src/HOL/Finite_Set.thy
author wenzelm
Thu Dec 06 00:38:55 2001 +0100 (2001-12-06)
changeset 12396 2298d5b8e530
child 12693 827818b891c7
permissions -rw-r--r--
renamed theory Finite to Finite_Set and converted;
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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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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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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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axclass finite \<subseteq> type
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  finite: "finite UNIV"
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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 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: "!!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)
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  apply blast
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  done
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lemma finite_imageI: "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)
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  apply simp
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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)
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    apply simp_all
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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])
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    apply 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])
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    apply 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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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)
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   apply 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)
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  apply 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) "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) "finite {..k}"
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  by (induct k) (simp_all add: atMost_Suc)
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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)
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  apply 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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   apply (auto simp add: finite_Field)
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  done
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subsection {* Finite cardinality *}
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text {*
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  This definition, although traditional, is ugly to work with: @{text
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  "card A == LEAST n. EX f. A = {f i | i. i < n}"}.  Therefore we have
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  switched to an inductive one:
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*}
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consts cardR :: "('a set \<times> nat) set"
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inductive cardR
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  intros
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    EmptyI: "({}, 0) : cardR"
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    InsertI: "(A, n) : cardR ==> a \<notin> A ==> (insert a A, Suc n) : cardR"
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constdefs
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  card :: "'a set => nat"
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  "card A == THE n. (A, n) : cardR"
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inductive_cases cardR_emptyE: "({}, n) : cardR"
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inductive_cases cardR_insertE: "(insert a A,n) : cardR"
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lemma cardR_SucD: "(A, n) : cardR ==> a : A --> (EX m. n = Suc m)"
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  by (induct set: cardR) simp_all
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lemma cardR_determ_aux1:
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    "(A, m): cardR ==> (!!n a. m = Suc n ==> a:A ==> (A - {a}, n) : cardR)"
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  apply (induct set: cardR)
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   apply auto
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  apply (simp add: insert_Diff_if)
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  apply auto
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   341
  apply (drule cardR_SucD)
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   342
  apply (blast intro!: cardR.intros)
wenzelm@12396
   343
  done
wenzelm@12396
   344
wenzelm@12396
   345
lemma cardR_determ_aux2: "(insert a A, Suc m) : cardR ==> a \<notin> A ==> (A, m) : cardR"
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   346
  by (drule cardR_determ_aux1) auto
wenzelm@12396
   347
wenzelm@12396
   348
lemma cardR_determ: "(A, m): cardR ==> (!!n. (A, n) : cardR ==> n = m)"
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   349
  apply (induct set: cardR)
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   350
   apply (safe elim!: cardR_emptyE cardR_insertE)
wenzelm@12396
   351
  apply (rename_tac B b m)
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   352
  apply (case_tac "a = b")
wenzelm@12396
   353
   apply (subgoal_tac "A = B")
wenzelm@12396
   354
    prefer 2 apply (blast elim: equalityE)
wenzelm@12396
   355
   apply blast
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   356
  apply (subgoal_tac "EX C. A = insert b C & B = insert a C")
wenzelm@12396
   357
   prefer 2
wenzelm@12396
   358
   apply (rule_tac x = "A Int B" in exI)
wenzelm@12396
   359
   apply (blast elim: equalityE)
wenzelm@12396
   360
  apply (frule_tac A = B in cardR_SucD)
wenzelm@12396
   361
  apply (blast intro!: cardR.intros dest!: cardR_determ_aux2)
wenzelm@12396
   362
  done
wenzelm@12396
   363
wenzelm@12396
   364
lemma cardR_imp_finite: "(A, n) : cardR ==> finite A"
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   365
  by (induct set: cardR) simp_all
wenzelm@12396
   366
wenzelm@12396
   367
lemma finite_imp_cardR: "finite A ==> EX n. (A, n) : cardR"
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   368
  by (induct set: Finites) (auto intro!: cardR.intros)
wenzelm@12396
   369
wenzelm@12396
   370
lemma card_equality: "(A,n) : cardR ==> card A = n"
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   371
  by (unfold card_def) (blast intro: cardR_determ)
wenzelm@12396
   372
wenzelm@12396
   373
lemma card_empty [simp]: "card {} = 0"
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   374
  by (unfold card_def) (blast intro!: cardR.intros elim!: cardR_emptyE)
wenzelm@12396
   375
wenzelm@12396
   376
lemma card_insert_disjoint [simp]:
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   377
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
wenzelm@12396
   378
proof -
wenzelm@12396
   379
  assume x: "x \<notin> A"
wenzelm@12396
   380
  hence aux: "!!n. ((insert x A, n) : cardR) = (EX m. (A, m) : cardR & n = Suc m)"
wenzelm@12396
   381
    apply (auto intro!: cardR.intros)
wenzelm@12396
   382
    apply (rule_tac A1 = A in finite_imp_cardR [THEN exE])
wenzelm@12396
   383
     apply (force dest: cardR_imp_finite)
wenzelm@12396
   384
    apply (blast intro!: cardR.intros intro: cardR_determ)
wenzelm@12396
   385
    done
wenzelm@12396
   386
  assume "finite A"
wenzelm@12396
   387
  thus ?thesis
wenzelm@12396
   388
    apply (simp add: card_def aux)
wenzelm@12396
   389
    apply (rule the_equality)
wenzelm@12396
   390
     apply (auto intro: finite_imp_cardR
wenzelm@12396
   391
       cong: conj_cong simp: card_def [symmetric] card_equality)
wenzelm@12396
   392
    done
wenzelm@12396
   393
qed
wenzelm@12396
   394
wenzelm@12396
   395
lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
wenzelm@12396
   396
  apply auto
wenzelm@12396
   397
  apply (drule_tac a = x in mk_disjoint_insert)
wenzelm@12396
   398
  apply clarify
wenzelm@12396
   399
  apply (rotate_tac -1)
wenzelm@12396
   400
  apply auto
wenzelm@12396
   401
  done
wenzelm@12396
   402
wenzelm@12396
   403
lemma card_insert_if:
wenzelm@12396
   404
    "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
wenzelm@12396
   405
  by (simp add: insert_absorb)
wenzelm@12396
   406
wenzelm@12396
   407
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
wenzelm@12396
   408
  apply (rule_tac t = A in insert_Diff [THEN subst])
wenzelm@12396
   409
   apply assumption
wenzelm@12396
   410
  apply simp
wenzelm@12396
   411
  done
wenzelm@12396
   412
wenzelm@12396
   413
lemma card_Diff_singleton:
wenzelm@12396
   414
    "finite A ==> x: A ==> card (A - {x}) = card A - 1"
wenzelm@12396
   415
  by (simp add: card_Suc_Diff1 [symmetric])
wenzelm@12396
   416
wenzelm@12396
   417
lemma card_Diff_singleton_if:
wenzelm@12396
   418
    "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
wenzelm@12396
   419
  by (simp add: card_Diff_singleton)
wenzelm@12396
   420
wenzelm@12396
   421
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
wenzelm@12396
   422
  by (simp add: card_insert_if card_Suc_Diff1)
wenzelm@12396
   423
wenzelm@12396
   424
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
wenzelm@12396
   425
  by (simp add: card_insert_if)
wenzelm@12396
   426
wenzelm@12396
   427
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
wenzelm@12396
   428
  apply (induct set: Finites)
wenzelm@12396
   429
   apply simp
wenzelm@12396
   430
  apply clarify
wenzelm@12396
   431
  apply (subgoal_tac "finite A & A - {x} <= F")
wenzelm@12396
   432
   prefer 2 apply (blast intro: finite_subset)
wenzelm@12396
   433
  apply atomize
wenzelm@12396
   434
  apply (drule_tac x = "A - {x}" in spec)
wenzelm@12396
   435
  apply (simp add: card_Diff_singleton_if split add: split_if_asm)
wenzelm@12396
   436
  apply (case_tac "card A")
wenzelm@12396
   437
   apply auto
wenzelm@12396
   438
  done
wenzelm@12396
   439
wenzelm@12396
   440
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
wenzelm@12396
   441
  apply (simp add: psubset_def linorder_not_le [symmetric])
wenzelm@12396
   442
  apply (blast dest: card_seteq)
wenzelm@12396
   443
  done
wenzelm@12396
   444
wenzelm@12396
   445
lemma card_mono: "finite B ==> A <= B ==> card A <= card B"
wenzelm@12396
   446
  apply (case_tac "A = B")
wenzelm@12396
   447
   apply simp
wenzelm@12396
   448
  apply (simp add: linorder_not_less [symmetric])
wenzelm@12396
   449
  apply (blast dest: card_seteq intro: order_less_imp_le)
wenzelm@12396
   450
  done
wenzelm@12396
   451
wenzelm@12396
   452
lemma card_Un_Int: "finite A ==> finite B
wenzelm@12396
   453
    ==> card A + card B = card (A Un B) + card (A Int B)"
wenzelm@12396
   454
  apply (induct set: Finites)
wenzelm@12396
   455
   apply simp
wenzelm@12396
   456
  apply (simp add: insert_absorb Int_insert_left)
wenzelm@12396
   457
  done
wenzelm@12396
   458
wenzelm@12396
   459
lemma card_Un_disjoint: "finite A ==> finite B
wenzelm@12396
   460
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
wenzelm@12396
   461
  by (simp add: card_Un_Int)
wenzelm@12396
   462
wenzelm@12396
   463
lemma card_Diff_subset:
wenzelm@12396
   464
    "finite A ==> B <= A ==> card A - card B = card (A - B)"
wenzelm@12396
   465
  apply (subgoal_tac "(A - B) Un B = A")
wenzelm@12396
   466
   prefer 2 apply blast
wenzelm@12396
   467
  apply (rule add_right_cancel [THEN iffD1])
wenzelm@12396
   468
  apply (rule card_Un_disjoint [THEN subst])
wenzelm@12396
   469
     apply (erule_tac [4] ssubst)
wenzelm@12396
   470
     prefer 3 apply blast
wenzelm@12396
   471
    apply (simp_all add: add_commute not_less_iff_le
wenzelm@12396
   472
      add_diff_inverse card_mono finite_subset)
wenzelm@12396
   473
  done
wenzelm@12396
   474
wenzelm@12396
   475
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
wenzelm@12396
   476
  apply (rule Suc_less_SucD)
wenzelm@12396
   477
  apply (simp add: card_Suc_Diff1)
wenzelm@12396
   478
  done
wenzelm@12396
   479
wenzelm@12396
   480
lemma card_Diff2_less:
wenzelm@12396
   481
    "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
wenzelm@12396
   482
  apply (case_tac "x = y")
wenzelm@12396
   483
   apply (simp add: card_Diff1_less)
wenzelm@12396
   484
  apply (rule less_trans)
wenzelm@12396
   485
   prefer 2 apply (auto intro!: card_Diff1_less)
wenzelm@12396
   486
  done
wenzelm@12396
   487
wenzelm@12396
   488
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
wenzelm@12396
   489
  apply (case_tac "x : A")
wenzelm@12396
   490
   apply (simp_all add: card_Diff1_less less_imp_le)
wenzelm@12396
   491
  done
wenzelm@12396
   492
wenzelm@12396
   493
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
wenzelm@12396
   494
  apply (erule psubsetI)
wenzelm@12396
   495
  apply blast
wenzelm@12396
   496
  done
wenzelm@12396
   497
wenzelm@12396
   498
wenzelm@12396
   499
subsubsection {* Cardinality of image *}
wenzelm@12396
   500
wenzelm@12396
   501
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
wenzelm@12396
   502
  apply (induct set: Finites)
wenzelm@12396
   503
   apply simp
wenzelm@12396
   504
  apply (simp add: le_SucI finite_imageI card_insert_if)
wenzelm@12396
   505
  done
wenzelm@12396
   506
wenzelm@12396
   507
lemma card_image: "finite A ==> inj_on f A ==> card (f ` A) = card A"
wenzelm@12396
   508
  apply (induct set: Finites)
wenzelm@12396
   509
   apply simp_all
wenzelm@12396
   510
  apply atomize
wenzelm@12396
   511
  apply safe
wenzelm@12396
   512
   apply (unfold inj_on_def)
wenzelm@12396
   513
   apply blast
wenzelm@12396
   514
  apply (subst card_insert_disjoint)
wenzelm@12396
   515
    apply (erule finite_imageI)
wenzelm@12396
   516
   apply blast
wenzelm@12396
   517
  apply blast
wenzelm@12396
   518
  done
wenzelm@12396
   519
wenzelm@12396
   520
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
wenzelm@12396
   521
  by (simp add: card_seteq card_image)
wenzelm@12396
   522
wenzelm@12396
   523
wenzelm@12396
   524
subsubsection {* Cardinality of the Powerset *}
wenzelm@12396
   525
wenzelm@12396
   526
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
wenzelm@12396
   527
  apply (induct set: Finites)
wenzelm@12396
   528
   apply (simp_all add: Pow_insert)
wenzelm@12396
   529
  apply (subst card_Un_disjoint)
wenzelm@12396
   530
     apply blast
wenzelm@12396
   531
    apply (blast intro: finite_imageI)
wenzelm@12396
   532
   apply blast
wenzelm@12396
   533
  apply (subgoal_tac "inj_on (insert x) (Pow F)")
wenzelm@12396
   534
   apply (simp add: card_image Pow_insert)
wenzelm@12396
   535
  apply (unfold inj_on_def)
wenzelm@12396
   536
  apply (blast elim!: equalityE)
wenzelm@12396
   537
  done
wenzelm@12396
   538
wenzelm@12396
   539
text {*
wenzelm@12396
   540
  \medskip Relates to equivalence classes.  Based on a theorem of
wenzelm@12396
   541
  F. Kammüller's.  The @{prop "finite C"} premise is redundant.
wenzelm@12396
   542
*}
wenzelm@12396
   543
wenzelm@12396
   544
lemma dvd_partition:
wenzelm@12396
   545
  "finite C ==> finite (Union C) ==>
wenzelm@12396
   546
    ALL c : C. k dvd card c ==>
wenzelm@12396
   547
    (ALL c1: C. ALL c2: C. c1 ~= c2 --> c1 Int c2 = {}) ==>
wenzelm@12396
   548
  k dvd card (Union C)"
wenzelm@12396
   549
  apply (induct set: Finites)
wenzelm@12396
   550
   apply simp_all
wenzelm@12396
   551
  apply clarify
wenzelm@12396
   552
  apply (subst card_Un_disjoint)
wenzelm@12396
   553
  apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
wenzelm@12396
   554
  done
wenzelm@12396
   555
wenzelm@12396
   556
wenzelm@12396
   557
subsection {* A fold functional for finite sets *}
wenzelm@12396
   558
wenzelm@12396
   559
text {*
wenzelm@12396
   560
  For @{text n} non-negative we have @{text "fold f e {x1, ..., xn} =
wenzelm@12396
   561
  f x1 (... (f xn e))"} where @{text f} is at least left-commutative.
wenzelm@12396
   562
*}
wenzelm@12396
   563
wenzelm@12396
   564
consts
wenzelm@12396
   565
  foldSet :: "('b => 'a => 'a) => 'a => ('b set \<times> 'a) set"
wenzelm@12396
   566
wenzelm@12396
   567
inductive "foldSet f e"
wenzelm@12396
   568
  intros
wenzelm@12396
   569
    emptyI [intro]: "({}, e) : foldSet f e"
wenzelm@12396
   570
    insertI [intro]: "x \<notin> A ==> (A, y) : foldSet f e ==> (insert x A, f x y) : foldSet f e"
wenzelm@12396
   571
wenzelm@12396
   572
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f e"
wenzelm@12396
   573
wenzelm@12396
   574
constdefs
wenzelm@12396
   575
  fold :: "('b => 'a => 'a) => 'a => 'b set => 'a"
wenzelm@12396
   576
  "fold f e A == THE x. (A, x) : foldSet f e"
wenzelm@12396
   577
wenzelm@12396
   578
lemma Diff1_foldSet: "(A - {x}, y) : foldSet f e ==> x: A ==> (A, f x y) : foldSet f e"
wenzelm@12396
   579
  apply (erule insert_Diff [THEN subst], rule foldSet.intros)
wenzelm@12396
   580
   apply auto
wenzelm@12396
   581
  done
wenzelm@12396
   582
wenzelm@12396
   583
lemma foldSet_imp_finite [simp]: "(A, x) : foldSet f e ==> finite A"
wenzelm@12396
   584
  by (induct set: foldSet) auto
wenzelm@12396
   585
wenzelm@12396
   586
lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f e"
wenzelm@12396
   587
  by (induct set: Finites) auto
wenzelm@12396
   588
wenzelm@12396
   589
wenzelm@12396
   590
subsubsection {* Left-commutative operations *}
wenzelm@12396
   591
wenzelm@12396
   592
locale LC =
wenzelm@12396
   593
  fixes f :: "'b => 'a => 'a"    (infixl "\<cdot>" 70)
wenzelm@12396
   594
  assumes left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
wenzelm@12396
   595
wenzelm@12396
   596
lemma (in LC) foldSet_determ_aux:
wenzelm@12396
   597
  "ALL A x. card A < n --> (A, x) : foldSet f e -->
wenzelm@12396
   598
    (ALL y. (A, y) : foldSet f e --> y = x)"
wenzelm@12396
   599
  apply (induct n)
wenzelm@12396
   600
   apply (auto simp add: less_Suc_eq)
wenzelm@12396
   601
  apply (erule foldSet.cases)
wenzelm@12396
   602
   apply blast
wenzelm@12396
   603
  apply (erule foldSet.cases)
wenzelm@12396
   604
   apply blast
wenzelm@12396
   605
  apply 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")
wenzelm@12396
   612
    prefer 2 apply (blast elim!: equalityE)
wenzelm@12396
   613
   apply blast
wenzelm@12396
   614
  txt {* case @{prop "xa \<notin> xb"}. *}
wenzelm@12396
   615
  apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb : Aa & xa : Ab")
wenzelm@12396
   616
   prefer 2 apply (blast elim!: equalityE)
wenzelm@12396
   617
  apply clarify
wenzelm@12396
   618
  apply (subgoal_tac "Aa = insert xb Ab - {xa}")
wenzelm@12396
   619
   prefer 2 apply blast
wenzelm@12396
   620
  apply (subgoal_tac "card Aa <= card Ab")
wenzelm@12396
   621
   prefer 2
wenzelm@12396
   622
   apply (rule Suc_le_mono [THEN subst])
wenzelm@12396
   623
   apply (simp add: card_Suc_Diff1)
wenzelm@12396
   624
  apply (rule_tac A1 = "Aa - {xb}" and f1 = f in finite_imp_foldSet [THEN exE])
wenzelm@12396
   625
  apply (blast intro: foldSet_imp_finite finite_Diff)
wenzelm@12396
   626
  apply (frule (1) Diff1_foldSet)
wenzelm@12396
   627
  apply (subgoal_tac "ya = f xb x")
wenzelm@12396
   628
   prefer 2 apply (blast del: equalityCE)
wenzelm@12396
   629
  apply (subgoal_tac "(Ab - {xa}, x) : foldSet f e")
wenzelm@12396
   630
   prefer 2 apply simp
wenzelm@12396
   631
  apply (subgoal_tac "yb = f xa x")
wenzelm@12396
   632
   prefer 2 apply (blast del: equalityCE dest: Diff1_foldSet)
wenzelm@12396
   633
  apply (simp (no_asm_simp) add: left_commute)
wenzelm@12396
   634
  done
wenzelm@12396
   635
wenzelm@12396
   636
lemma (in LC) foldSet_determ: "(A, x) : foldSet f e ==> (A, y) : foldSet f e ==> y = x"
wenzelm@12396
   637
  by (blast intro: foldSet_determ_aux [rule_format])
wenzelm@12396
   638
wenzelm@12396
   639
lemma (in LC) fold_equality: "(A, y) : foldSet f e ==> fold f e A = y"
wenzelm@12396
   640
  by (unfold fold_def) (blast intro: foldSet_determ)
wenzelm@12396
   641
wenzelm@12396
   642
lemma fold_empty [simp]: "fold f e {} = e"
wenzelm@12396
   643
  by (unfold fold_def) blast
wenzelm@12396
   644
wenzelm@12396
   645
lemma (in LC) fold_insert_aux: "x \<notin> A ==>
wenzelm@12396
   646
    ((insert x A, v) : foldSet f e) =
wenzelm@12396
   647
    (EX y. (A, y) : foldSet f e & v = f x y)"
wenzelm@12396
   648
  apply auto
wenzelm@12396
   649
  apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
wenzelm@12396
   650
   apply (fastsimp dest: foldSet_imp_finite)
wenzelm@12396
   651
  apply (blast intro: foldSet_determ)
wenzelm@12396
   652
  done
wenzelm@12396
   653
wenzelm@12396
   654
lemma (in LC) fold_insert:
wenzelm@12396
   655
    "finite A ==> x \<notin> A ==> fold f e (insert x A) = f x (fold f e A)"
wenzelm@12396
   656
  apply (unfold fold_def)
wenzelm@12396
   657
  apply (simp add: fold_insert_aux)
wenzelm@12396
   658
  apply (rule the_equality)
wenzelm@12396
   659
  apply (auto intro: finite_imp_foldSet
wenzelm@12396
   660
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
wenzelm@12396
   661
  done
wenzelm@12396
   662
wenzelm@12396
   663
lemma (in LC) fold_commute: "finite A ==> (!!e. f x (fold f e A) = fold f (f x e) A)"
wenzelm@12396
   664
  apply (induct set: Finites)
wenzelm@12396
   665
   apply simp
wenzelm@12396
   666
  apply (simp add: left_commute fold_insert)
wenzelm@12396
   667
  done
wenzelm@12396
   668
wenzelm@12396
   669
lemma (in LC) fold_nest_Un_Int:
wenzelm@12396
   670
  "finite A ==> finite B
wenzelm@12396
   671
    ==> fold f (fold f e B) A = fold f (fold f e (A Int B)) (A Un B)"
wenzelm@12396
   672
  apply (induct set: Finites)
wenzelm@12396
   673
   apply simp
wenzelm@12396
   674
  apply (simp add: fold_insert fold_commute Int_insert_left insert_absorb)
wenzelm@12396
   675
  done
wenzelm@12396
   676
wenzelm@12396
   677
lemma (in LC) fold_nest_Un_disjoint:
wenzelm@12396
   678
  "finite A ==> finite B ==> A Int B = {}
wenzelm@12396
   679
    ==> fold f e (A Un B) = fold f (fold f e B) A"
wenzelm@12396
   680
  by (simp add: fold_nest_Un_Int)
wenzelm@12396
   681
wenzelm@12396
   682
declare foldSet_imp_finite [simp del]
wenzelm@12396
   683
    empty_foldSetE [rule del]  foldSet.intros [rule del]
wenzelm@12396
   684
  -- {* Delete rules to do with @{text foldSet} relation. *}
wenzelm@12396
   685
wenzelm@12396
   686
wenzelm@12396
   687
wenzelm@12396
   688
subsubsection {* Commutative monoids *}
wenzelm@12396
   689
wenzelm@12396
   690
text {*
wenzelm@12396
   691
  We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"}
wenzelm@12396
   692
  instead of @{text "'b => 'a => 'a"}.
wenzelm@12396
   693
*}
wenzelm@12396
   694
wenzelm@12396
   695
locale ACe =
wenzelm@12396
   696
  fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
wenzelm@12396
   697
    and e :: 'a
wenzelm@12396
   698
  assumes ident [simp]: "x \<cdot> e = x"
wenzelm@12396
   699
    and commute: "x \<cdot> y = y \<cdot> x"
wenzelm@12396
   700
    and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
wenzelm@12396
   701
wenzelm@12396
   702
lemma (in ACe) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
wenzelm@12396
   703
proof -
wenzelm@12396
   704
  have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
wenzelm@12396
   705
  also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
wenzelm@12396
   706
  also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
wenzelm@12396
   707
  finally show ?thesis .
wenzelm@12396
   708
qed
wenzelm@12396
   709
wenzelm@12396
   710
lemma (in ACe)
wenzelm@12396
   711
    AC: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"  "x \<cdot> y = y \<cdot> x"  "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
wenzelm@12396
   712
  by (rule assoc, rule commute, rule left_commute)  (* FIXME localize "lemmas" (!??) *)
wenzelm@12396
   713
wenzelm@12396
   714
lemma (in ACe [simp]) left_ident: "e \<cdot> x = x"
wenzelm@12396
   715
proof -
wenzelm@12396
   716
  have "x \<cdot> e = x" by (rule ident)
wenzelm@12396
   717
  thus ?thesis by (subst commute)
wenzelm@12396
   718
qed
wenzelm@12396
   719
wenzelm@12396
   720
lemma (in ACe) fold_Un_Int:
wenzelm@12396
   721
  "finite A ==> finite B ==>
wenzelm@12396
   722
    fold f e A \<cdot> fold f e B = fold f e (A Un B) \<cdot> fold f e (A Int B)"
wenzelm@12396
   723
  apply (induct set: Finites)
wenzelm@12396
   724
   apply simp
wenzelm@12396
   725
  apply (simp add: AC fold_insert insert_absorb Int_insert_left)
wenzelm@12396
   726
  done
wenzelm@12396
   727
wenzelm@12396
   728
lemma (in ACe) fold_Un_disjoint:
wenzelm@12396
   729
  "finite A ==> finite B ==> A Int B = {} ==>
wenzelm@12396
   730
    fold f e (A Un B) = fold f e A \<cdot> fold f e B"
wenzelm@12396
   731
  by (simp add: fold_Un_Int)
wenzelm@12396
   732
wenzelm@12396
   733
lemma (in ACe) fold_Un_disjoint2:
wenzelm@12396
   734
  "finite A ==> finite B ==> A Int B = {} ==>
wenzelm@12396
   735
    fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B"
wenzelm@12396
   736
proof -
wenzelm@12396
   737
  assume b: "finite B"
wenzelm@12396
   738
  assume "finite A"
wenzelm@12396
   739
  thus "A Int B = {} ==>
wenzelm@12396
   740
    fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B"
wenzelm@12396
   741
  proof induct
wenzelm@12396
   742
    case empty
wenzelm@12396
   743
    thus ?case by simp
wenzelm@12396
   744
  next
wenzelm@12396
   745
    case (insert F x)
wenzelm@12396
   746
    have "fold (f \<circ> g) e (insert x F \<union> B) = fold (f \<circ> g) e (insert x (F \<union> B))"
wenzelm@12396
   747
      by simp
wenzelm@12396
   748
    also have "... = (f \<circ> g) x (fold (f \<circ> g) e (F \<union> B))"
wenzelm@12396
   749
      by (rule fold_insert) (insert b insert, auto simp add: left_commute)  (* FIXME import of fold_insert (!?) *)
wenzelm@12396
   750
    also from insert have "fold (f \<circ> g) e (F \<union> B) =
wenzelm@12396
   751
      fold (f \<circ> g) e F \<cdot> fold (f \<circ> g) e B" by blast
wenzelm@12396
   752
    also have "(f \<circ> g) x ... = (f \<circ> g) x (fold (f \<circ> g) e F) \<cdot> fold (f \<circ> g) e B"
wenzelm@12396
   753
      by (simp add: AC)
wenzelm@12396
   754
    also have "(f \<circ> g) x (fold (f \<circ> g) e F) = fold (f \<circ> g) e (insert x F)"
wenzelm@12396
   755
      by (rule fold_insert [symmetric]) (insert b insert, auto simp add: left_commute)
wenzelm@12396
   756
    finally show ?case .
wenzelm@12396
   757
  qed
wenzelm@12396
   758
qed
wenzelm@12396
   759
wenzelm@12396
   760
wenzelm@12396
   761
subsection {* Generalized summation over a set *}
wenzelm@12396
   762
wenzelm@12396
   763
constdefs
wenzelm@12396
   764
  setsum :: "('a => 'b) => 'a set => 'b::plus_ac0"
wenzelm@12396
   765
  "setsum f A == if finite A then fold (op + o f) 0 A else 0"
wenzelm@12396
   766
wenzelm@12396
   767
syntax
wenzelm@12396
   768
  "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0"    ("\<Sum>_:_. _" [0, 51, 10] 10)
wenzelm@12396
   769
syntax (xsymbols)
wenzelm@12396
   770
  "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0"    ("\<Sum>_\<in>_. _" [0, 51, 10] 10)
wenzelm@12396
   771
translations
wenzelm@12396
   772
  "\<Sum>i:A. b" == "setsum (%i. b) A"  -- {* Beware of argument permutation! *}
wenzelm@12396
   773
wenzelm@12396
   774
wenzelm@12396
   775
lemma setsum_empty [simp]: "setsum f {} = 0"
wenzelm@12396
   776
  by (simp add: setsum_def)
wenzelm@12396
   777
wenzelm@12396
   778
lemma setsum_insert [simp]:
wenzelm@12396
   779
    "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
wenzelm@12396
   780
  by (simp add: setsum_def fold_insert plus_ac0_left_commute)
wenzelm@12396
   781
wenzelm@12396
   782
lemma setsum_0: "setsum (\<lambda>i. 0) A = 0"
wenzelm@12396
   783
  apply (case_tac "finite A")
wenzelm@12396
   784
   prefer 2 apply (simp add: setsum_def)
wenzelm@12396
   785
  apply (erule finite_induct)
wenzelm@12396
   786
   apply auto
wenzelm@12396
   787
  done
wenzelm@12396
   788
wenzelm@12396
   789
lemma setsum_eq_0_iff [simp]:
wenzelm@12396
   790
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
wenzelm@12396
   791
  by (induct set: Finites) auto
wenzelm@12396
   792
wenzelm@12396
   793
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
wenzelm@12396
   794
  apply (case_tac "finite A")
wenzelm@12396
   795
   prefer 2 apply (simp add: setsum_def)
wenzelm@12396
   796
  apply (erule rev_mp)
wenzelm@12396
   797
  apply (erule finite_induct)
wenzelm@12396
   798
   apply auto
wenzelm@12396
   799
  done
wenzelm@12396
   800
wenzelm@12396
   801
lemma card_eq_setsum: "finite A ==> card A = setsum (\<lambda>x. 1) A"
wenzelm@12396
   802
  -- {* Could allow many @{text "card"} proofs to be simplified. *}
wenzelm@12396
   803
  by (induct set: Finites) auto
wenzelm@12396
   804
wenzelm@12396
   805
lemma setsum_Un_Int: "finite A ==> finite B
wenzelm@12396
   806
    ==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
wenzelm@12396
   807
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
wenzelm@12396
   808
  apply (induct set: Finites)
wenzelm@12396
   809
   apply simp
wenzelm@12396
   810
  apply (simp add: plus_ac0 Int_insert_left insert_absorb)
wenzelm@12396
   811
  done
wenzelm@12396
   812
wenzelm@12396
   813
lemma setsum_Un_disjoint: "finite A ==> finite B
wenzelm@12396
   814
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
wenzelm@12396
   815
  apply (subst setsum_Un_Int [symmetric])
wenzelm@12396
   816
    apply auto
wenzelm@12396
   817
  done
wenzelm@12396
   818
wenzelm@12396
   819
lemma setsum_UN_disjoint: (fixes f :: "'a => 'b::plus_ac0")
wenzelm@12396
   820
  "finite I ==> (ALL i:I. finite (A i)) ==>
wenzelm@12396
   821
      (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
wenzelm@12396
   822
    setsum f (UNION I A) = setsum (\<lambda>i. setsum f (A i)) I"
wenzelm@12396
   823
  apply (induct set: Finites)
wenzelm@12396
   824
   apply simp
wenzelm@12396
   825
  apply atomize
wenzelm@12396
   826
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
wenzelm@12396
   827
   prefer 2 apply blast
wenzelm@12396
   828
  apply (subgoal_tac "A x Int UNION F A = {}")
wenzelm@12396
   829
   prefer 2 apply blast
wenzelm@12396
   830
  apply (simp add: setsum_Un_disjoint)
wenzelm@12396
   831
  done
wenzelm@12396
   832
wenzelm@12396
   833
lemma setsum_addf: "setsum (\<lambda>x. f x + g x) A = (setsum f A + setsum g A)"
wenzelm@12396
   834
  apply (case_tac "finite A")
wenzelm@12396
   835
   prefer 2 apply (simp add: setsum_def)
wenzelm@12396
   836
  apply (erule finite_induct)
wenzelm@12396
   837
   apply auto
wenzelm@12396
   838
  apply (simp add: plus_ac0)
wenzelm@12396
   839
  done
wenzelm@12396
   840
wenzelm@12396
   841
lemma setsum_Un: "finite A ==> finite B ==>
wenzelm@12396
   842
    (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
wenzelm@12396
   843
  -- {* For the natural numbers, we have subtraction. *}
wenzelm@12396
   844
  apply (subst setsum_Un_Int [symmetric])
wenzelm@12396
   845
    apply auto
wenzelm@12396
   846
  done
wenzelm@12396
   847
wenzelm@12396
   848
lemma setsum_diff1: "(setsum f (A - {a}) :: nat) =
wenzelm@12396
   849
    (if a:A then setsum f A - f a else setsum f A)"
wenzelm@12396
   850
  apply (case_tac "finite A")
wenzelm@12396
   851
   prefer 2 apply (simp add: setsum_def)
wenzelm@12396
   852
  apply (erule finite_induct)
wenzelm@12396
   853
   apply (auto simp add: insert_Diff_if)
wenzelm@12396
   854
  apply (drule_tac a = a in mk_disjoint_insert)
wenzelm@12396
   855
  apply auto
wenzelm@12396
   856
  done
wenzelm@12396
   857
wenzelm@12396
   858
lemma setsum_cong:
wenzelm@12396
   859
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
wenzelm@12396
   860
  apply (case_tac "finite B")
wenzelm@12396
   861
   prefer 2 apply (simp add: setsum_def)
wenzelm@12396
   862
  apply simp
wenzelm@12396
   863
  apply (subgoal_tac "ALL C. C <= B --> (ALL x:C. f x = g x) --> setsum f C = setsum g C")
wenzelm@12396
   864
   apply simp
wenzelm@12396
   865
  apply (erule finite_induct)
wenzelm@12396
   866
  apply simp
wenzelm@12396
   867
  apply (simp add: subset_insert_iff)
wenzelm@12396
   868
  apply clarify
wenzelm@12396
   869
  apply (subgoal_tac "finite C")
wenzelm@12396
   870
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
wenzelm@12396
   871
  apply (subgoal_tac "C = insert x (C - {x})")
wenzelm@12396
   872
   prefer 2 apply blast
wenzelm@12396
   873
  apply (erule ssubst)
wenzelm@12396
   874
  apply (drule spec)
wenzelm@12396
   875
  apply (erule (1) notE impE)
wenzelm@12396
   876
  apply (simp add: Ball_def)
wenzelm@12396
   877
  done
wenzelm@12396
   878
wenzelm@12396
   879
wenzelm@12396
   880
text {*
wenzelm@12396
   881
  \medskip Basic theorem about @{text "choose"}.  By Florian
wenzelm@12396
   882
  Kammüller, tidied by LCP.
wenzelm@12396
   883
*}
wenzelm@12396
   884
wenzelm@12396
   885
lemma card_s_0_eq_empty:
wenzelm@12396
   886
    "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
wenzelm@12396
   887
  apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
wenzelm@12396
   888
  apply (simp cong add: rev_conj_cong)
wenzelm@12396
   889
  done
wenzelm@12396
   890
wenzelm@12396
   891
lemma choose_deconstruct: "finite M ==> x \<notin> M
wenzelm@12396
   892
  ==> {s. s <= insert x M & card(s) = Suc k}
wenzelm@12396
   893
       = {s. s <= M & card(s) = Suc k} Un
wenzelm@12396
   894
         {s. EX t. t <= M & card(t) = k & s = insert x t}"
wenzelm@12396
   895
  apply safe
wenzelm@12396
   896
   apply (auto intro: finite_subset [THEN card_insert_disjoint])
wenzelm@12396
   897
  apply (drule_tac x = "xa - {x}" in spec)
wenzelm@12396
   898
  apply (subgoal_tac "x ~: xa")
wenzelm@12396
   899
   apply auto
wenzelm@12396
   900
  apply (erule rev_mp, subst card_Diff_singleton)
wenzelm@12396
   901
  apply (auto intro: finite_subset)
wenzelm@12396
   902
  done
wenzelm@12396
   903
wenzelm@12396
   904
lemma card_inj_on_le:
wenzelm@12396
   905
    "finite A ==> finite B ==> f ` A \<subseteq> B ==> inj_on f A ==> card A <= card B"
wenzelm@12396
   906
  by (auto intro: card_mono simp add: card_image [symmetric])
wenzelm@12396
   907
wenzelm@12396
   908
lemma card_bij_eq: "finite A ==> finite B ==>
wenzelm@12396
   909
  f ` A \<subseteq> B ==> inj_on f A ==> g ` B \<subseteq> A ==> inj_on g B ==> card A = card B"
wenzelm@12396
   910
  by (auto intro: le_anti_sym card_inj_on_le)
wenzelm@12396
   911
wenzelm@12396
   912
lemma constr_bij: "finite A ==> x \<notin> A ==>
wenzelm@12396
   913
  card {B. EX C. C <= A & card(C) = k & B = insert x C} =
wenzelm@12396
   914
    card {B. B <= A & card(B) = k}"
wenzelm@12396
   915
  apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
wenzelm@12396
   916
       apply (rule_tac B = "Pow (insert x A) " in finite_subset)
wenzelm@12396
   917
        apply (rule_tac [3] B = "Pow (A) " in finite_subset)
wenzelm@12396
   918
         apply fast+
wenzelm@12396
   919
     txt {* arity *}
wenzelm@12396
   920
     apply (auto elim!: equalityE simp add: inj_on_def)
wenzelm@12396
   921
  apply (subst Diff_insert0)
wenzelm@12396
   922
  apply auto
wenzelm@12396
   923
  done
wenzelm@12396
   924
wenzelm@12396
   925
text {*
wenzelm@12396
   926
  Main theorem: combinatorial statement about number of subsets of a set.
wenzelm@12396
   927
*}
wenzelm@12396
   928
wenzelm@12396
   929
lemma n_sub_lemma:
wenzelm@12396
   930
  "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
wenzelm@12396
   931
  apply (induct k)
wenzelm@12396
   932
   apply (simp add: card_s_0_eq_empty)
wenzelm@12396
   933
  apply atomize
wenzelm@12396
   934
  apply (rotate_tac -1, erule finite_induct)
wenzelm@12396
   935
   apply (simp_all (no_asm_simp) cong add: conj_cong add: card_s_0_eq_empty choose_deconstruct)
wenzelm@12396
   936
  apply (subst card_Un_disjoint)
wenzelm@12396
   937
     prefer 4 apply (force simp add: constr_bij)
wenzelm@12396
   938
    prefer 3 apply force
wenzelm@12396
   939
   prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
wenzelm@12396
   940
     finite_subset [of _ "Pow (insert x F)", standard])
wenzelm@12396
   941
  apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
wenzelm@12396
   942
  done
wenzelm@12396
   943
wenzelm@12396
   944
theorem n_subsets: "finite A ==> card {B. B <= A & card(B) = k} = (card A choose k)"
wenzelm@12396
   945
  by (simp add: n_sub_lemma)
wenzelm@12396
   946
wenzelm@12396
   947
end