkow@13490
   866
subsubsection{* Min and Max of finite linearly ordered sets *}
nipkow@13490
   867
nipkow@13490
   868
text{* Seemed easier to define directly than via fold. *}
nipkow@13490
   869
nipkow@13490
   870
lemma ex_Max: fixes S :: "('a::linorder)set"
nipkow@13490
   871
  assumes fin: "finite S" shows "S \<noteq> {} \<Longrightarrow> \<exists>m\<in>S. \<forall>s \<in> S. s \<le> m"
nipkow@13490
   872
using fin
nipkow@13490
   873
proof (induct)
nipkow@13490
   874
  case empty thus ?case by simp
nipkow@13490
   875
next
nipkow@13490
   876
  case (insert S x)
nipkow@13490
   877
  show ?case
nipkow@13490
   878
  proof (cases)
nipkow@13490
   879
    assume "S = {}" thus ?thesis by simp
nipkow@13490
   880
  next
nipkow@13490
   881
    assume nonempty: "S \<noteq> {}"
nipkow@13490
   882
    then obtain m where m: "m\<in>S" "\<forall>s\<in>S. s \<le> m" using insert by blast
nipkow@13490
   883
    show ?thesis
nipkow@13490
   884
    proof (cases)
nipkow@13490
   885
      assume "x \<le> m" thus ?thesis using m by blast
nipkow@13490
   886
    next
nipkow@13490
   887
      assume "\<not> x \<le> m" thus ?thesis using m
nipkow@13490
   888
	by(simp add:linorder_not_le order_less_le)(blast intro: order_trans)
nipkow@13490
   889
    qed
nipkow@13490
   890
  qed
nipkow@13490
   891
qed
nipkow@13490
   892
nipkow@13490
   893
lemma ex_Min: fixes S :: "('a::linorder)set"
nipkow@13490
   894
  assumes fin: "finite S" shows "S \<noteq> {} \<Longrightarrow> \<exists>m\<in>S. \<forall>s \<in> S. m \<le> s"
nipkow@13490
   895
using fin
nipkow@13490
   896
proof (induct)
nipkow@13490
   897
  case empty thus ?case by simp
nipkow@13490
   898
next
nipkow@13490
   899
  case (insert S x)
nipkow@13490
   900
  show ?case
nipkow@13490
   901
  proof (cases)
nipkow@13490
   902
    assume "S = {}" thus ?thesis by simp
nipkow@13490
   903
  next
nipkow@13490
   904
    assume nonempty: "S \<noteq> {}"
nipkow@13490
   905
    then obtain m where m: "m\<in>S" "\<forall>s\<in>S. m \<le> s" using insert by blast
nipkow@13490
   906
    show ?thesis
nipkow@13490
   907
    proof (cases)
nipkow@13490
   908
      assume "m \<le> x" thus ?thesis using m by blast
nipkow@13490
   909
    next
nipkow@13490
   910
      assume "\<not> m \<le> x" thus ?thesis using m
nipkow@13490
   911
	by(simp add:linorder_not_le order_less_le)(blast intro: order_trans)
nipkow@13490
   912
    qed
nipkow@13490
   913
  qed
nipkow@13490
   914
qed
nipkow@13490
   915
nipkow@13490
   916
constdefs
nipkow@13490
   917
 Min :: "('a::linorder)set \<Rightarrow> 'a"
nipkow@13490
   918
"Min S  \<equiv>  THE m. m \<in> S \<and> (\<forall>s \<in> S. m \<le> s)"
nipkow@13490
   919
nipkow@13490
   920
 Max :: "('a::linorder)set \<Rightarrow> 'a"
nipkow@13490
   921
"Max S  \<equiv>  THE m. m \<in> S \<and> (\<forall>s \<in> S. s \<le> m)"
nipkow@13490
   922
nipkow@13490
   923
lemma Min[simp]: assumes a: "finite S" "S \<noteq> {}"
nipkow@13490
   924
                 shows "Min S \<in> S \<and> (\<forall>s \<in> S. Min S \<le> s)" (is "?P(Min S)")
nipkow@13490
   925
proof (unfold Min_def, rule theI')
nipkow@13490
   926
  show "\<exists>!m. ?P m"
nipkow@13490
   927
  proof (rule ex_ex1I)
nipkow@13490
   928
    show "\<exists>m. ?P m" using ex_Min[OF a] by blast
nipkow@13490
   929
  next
nipkow@13490
   930
    fix m1 m2 assume "?P m1" "?P m2"
nipkow@13490
   931
    thus "m1 = m2" by (blast dest:order_antisym)
nipkow@13490
   932
  qed
nipkow@13490
   933
qed
nipkow@13490
   934
nipkow@13490
   935
lemma Max[simp]: assumes a: "finite S" "S \<noteq> {}"
nipkow@13490
   936
                 shows "Max S \<in> S \<and> (\<forall>s \<in> S. s \<le> Max S)" (is "?P(Max S)")
nipkow@13490
   937
proof (unfold Max_def, rule theI')
nipkow@13490
   938
  show "\<exists>!m. ?P m"
nipkow@13490
   939
  proof (rule ex_ex1I)
nipkow@13490
   940
    show "\<exists>m. ?P m" using ex_Max[OF a] by blast
nipkow@13490
   941
  next
nipkow@13490
   942
    fix m1 m2 assume "?P m1" "?P m2"
nipkow@13490
   943
    thus "m1 = m2" by (blast dest:order_antisym)
nipkow@13490
   944
  qed
nipkow@13490
   945
qed
nipkow@13490
   946
wenzelm@12396
   947
wenzelm@12396
   948
text {*
wenzelm@12396
   949
  \medskip Basic theorem about @{text "choose"}.  By Florian
wenzelm@12396
   950
  Kammüller, tidied by LCP.
wenzelm@12396
   951
*}
wenzelm@12396
   952
wenzelm@12396
   953
lemma card_s_0_eq_empty:
wenzelm@12396
   954
    "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
wenzelm@12396
   955
  apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
wenzelm@12396
   956
  apply (simp cong add: rev_conj_cong)
wenzelm@12396
   957
  done
wenzelm@12396
   958
wenzelm@12396
   959
lemma choose_deconstruct: "finite M ==> x \<notin> M
wenzelm@12396
   960
  ==> {s. s <= insert x M & card(s) = Suc k}
wenzelm@12396
   961
       = {s. s <= M & card(s) = Suc k} Un
wenzelm@12396
   962
         {s. EX t. t <= M & card(t) = k & s = insert x t}"
wenzelm@12396
   963
  apply safe
wenzelm@12396
   964
   apply (auto intro: finite_subset [THEN card_insert_disjoint])
wenzelm@12396
   965
  apply (drule_tac x = "xa - {x}" in spec)
paulson@14208
   966
  apply (subgoal_tac "x ~: xa", auto)
wenzelm@12396
   967
  apply (erule rev_mp, subst card_Diff_singleton)
wenzelm@12396
   968
  apply (auto intro: finite_subset)
wenzelm@12396
   969
  done
wenzelm@12396
   970
wenzelm@12396
   971
lemma card_inj_on_le:
paulson@13595
   972
    "[|inj_on f A; f ` A \<subseteq> B; finite A; finite B |] ==> card A <= card B"
wenzelm@12396
   973
  by (auto intro: card_mono simp add: card_image [symmetric])
wenzelm@12396
   974
paulson@13595
   975
lemma card_bij_eq: 
paulson@13595
   976
    "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A; 
paulson@13595
   977
       finite A; finite B |] ==> card A = card B"
wenzelm@12396
   978
  by (auto intro: le_anti_sym card_inj_on_le)
wenzelm@12396
   979
paulson@13595
   980
text{*There are as many subsets of @{term A} having cardinality @{term k}
paulson@13595
   981
 as there are sets obtained from the former by inserting a fixed element
paulson@13595
   982
 @{term x} into each.*}
paulson@13595
   983
lemma constr_bij:
paulson@13595
   984
   "[|finite A; x \<notin> A|] ==>
paulson@13595
   985
    card {B. EX C. C <= A & card(C) = k & B = insert x C} =
wenzelm@12396
   986
    card {B. B <= A & card(B) = k}"
wenzelm@12396
   987
  apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
paulson@13595
   988
       apply (auto elim!: equalityE simp add: inj_on_def)
paulson@13595
   989
    apply (subst Diff_insert0, auto)
paulson@13595
   990
   txt {* finiteness of the two sets *}
paulson@13595
   991
   apply (rule_tac [2] B = "Pow (A)" in finite_subset)
paulson@13595
   992
   apply (rule_tac B = "Pow (insert x A)" in finite_subset)
paulson@13595
   993
   apply fast+
wenzelm@12396
   994
  done
wenzelm@12396
   995
wenzelm@12396
   996
text {*
wenzelm@12396
   997
  Main theorem: combinatorial statement about number of subsets of a set.
wenzelm@12396
   998
*}
wenzelm@12396
   999
wenzelm@12396
  1000
lemma n_sub_lemma:
wenzelm@12396
  1001
  "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
wenzelm@12396
  1002
  apply (induct k)
paulson@14208
  1003
   apply (simp add: card_s_0_eq_empty, atomize)
wenzelm@12396
  1004
  apply (rotate_tac -1, erule finite_induct)
wenzelm@13421
  1005
   apply (simp_all (no_asm_simp) cong add: conj_cong
wenzelm@13421
  1006
     add: card_s_0_eq_empty choose_deconstruct)
wenzelm@12396
  1007
  apply (subst card_Un_disjoint)
wenzelm@12396
  1008
     prefer 4 apply (force simp add: constr_bij)
wenzelm@12396
  1009
    prefer 3 apply force
wenzelm@12396
  1010
   prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
wenzelm@12396
  1011
     finite_subset [of _ "Pow (insert x F)", standard])
wenzelm@12396
  1012
  apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
wenzelm@12396
  1013
  done
wenzelm@12396
  1014
wenzelm@13421
  1015
theorem n_subsets:
wenzelm@13421
  1016
    "finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
wenzelm@12396
  1017
  by (simp add: n_sub_lemma)
wenzelm@12396
  1018
wenzelm@12396
  1019
end