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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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apply (drule cardR_SucD)
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apply (blast intro!: cardR.intros)
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done
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lemma cardR_determ_aux2: "(insert a A, Suc m) : cardR ==> a \<notin> A ==> (A, m) : cardR"
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by (drule cardR_determ_aux1) auto
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lemma cardR_determ: "(A, m): cardR ==> (!!n. (A, n) : cardR ==> n = m)"
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apply (induct set: cardR)
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apply (safe elim!: cardR_emptyE cardR_insertE)
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apply (rename_tac B b m)
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apply (case_tac "a = b")
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apply (subgoal_tac "A = B")
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prefer 2 apply (blast elim: equalityE)
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apply blast
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apply (subgoal_tac "EX C. A = insert b C & B = insert a C")
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prefer 2
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apply (rule_tac x = "A Int B" in exI)
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apply (blast elim: equalityE)
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apply (frule_tac A = B in cardR_SucD)
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apply (blast intro!: cardR.intros dest!: cardR_determ_aux2)
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done
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lemma cardR_imp_finite: "(A, n) : cardR ==> finite A"
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by (induct set: cardR) simp_all
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lemma finite_imp_cardR: "finite A ==> EX n. (A, n) : cardR"
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by (induct set: Finites) (auto intro!: cardR.intros)
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lemma card_equality: "(A,n) : cardR ==> card A = n"
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by (unfold card_def) (blast intro: cardR_determ)
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lemma card_empty [simp]: "card {} = 0"
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by (unfold card_def) (blast intro!: cardR.intros elim!: cardR_emptyE)
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lemma card_insert_disjoint [simp]:
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"finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
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proof -
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assume x: "x \<notin> A"
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hence aux: "!!n. ((insert x A, n) : cardR) = (EX m. (A, m) : cardR & n = Suc m)"
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381 |
apply (auto intro!: cardR.intros)
|
|
382 |
apply (rule_tac A1 = A in finite_imp_cardR [THEN exE])
|
|
383 |
apply (force dest: cardR_imp_finite)
|
|
384 |
apply (blast intro!: cardR.intros intro: cardR_determ)
|
|
385 |
done
|
|
386 |
assume "finite A"
|
|
387 |
thus ?thesis
|
|
388 |
apply (simp add: card_def aux)
|
|
389 |
apply (rule the_equality)
|
|
390 |
apply (auto intro: finite_imp_cardR
|
|
391 |
cong: conj_cong simp: card_def [symmetric] card_equality)
|
|
392 |
done
|
|
393 |
qed
|
|
394 |
|
|
395 |
lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
|
|
396 |
apply auto
|
|
397 |
apply (drule_tac a = x in mk_disjoint_insert)
|
|
398 |
apply clarify
|
|
399 |
apply (rotate_tac -1)
|
|
400 |
apply auto
|
|
401 |
done
|
|
402 |
|
|
403 |
lemma card_insert_if:
|
|
404 |
"finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
|
|
405 |
by (simp add: insert_absorb)
|
|
406 |
|
|
407 |
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
|
|
408 |
apply (rule_tac t = A in insert_Diff [THEN subst])
|
|
409 |
apply assumption
|
|
410 |
apply simp
|
|
411 |
done
|
|
412 |
|
|
413 |
lemma card_Diff_singleton:
|
|
414 |
"finite A ==> x: A ==> card (A - {x}) = card A - 1"
|
|
415 |
by (simp add: card_Suc_Diff1 [symmetric])
|
|
416 |
|
|
417 |
lemma card_Diff_singleton_if:
|
|
418 |
"finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
|
|
419 |
by (simp add: card_Diff_singleton)
|
|
420 |
|
|
421 |
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
|
|
422 |
by (simp add: card_insert_if card_Suc_Diff1)
|
|
423 |
|
|
424 |
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
|
|
425 |
by (simp add: card_insert_if)
|
|
426 |
|
|
427 |
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
|
|
428 |
apply (induct set: Finites)
|
|
429 |
apply simp
|
|
430 |
apply clarify
|
|
431 |
apply (subgoal_tac "finite A & A - {x} <= F")
|
|
432 |
prefer 2 apply (blast intro: finite_subset)
|
|
433 |
apply atomize
|
|
434 |
apply (drule_tac x = "A - {x}" in spec)
|
|
435 |
apply (simp add: card_Diff_singleton_if split add: split_if_asm)
|
|
436 |
apply (case_tac "card A")
|
|
437 |
apply auto
|
|
438 |
done
|
|
439 |
|
|
440 |
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
|
|
441 |
apply (simp add: psubset_def linorder_not_le [symmetric])
|
|
442 |
apply (blast dest: card_seteq)
|
|
443 |
done
|
|
444 |
|
|
445 |
lemma card_mono: "finite B ==> A <= B ==> card A <= card B"
|
|
446 |
apply (case_tac "A = B")
|
|
447 |
apply simp
|
|
448 |
apply (simp add: linorder_not_less [symmetric])
|
|
449 |
apply (blast dest: card_seteq intro: order_less_imp_le)
|
|
450 |
done
|
|
451 |
|
|
452 |
lemma card_Un_Int: "finite A ==> finite B
|
|
453 |
==> card A + card B = card (A Un B) + card (A Int B)"
|
|
454 |
apply (induct set: Finites)
|
|
455 |
apply simp
|
|
456 |
apply (simp add: insert_absorb Int_insert_left)
|
|
457 |
done
|
|
458 |
|
|
459 |
lemma card_Un_disjoint: "finite A ==> finite B
|
|
460 |
==> A Int B = {} ==> card (A Un B) = card A + card B"
|
|
461 |
by (simp add: card_Un_Int)
|
|
462 |
|
|
463 |
lemma card_Diff_subset:
|
|
464 |
"finite A ==> B <= A ==> card A - card B = card (A - B)"
|
|
465 |
apply (subgoal_tac "(A - B) Un B = A")
|
|
466 |
prefer 2 apply blast
|
|
467 |
apply (rule add_right_cancel [THEN iffD1])
|
|
468 |
apply (rule card_Un_disjoint [THEN subst])
|
|
469 |
apply (erule_tac [4] ssubst)
|
|
470 |
prefer 3 apply blast
|
|
471 |
apply (simp_all add: add_commute not_less_iff_le
|
|
472 |
add_diff_inverse card_mono finite_subset)
|
|
473 |
done
|
|
474 |
|
|
475 |
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
|
|
476 |
apply (rule Suc_less_SucD)
|
|
477 |
apply (simp add: card_Suc_Diff1)
|
|
478 |
done
|
|
479 |
|
|
480 |
lemma card_Diff2_less:
|
|
481 |
"finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
|
|
482 |
apply (case_tac "x = y")
|
|
483 |
apply (simp add: card_Diff1_less)
|
|
484 |
apply (rule less_trans)
|
|
485 |
prefer 2 apply (auto intro!: card_Diff1_less)
|
|
486 |
done
|
|
487 |
|
|
488 |
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
|
|
489 |
apply (case_tac "x : A")
|
|
490 |
apply (simp_all add: card_Diff1_less less_imp_le)
|
|
491 |
done
|
|
492 |
|
|
493 |
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
|
|
494 |
apply (erule psubsetI)
|
|
495 |
apply blast
|
|
496 |
done
|
|
497 |
|
|
498 |
|
|
499 |
subsubsection {* Cardinality of image *}
|
|
500 |
|
|
501 |
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
|
|
502 |
apply (induct set: Finites)
|
|
503 |
apply simp
|
|
504 |
apply (simp add: le_SucI finite_imageI card_insert_if)
|
|
505 |
done
|
|
506 |
|
|
507 |
lemma card_image: "finite A ==> inj_on f A ==> card (f ` A) = card A"
|
|
508 |
apply (induct set: Finites)
|
|
509 |
apply simp_all
|
|
510 |
apply atomize
|
|
511 |
apply safe
|
|
512 |
apply (unfold inj_on_def)
|
|
513 |
apply blast
|
|
514 |
apply (subst card_insert_disjoint)
|
|
515 |
apply (erule finite_imageI)
|
|
516 |
apply blast
|
|
517 |
apply blast
|
|
518 |
done
|
|
519 |
|
|
520 |
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
|
|
521 |
by (simp add: card_seteq card_image)
|
|
522 |
|
|
523 |
|
|
524 |
subsubsection {* Cardinality of the Powerset *}
|
|
525 |
|
|
526 |
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *)
|
|
527 |
apply (induct set: Finites)
|
|
528 |
apply (simp_all add: Pow_insert)
|
|
529 |
apply (subst card_Un_disjoint)
|
|
530 |
apply blast
|
|
531 |
apply (blast intro: finite_imageI)
|
|
532 |
apply blast
|
|
533 |
apply (subgoal_tac "inj_on (insert x) (Pow F)")
|
|
534 |
apply (simp add: card_image Pow_insert)
|
|
535 |
apply (unfold inj_on_def)
|
|
536 |
apply (blast elim!: equalityE)
|
|
537 |
done
|
|
538 |
|
|
539 |
text {*
|
|
540 |
\medskip Relates to equivalence classes. Based on a theorem of
|
|
541 |
F. Kammüller's. The @{prop "finite C"} premise is redundant.
|
|
542 |
*}
|
|
543 |
|
|
544 |
lemma dvd_partition:
|
|
545 |
"finite C ==> finite (Union C) ==>
|
|
546 |
ALL c : C. k dvd card c ==>
|
|
547 |
(ALL c1: C. ALL c2: C. c1 ~= c2 --> c1 Int c2 = {}) ==>
|
|
548 |
k dvd card (Union C)"
|
|
549 |
apply (induct set: Finites)
|
|
550 |
apply simp_all
|
|
551 |
apply clarify
|
|
552 |
apply (subst card_Un_disjoint)
|
|
553 |
apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
|
|
554 |
done
|
|
555 |
|
|
556 |
|
|
557 |
subsection {* A fold functional for finite sets *}
|
|
558 |
|
|
559 |
text {*
|
|
560 |
For @{text n} non-negative we have @{text "fold f e {x1, ..., xn} =
|
|
561 |
f x1 (... (f xn e))"} where @{text f} is at least left-commutative.
|
|
562 |
*}
|
|
563 |
|
|
564 |
consts
|
|
565 |
foldSet :: "('b => 'a => 'a) => 'a => ('b set \<times> 'a) set"
|
|
566 |
|
|
567 |
inductive "foldSet f e"
|
|
568 |
intros
|
|
569 |
emptyI [intro]: "({}, e) : foldSet f e"
|
|
570 |
insertI [intro]: "x \<notin> A ==> (A, y) : foldSet f e ==> (insert x A, f x y) : foldSet f e"
|
|
571 |
|
|
572 |
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f e"
|
|
573 |
|
|
574 |
constdefs
|
|
575 |
fold :: "('b => 'a => 'a) => 'a => 'b set => 'a"
|
|
576 |
"fold f e A == THE x. (A, x) : foldSet f e"
|
|
577 |
|
|
578 |
lemma Diff1_foldSet: "(A - {x}, y) : foldSet f e ==> x: A ==> (A, f x y) : foldSet f e"
|
|
579 |
apply (erule insert_Diff [THEN subst], rule foldSet.intros)
|
|
580 |
apply auto
|
|
581 |
done
|
|
582 |
|
|
583 |
lemma foldSet_imp_finite [simp]: "(A, x) : foldSet f e ==> finite A"
|
|
584 |
by (induct set: foldSet) auto
|
|
585 |
|
|
586 |
lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f e"
|
|
587 |
by (induct set: Finites) auto
|
|
588 |
|
|
589 |
|
|
590 |
subsubsection {* Left-commutative operations *}
|
|
591 |
|
|
592 |
locale LC =
|
|
593 |
fixes f :: "'b => 'a => 'a" (infixl "\<cdot>" 70)
|
|
594 |
assumes left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
|
|
595 |
|
|
596 |
lemma (in LC) foldSet_determ_aux:
|
|
597 |
"ALL A x. card A < n --> (A, x) : foldSet f e -->
|
|
598 |
(ALL y. (A, y) : foldSet f e --> y = x)"
|
|
599 |
apply (induct n)
|
|
600 |
apply (auto simp add: less_Suc_eq)
|
|
601 |
apply (erule foldSet.cases)
|
|
602 |
apply blast
|
|
603 |
apply (erule foldSet.cases)
|
|
604 |
apply blast
|
|
605 |
apply clarify
|
|
606 |
txt {* force simplification of @{text "card A < card (insert ...)"}. *}
|
|
607 |
apply (erule rev_mp)
|
|
608 |
apply (simp add: less_Suc_eq_le)
|
|
609 |
apply (rule impI)
|
|
610 |
apply (rename_tac Aa xa ya Ab xb yb, case_tac "xa = xb")
|
|
611 |
apply (subgoal_tac "Aa = Ab")
|
|
612 |
prefer 2 apply (blast elim!: equalityE)
|
|
613 |
apply blast
|
|
614 |
txt {* case @{prop "xa \<notin> xb"}. *}
|
|
615 |
apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb : Aa & xa : Ab")
|
|
616 |
prefer 2 apply (blast elim!: equalityE)
|
|
617 |
apply clarify
|
|
618 |
apply (subgoal_tac "Aa = insert xb Ab - {xa}")
|
|
619 |
prefer 2 apply blast
|
|
620 |
apply (subgoal_tac "card Aa <= card Ab")
|
|
621 |
prefer 2
|
|
622 |
apply (rule Suc_le_mono [THEN subst])
|
|
623 |
apply (simp add: card_Suc_Diff1)
|
|
624 |
apply (rule_tac A1 = "Aa - {xb}" and f1 = f in finite_imp_foldSet [THEN exE])
|
|
625 |
apply (blast intro: foldSet_imp_finite finite_Diff)
|
|
626 |
apply (frule (1) Diff1_foldSet)
|
|
627 |
apply (subgoal_tac "ya = f xb x")
|
|
628 |
prefer 2 apply (blast del: equalityCE)
|
|
629 |
apply (subgoal_tac "(Ab - {xa}, x) : foldSet f e")
|
|
630 |
prefer 2 apply simp
|
|
631 |
apply (subgoal_tac "yb = f xa x")
|
|
632 |
prefer 2 apply (blast del: equalityCE dest: Diff1_foldSet)
|
|
633 |
apply (simp (no_asm_simp) add: left_commute)
|
|
634 |
done
|
|
635 |
|
|
636 |
lemma (in LC) foldSet_determ: "(A, x) : foldSet f e ==> (A, y) : foldSet f e ==> y = x"
|
|
637 |
by (blast intro: foldSet_determ_aux [rule_format])
|
|
638 |
|
|
639 |
lemma (in LC) fold_equality: "(A, y) : foldSet f e ==> fold f e A = y"
|
|
640 |
by (unfold fold_def) (blast intro: foldSet_determ)
|
|
641 |
|
|
642 |
lemma fold_empty [simp]: "fold f e {} = e"
|
|
643 |
by (unfold fold_def) blast
|
|
644 |
|
|
645 |
lemma (in LC) fold_insert_aux: "x \<notin> A ==>
|
|
646 |
((insert x A, v) : foldSet f e) =
|
|
647 |
(EX y. (A, y) : foldSet f e & v = f x y)"
|
|
648 |
apply auto
|
|
649 |
apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
|
|
650 |
apply (fastsimp dest: foldSet_imp_finite)
|
|
651 |
apply (blast intro: foldSet_determ)
|
|
652 |
done
|
|
653 |
|
|
654 |
lemma (in LC) fold_insert:
|
|
655 |
"finite A ==> x \<notin> A ==> fold f e (insert x A) = f x (fold f e A)"
|
|
656 |
apply (unfold fold_def)
|
|
657 |
apply (simp add: fold_insert_aux)
|
|
658 |
apply (rule the_equality)
|
|
659 |
apply (auto intro: finite_imp_foldSet
|
|
660 |
cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
|
|
661 |
done
|
|
662 |
|
|
663 |
lemma (in LC) fold_commute: "finite A ==> (!!e. f x (fold f e A) = fold f (f x e) A)"
|
|
664 |
apply (induct set: Finites)
|
|
665 |
apply simp
|
|
666 |
apply (simp add: left_commute fold_insert)
|
|
667 |
done
|
|
668 |
|
|
669 |
lemma (in LC) fold_nest_Un_Int:
|
|
670 |
"finite A ==> finite B
|
|
671 |
==> fold f (fold f e B) A = fold f (fold f e (A Int B)) (A Un B)"
|
|
672 |
apply (induct set: Finites)
|
|
673 |
apply simp
|
|
674 |
apply (simp add: fold_insert fold_commute Int_insert_left insert_absorb)
|
|
675 |
done
|
|
676 |
|
|
677 |
lemma (in LC) fold_nest_Un_disjoint:
|
|
678 |
"finite A ==> finite B ==> A Int B = {}
|
|
679 |
==> fold f e (A Un B) = fold f (fold f e B) A"
|
|
680 |
by (simp add: fold_nest_Un_Int)
|
|
681 |
|
|
682 |
declare foldSet_imp_finite [simp del]
|
|
683 |
empty_foldSetE [rule del] foldSet.intros [rule del]
|
|
684 |
-- {* Delete rules to do with @{text foldSet} relation. *}
|
|
685 |
|
|
686 |
|
|
687 |
|
|
688 |
subsubsection {* Commutative monoids *}
|
|
689 |
|
|
690 |
text {*
|
|
691 |
We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"}
|
|
692 |
instead of @{text "'b => 'a => 'a"}.
|
|
693 |
*}
|
|
694 |
|
|
695 |
locale ACe =
|
|
696 |
fixes f :: "'a => 'a => 'a" (infixl "\<cdot>" 70)
|
|
697 |
and e :: 'a
|
|
698 |
assumes ident [simp]: "x \<cdot> e = x"
|
|
699 |
and commute: "x \<cdot> y = y \<cdot> x"
|
|
700 |
and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
|
|
701 |
|
|
702 |
lemma (in ACe) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
|
|
703 |
proof -
|
|
704 |
have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
|
|
705 |
also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
|
|
706 |
also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
|
|
707 |
finally show ?thesis .
|
|
708 |
qed
|
|
709 |
|
|
710 |
lemma (in ACe)
|
|
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)"
|
|
712 |
by (rule assoc, rule commute, rule left_commute) (* FIXME localize "lemmas" (!??) *)
|
|
713 |
|
|
714 |
lemma (in ACe [simp]) left_ident: "e \<cdot> x = x"
|
|
715 |
proof -
|
|
716 |
have "x \<cdot> e = x" by (rule ident)
|
|
717 |
thus ?thesis by (subst commute)
|
|
718 |
qed
|
|
719 |
|
|
720 |
lemma (in ACe) fold_Un_Int:
|
|
721 |
"finite A ==> finite B ==>
|
|
722 |
fold f e A \<cdot> fold f e B = fold f e (A Un B) \<cdot> fold f e (A Int B)"
|
|
723 |
apply (induct set: Finites)
|
|
724 |
apply simp
|
|
725 |
apply (simp add: AC fold_insert insert_absorb Int_insert_left)
|
|
726 |
done
|
|
727 |
|
|
728 |
lemma (in ACe) fold_Un_disjoint:
|
|
729 |
"finite A ==> finite B ==> A Int B = {} ==>
|
|
730 |
fold f e (A Un B) = fold f e A \<cdot> fold f e B"
|
|
731 |
by (simp add: fold_Un_Int)
|
|
732 |
|
|
733 |
lemma (in ACe) fold_Un_disjoint2:
|
|
734 |
"finite A ==> finite B ==> A Int B = {} ==>
|
|
735 |
fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B"
|
|
736 |
proof -
|
|
737 |
assume b: "finite B"
|
|
738 |
assume "finite A"
|
|
739 |
thus "A Int B = {} ==>
|
|
740 |
fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B"
|
|
741 |
proof induct
|
|
742 |
case empty
|
|
743 |
thus ?case by simp
|
|
744 |
next
|
|
745 |
case (insert F x)
|
|
746 |
have "fold (f \<circ> g) e (insert x F \<union> B) = fold (f \<circ> g) e (insert x (F \<union> B))"
|
|
747 |
by simp
|
|
748 |
also have "... = (f \<circ> g) x (fold (f \<circ> g) e (F \<union> B))"
|
|
749 |
by (rule fold_insert) (insert b insert, auto simp add: left_commute) (* FIXME import of fold_insert (!?) *)
|
|
750 |
also from insert have "fold (f \<circ> g) e (F \<union> B) =
|
|
751 |
fold (f \<circ> g) e F \<cdot> fold (f \<circ> g) e B" by blast
|
|
752 |
also have "(f \<circ> g) x ... = (f \<circ> g) x (fold (f \<circ> g) e F) \<cdot> fold (f \<circ> g) e B"
|
|
753 |
by (simp add: AC)
|
|
754 |
also have "(f \<circ> g) x (fold (f \<circ> g) e F) = fold (f \<circ> g) e (insert x F)"
|
|
755 |
by (rule fold_insert [symmetric]) (insert b insert, auto simp add: left_commute)
|
|
756 |
finally show ?case .
|
|
757 |
qed
|
|
758 |
qed
|
|
759 |
|
|
760 |
|
|
761 |
subsection {* Generalized summation over a set *}
|
|
762 |
|
|
763 |
constdefs
|
|
764 |
setsum :: "('a => 'b) => 'a set => 'b::plus_ac0"
|
|
765 |
"setsum f A == if finite A then fold (op + o f) 0 A else 0"
|
|
766 |
|
|
767 |
syntax
|
|
768 |
"_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0" ("\<Sum>_:_. _" [0, 51, 10] 10)
|
|
769 |
syntax (xsymbols)
|
|
770 |
"_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0" ("\<Sum>_\<in>_. _" [0, 51, 10] 10)
|
|
771 |
translations
|
|
772 |
"\<Sum>i:A. b" == "setsum (%i. b) A" -- {* Beware of argument permutation! *}
|
|
773 |
|
|
774 |
|
|
775 |
lemma setsum_empty [simp]: "setsum f {} = 0"
|
|
776 |
by (simp add: setsum_def)
|
|
777 |
|
|
778 |
lemma setsum_insert [simp]:
|
|
779 |
"finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
|
|
780 |
by (simp add: setsum_def fold_insert plus_ac0_left_commute)
|
|
781 |
|
|
782 |
lemma setsum_0: "setsum (\<lambda>i. 0) A = 0"
|
|
783 |
apply (case_tac "finite A")
|
|
784 |
prefer 2 apply (simp add: setsum_def)
|
|
785 |
apply (erule finite_induct)
|
|
786 |
apply auto
|
|
787 |
done
|
|
788 |
|
|
789 |
lemma setsum_eq_0_iff [simp]:
|
|
790 |
"finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
|
|
791 |
by (induct set: Finites) auto
|
|
792 |
|
|
793 |
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
|
|
794 |
apply (case_tac "finite A")
|
|
795 |
prefer 2 apply (simp add: setsum_def)
|
|
796 |
apply (erule rev_mp)
|
|
797 |
apply (erule finite_induct)
|
|
798 |
apply auto
|
|
799 |
done
|
|
800 |
|
|
801 |
lemma card_eq_setsum: "finite A ==> card A = setsum (\<lambda>x. 1) A"
|
|
802 |
-- {* Could allow many @{text "card"} proofs to be simplified. *}
|
|
803 |
by (induct set: Finites) auto
|
|
804 |
|
|
805 |
lemma setsum_Un_Int: "finite A ==> finite B
|
|
806 |
==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
|
|
807 |
-- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
|
|
808 |
apply (induct set: Finites)
|
|
809 |
apply simp
|
|
810 |
apply (simp add: plus_ac0 Int_insert_left insert_absorb)
|
|
811 |
done
|
|
812 |
|
|
813 |
lemma setsum_Un_disjoint: "finite A ==> finite B
|
|
814 |
==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
|
|
815 |
apply (subst setsum_Un_Int [symmetric])
|
|
816 |
apply auto
|
|
817 |
done
|
|
818 |
|
|
819 |
lemma setsum_UN_disjoint: (fixes f :: "'a => 'b::plus_ac0")
|
|
820 |
"finite I ==> (ALL i:I. finite (A i)) ==>
|
|
821 |
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
|
|
822 |
setsum f (UNION I A) = setsum (\<lambda>i. setsum f (A i)) I"
|
|
823 |
apply (induct set: Finites)
|
|
824 |
apply simp
|
|
825 |
apply atomize
|
|
826 |
apply (subgoal_tac "ALL i:F. x \<noteq> i")
|
|
827 |
prefer 2 apply blast
|
|
828 |
apply (subgoal_tac "A x Int UNION F A = {}")
|
|
829 |
prefer 2 apply blast
|
|
830 |
apply (simp add: setsum_Un_disjoint)
|
|
831 |
done
|
|
832 |
|
|
833 |
lemma setsum_addf: "setsum (\<lambda>x. f x + g x) A = (setsum f A + setsum g A)"
|
|
834 |
apply (case_tac "finite A")
|
|
835 |
prefer 2 apply (simp add: setsum_def)
|
|
836 |
apply (erule finite_induct)
|
|
837 |
apply auto
|
|
838 |
apply (simp add: plus_ac0)
|
|
839 |
done
|
|
840 |
|
|
841 |
lemma setsum_Un: "finite A ==> finite B ==>
|
|
842 |
(setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
|
|
843 |
-- {* For the natural numbers, we have subtraction. *}
|
|
844 |
apply (subst setsum_Un_Int [symmetric])
|
|
845 |
apply auto
|
|
846 |
done
|
|
847 |
|
|
848 |
lemma setsum_diff1: "(setsum f (A - {a}) :: nat) =
|
|
849 |
(if a:A then setsum f A - f a else setsum f A)"
|
|
850 |
apply (case_tac "finite A")
|
|
851 |
prefer 2 apply (simp add: setsum_def)
|
|
852 |
apply (erule finite_induct)
|
|
853 |
apply (auto simp add: insert_Diff_if)
|
|
854 |
apply (drule_tac a = a in mk_disjoint_insert)
|
|
855 |
apply auto
|
|
856 |
done
|
|
857 |
|
|
858 |
lemma setsum_cong:
|
|
859 |
"A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
|
|
860 |
apply (case_tac "finite B")
|
|
861 |
prefer 2 apply (simp add: setsum_def)
|
|
862 |
apply simp
|
|
863 |
apply (subgoal_tac "ALL C. C <= B --> (ALL x:C. f x = g x) --> setsum f C = setsum g C")
|
|
864 |
apply simp
|
|
865 |
apply (erule finite_induct)
|
|
866 |
apply simp
|
|
867 |
apply (simp add: subset_insert_iff)
|
|
868 |
apply clarify
|
|
869 |
apply (subgoal_tac "finite C")
|
|
870 |
prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
|
|
871 |
apply (subgoal_tac "C = insert x (C - {x})")
|
|
872 |
prefer 2 apply blast
|
|
873 |
apply (erule ssubst)
|
|
874 |
apply (drule spec)
|
|
875 |
apply (erule (1) notE impE)
|
|
876 |
apply (simp add: Ball_def)
|
|
877 |
done
|
|
878 |
|
|
879 |
|
|
880 |
text {*
|
|
881 |
\medskip Basic theorem about @{text "choose"}. By Florian
|
|
882 |
Kammüller, tidied by LCP.
|
|
883 |
*}
|
|
884 |
|
|
885 |
lemma card_s_0_eq_empty:
|
|
886 |
"finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
|
|
887 |
apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
|
|
888 |
apply (simp cong add: rev_conj_cong)
|
|
889 |
done
|
|
890 |
|
|
891 |
lemma choose_deconstruct: "finite M ==> x \<notin> M
|
|
892 |
==> {s. s <= insert x M & card(s) = Suc k}
|
|
893 |
= {s. s <= M & card(s) = Suc k} Un
|
|
894 |
{s. EX t. t <= M & card(t) = k & s = insert x t}"
|
|
895 |
apply safe
|
|
896 |
apply (auto intro: finite_subset [THEN card_insert_disjoint])
|
|
897 |
apply (drule_tac x = "xa - {x}" in spec)
|
|
898 |
apply (subgoal_tac "x ~: xa")
|
|
899 |
apply auto
|
|
900 |
apply (erule rev_mp, subst card_Diff_singleton)
|
|
901 |
apply (auto intro: finite_subset)
|
|
902 |
done
|
|
903 |
|
|
904 |
lemma card_inj_on_le:
|
|
905 |
"finite A ==> finite B ==> f ` A \<subseteq> B ==> inj_on f A ==> card A <= card B"
|
|
906 |
by (auto intro: card_mono simp add: card_image [symmetric])
|
|
907 |
|
|
908 |
lemma card_bij_eq: "finite A ==> finite B ==>
|
|
909 |
f ` A \<subseteq> B ==> inj_on f A ==> g ` B \<subseteq> A ==> inj_on g B ==> card A = card B"
|
|
910 |
by (auto intro: le_anti_sym card_inj_on_le)
|
|
911 |
|
|
912 |
lemma constr_bij: "finite A ==> x \<notin> A ==>
|
|
913 |
card {B. EX C. C <= A & card(C) = k & B = insert x C} =
|
|
914 |
card {B. B <= A & card(B) = k}"
|
|
915 |
apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
|
|
916 |
apply (rule_tac B = "Pow (insert x A) " in finite_subset)
|
|
917 |
apply (rule_tac [3] B = "Pow (A) " in finite_subset)
|
|
918 |
apply fast+
|
|
919 |
txt {* arity *}
|
|
920 |
apply (auto elim!: equalityE simp add: inj_on_def)
|
|
921 |
apply (subst Diff_insert0)
|
|
922 |
apply auto
|
|
923 |
done
|
|
924 |
|
|
925 |
text {*
|
|
926 |
Main theorem: combinatorial statement about number of subsets of a set.
|
|
927 |
*}
|
|
928 |
|
|
929 |
lemma n_sub_lemma:
|
|
930 |
"!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
|
|
931 |
apply (induct k)
|
|
932 |
apply (simp add: card_s_0_eq_empty)
|
|
933 |
apply atomize
|
|
934 |
apply (rotate_tac -1, erule finite_induct)
|
|
935 |
apply (simp_all (no_asm_simp) cong add: conj_cong add: card_s_0_eq_empty choose_deconstruct)
|
|
936 |
apply (subst card_Un_disjoint)
|
|
937 |
prefer 4 apply (force simp add: constr_bij)
|
|
938 |
prefer 3 apply force
|
|
939 |
prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
|
|
940 |
finite_subset [of _ "Pow (insert x F)", standard])
|
|
941 |
apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
|
|
942 |
done
|
|
943 |
|
|
944 |
theorem n_subsets: "finite A ==> card {B. B <= A & card(B) = k} = (card A choose k)"
|
|
945 |
by (simp add: n_sub_lemma)
|
|
946 |
|
|
947 |
end
|