src/HOL/Finite_Set.thy
author haftmann
Tue Mar 26 21:53:56 2013 +0100 (2013-03-26)
changeset 51546 2e26df807dc7
parent 51489 f738e6dbd844
child 51598 5dbe537087aa
permissions -rw-r--r--
more uniform style for interpretation and sublocale declarations
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(*  Title:      HOL/Finite_Set.thy
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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                with contributions by Jeremy Avigad
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*)
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header {* Finite sets *}
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theory Finite_Set
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imports Option Power
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begin
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subsection {* Predicate for finite sets *}
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inductive finite :: "'a set \<Rightarrow> bool"
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  where
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    emptyI [simp, intro!]: "finite {}"
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  | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
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simproc_setup finite_Collect ("finite (Collect P)") = {* K Set_Comprehension_Pointfree.simproc *}
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lemma finite_induct [case_names empty insert, induct set: finite]:
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  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
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  assumes "finite F"
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  assumes "P {}"
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    and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
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  shows "P F"
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using `finite F`
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proof induct
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  show "P {}" by fact
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  fix x F 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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subsubsection {* Choice principles *}
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lemma ex_new_if_finite: -- "does not depend on def of finite at all"
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  assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
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  shows "\<exists>a::'a. a \<notin> A"
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proof -
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  from assms have "A \<noteq> UNIV" by blast
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  then show ?thesis by blast
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qed
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text {* A finite choice principle. Does not need the SOME choice operator. *}
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lemma finite_set_choice:
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  "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
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proof (induct rule: finite_induct)
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  case empty then show ?case by simp
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next
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  case (insert a A)
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  then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
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  show ?case (is "EX f. ?P f")
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  proof
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    show "?P(%x. if x = a then b else f x)" using f ab by auto
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  qed
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qed
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subsubsection {* Finite sets are the images of initial segments of natural numbers *}
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lemma finite_imp_nat_seg_image_inj_on:
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  assumes "finite A" 
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  shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
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using assms
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proof induct
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  case empty
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  show ?case
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  proof
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    show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp 
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  qed
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next
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  case (insert a A)
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  have notinA: "a \<notin> A" by fact
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  from insert.hyps obtain n f
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    where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
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  hence "insert a A = f(n:=a) ` {i. i < Suc n}"
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        "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
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    by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
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  thus ?case by blast
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qed
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lemma nat_seg_image_imp_finite:
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  "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
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proof (induct n arbitrary: A)
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  case 0 thus ?case by simp
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next
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  case (Suc n)
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  let ?B = "f ` {i. i < n}"
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  have finB: "finite ?B" by(rule Suc.hyps[OF refl])
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  show ?case
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  proof cases
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    assume "\<exists>k<n. f n = f k"
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    hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  next
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    assume "\<not>(\<exists> k<n. f n = f k)"
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    hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  qed
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qed
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lemma finite_conv_nat_seg_image:
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  "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
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  by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
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lemma finite_imp_inj_to_nat_seg:
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  assumes "finite A"
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  shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
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proof -
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  from finite_imp_nat_seg_image_inj_on[OF `finite A`]
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  obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
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    by (auto simp:bij_betw_def)
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  let ?f = "the_inv_into {i. i<n} f"
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  have "inj_on ?f A & ?f ` A = {i. i<n}"
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    by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
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  thus ?thesis by blast
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qed
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lemma finite_Collect_less_nat [iff]:
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  "finite {n::nat. n < k}"
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  by (fastforce simp: finite_conv_nat_seg_image)
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lemma finite_Collect_le_nat [iff]:
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  "finite {n::nat. n \<le> k}"
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  by (simp add: le_eq_less_or_eq Collect_disj_eq)
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subsubsection {* Finiteness and common set operations *}
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lemma rev_finite_subset:
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  "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
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proof (induct arbitrary: A rule: finite_induct)
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  case empty
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  then show ?case by simp
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next
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  case (insert x F A)
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  have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
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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" using x 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" by fact
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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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lemma finite_subset:
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  "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
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  by (rule rev_finite_subset)
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lemma finite_UnI:
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  assumes "finite F" and "finite G"
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  shows "finite (F \<union> G)"
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  using assms by induct simp_all
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lemma finite_Un [iff]:
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  "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
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  by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
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lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
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proof -
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  have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
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  then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
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  then show ?thesis by simp
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qed
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lemma finite_Int [simp, intro]:
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  "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
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  by (blast intro: finite_subset)
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lemma finite_Collect_conjI [simp, intro]:
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  "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
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  by (simp add: Collect_conj_eq)
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lemma finite_Collect_disjI [simp]:
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  "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
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  by (simp add: Collect_disj_eq)
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lemma finite_Diff [simp, intro]:
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  "finite A \<Longrightarrow> finite (A - B)"
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  by (rule finite_subset, rule Diff_subset)
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lemma finite_Diff2 [simp]:
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  assumes "finite B"
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  shows "finite (A - B) \<longleftrightarrow> finite A"
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proof -
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  have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
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  also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
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  finally show ?thesis ..
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qed
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lemma finite_Diff_insert [iff]:
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  "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
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proof -
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  have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
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  moreover have "A - insert a B = A - B - {a}" by auto
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  ultimately show ?thesis by simp
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qed
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lemma finite_compl[simp]:
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  "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
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  by (simp add: Compl_eq_Diff_UNIV)
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lemma finite_Collect_not[simp]:
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  "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
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  by (simp add: Collect_neg_eq)
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lemma finite_Union [simp, intro]:
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  "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
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  by (induct rule: finite_induct) simp_all
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lemma finite_UN_I [intro]:
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  "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
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  by (induct rule: finite_induct) simp_all
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lemma finite_UN [simp]:
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  "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
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  by (blast intro: finite_subset)
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lemma finite_Inter [intro]:
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  "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
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  by (blast intro: Inter_lower finite_subset)
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lemma finite_INT [intro]:
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  "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
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  by (blast intro: INT_lower finite_subset)
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lemma finite_imageI [simp, intro]:
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  "finite F \<Longrightarrow> finite (h ` F)"
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  by (induct rule: finite_induct) simp_all
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lemma finite_image_set [simp]:
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  "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
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  by (simp add: image_Collect [symmetric])
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lemma finite_imageD:
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  assumes "finite (f ` A)" and "inj_on f A"
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  shows "finite A"
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using assms
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proof (induct "f ` A" arbitrary: A)
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  case empty then show ?case by simp
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next
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  case (insert x B)
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  then have B_A: "insert x B = f ` A" by simp
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  then obtain y where "x = f y" and "y \<in> A" by blast
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  from B_A `x \<notin> B` have "B = f ` A - {x}" by blast
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  with B_A `x \<notin> B` `x = f y` `inj_on f A` `y \<in> A` have "B = f ` (A - {y})" by (simp add: inj_on_image_set_diff)
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  moreover from `inj_on f A` have "inj_on f (A - {y})" by (rule inj_on_diff)
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  ultimately have "finite (A - {y})" by (rule insert.hyps)
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  then show "finite A" by simp
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qed
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lemma finite_surj:
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  "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
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  by (erule finite_subset) (rule finite_imageI)
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lemma finite_range_imageI:
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  "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
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  by (drule finite_imageI) (simp add: range_composition)
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lemma finite_subset_image:
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  assumes "finite B"
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  shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
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using assms
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proof induct
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  case empty then show ?case by simp
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next
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  case insert then show ?case
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    by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
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       blast
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qed
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lemma finite_vimage_IntI:
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  "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
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  apply (induct rule: finite_induct)
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   apply simp_all
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  apply (subst vimage_insert)
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  apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
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  done
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lemma finite_vimageI:
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  "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
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  using finite_vimage_IntI[of F h UNIV] by auto
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lemma finite_vimageD:
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  assumes fin: "finite (h -` F)" and surj: "surj h"
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  shows "finite F"
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proof -
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  have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
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  also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
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  finally show "finite F" .
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qed
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lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
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  unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
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lemma finite_Collect_bex [simp]:
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  assumes "finite A"
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  shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
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proof -
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  have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
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  with assms show ?thesis by simp
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qed
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haftmann@41656
   321
lemma finite_Collect_bounded_ex [simp]:
haftmann@41656
   322
  assumes "finite {y. P y}"
haftmann@41656
   323
  shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
haftmann@41656
   324
proof -
haftmann@41656
   325
  have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
haftmann@41656
   326
  with assms show ?thesis by simp
haftmann@41656
   327
qed
nipkow@29920
   328
haftmann@41656
   329
lemma finite_Plus:
haftmann@41656
   330
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
haftmann@41656
   331
  by (simp add: Plus_def)
nipkow@17022
   332
nipkow@31080
   333
lemma finite_PlusD: 
nipkow@31080
   334
  fixes A :: "'a set" and B :: "'b set"
nipkow@31080
   335
  assumes fin: "finite (A <+> B)"
nipkow@31080
   336
  shows "finite A" "finite B"
nipkow@31080
   337
proof -
nipkow@31080
   338
  have "Inl ` A \<subseteq> A <+> B" by auto
haftmann@41656
   339
  then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
haftmann@41656
   340
  then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
nipkow@31080
   341
next
nipkow@31080
   342
  have "Inr ` B \<subseteq> A <+> B" by auto
haftmann@41656
   343
  then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
haftmann@41656
   344
  then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
nipkow@31080
   345
qed
nipkow@31080
   346
haftmann@41656
   347
lemma finite_Plus_iff [simp]:
haftmann@41656
   348
  "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
haftmann@41656
   349
  by (auto intro: finite_PlusD finite_Plus)
nipkow@31080
   350
haftmann@41656
   351
lemma finite_Plus_UNIV_iff [simp]:
haftmann@41656
   352
  "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
haftmann@41656
   353
  by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
wenzelm@12396
   354
nipkow@40786
   355
lemma finite_SigmaI [simp, intro]:
haftmann@41656
   356
  "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
nipkow@40786
   357
  by (unfold Sigma_def) blast
wenzelm@12396
   358
Andreas@51290
   359
lemma finite_SigmaI2:
Andreas@51290
   360
  assumes "finite {x\<in>A. B x \<noteq> {}}"
Andreas@51290
   361
  and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)"
Andreas@51290
   362
  shows "finite (Sigma A B)"
Andreas@51290
   363
proof -
Andreas@51290
   364
  from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)" by auto
Andreas@51290
   365
  also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B" by auto
Andreas@51290
   366
  finally show ?thesis .
Andreas@51290
   367
qed
Andreas@51290
   368
haftmann@41656
   369
lemma finite_cartesian_product:
haftmann@41656
   370
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
nipkow@15402
   371
  by (rule finite_SigmaI)
nipkow@15402
   372
wenzelm@12396
   373
lemma finite_Prod_UNIV:
haftmann@41656
   374
  "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
haftmann@41656
   375
  by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
wenzelm@12396
   376
paulson@15409
   377
lemma finite_cartesian_productD1:
haftmann@42207
   378
  assumes "finite (A \<times> B)" and "B \<noteq> {}"
haftmann@42207
   379
  shows "finite A"
haftmann@42207
   380
proof -
haftmann@42207
   381
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
haftmann@42207
   382
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   383
  then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
haftmann@42207
   384
  with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
haftmann@42207
   385
    by (simp add: image_compose)
haftmann@42207
   386
  then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
haftmann@42207
   387
  then show ?thesis
haftmann@42207
   388
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   389
qed
paulson@15409
   390
paulson@15409
   391
lemma finite_cartesian_productD2:
haftmann@42207
   392
  assumes "finite (A \<times> B)" and "A \<noteq> {}"
haftmann@42207
   393
  shows "finite B"
haftmann@42207
   394
proof -
haftmann@42207
   395
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
haftmann@42207
   396
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   397
  then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
haftmann@42207
   398
  with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
haftmann@42207
   399
    by (simp add: image_compose)
haftmann@42207
   400
  then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
haftmann@42207
   401
  then show ?thesis
haftmann@42207
   402
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   403
qed
paulson@15409
   404
Andreas@48175
   405
lemma finite_prod: 
Andreas@48175
   406
  "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
Andreas@48175
   407
by(auto simp add: UNIV_Times_UNIV[symmetric] simp del: UNIV_Times_UNIV 
Andreas@48175
   408
   dest: finite_cartesian_productD1 finite_cartesian_productD2)
Andreas@48175
   409
haftmann@41656
   410
lemma finite_Pow_iff [iff]:
haftmann@41656
   411
  "finite (Pow A) \<longleftrightarrow> finite A"
wenzelm@12396
   412
proof
wenzelm@12396
   413
  assume "finite (Pow A)"
haftmann@41656
   414
  then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
haftmann@41656
   415
  then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
wenzelm@12396
   416
next
wenzelm@12396
   417
  assume "finite A"
haftmann@41656
   418
  then show "finite (Pow A)"
huffman@35216
   419
    by induct (simp_all add: Pow_insert)
wenzelm@12396
   420
qed
wenzelm@12396
   421
haftmann@41656
   422
corollary finite_Collect_subsets [simp, intro]:
haftmann@41656
   423
  "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
haftmann@41656
   424
  by (simp add: Pow_def [symmetric])
nipkow@29918
   425
Andreas@48175
   426
lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
Andreas@48175
   427
by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
Andreas@48175
   428
nipkow@15392
   429
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
haftmann@41656
   430
  by (blast intro: finite_subset [OF subset_Pow_Union])
nipkow@15392
   431
nipkow@15392
   432
haftmann@41656
   433
subsubsection {* Further induction rules on finite sets *}
haftmann@41656
   434
haftmann@41656
   435
lemma finite_ne_induct [case_names singleton insert, consumes 2]:
haftmann@41656
   436
  assumes "finite F" and "F \<noteq> {}"
haftmann@41656
   437
  assumes "\<And>x. P {x}"
haftmann@41656
   438
    and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
haftmann@41656
   439
  shows "P F"
wenzelm@46898
   440
using assms
wenzelm@46898
   441
proof induct
haftmann@41656
   442
  case empty then show ?case by simp
haftmann@41656
   443
next
haftmann@41656
   444
  case (insert x F) then show ?case by cases auto
haftmann@41656
   445
qed
haftmann@41656
   446
haftmann@41656
   447
lemma finite_subset_induct [consumes 2, case_names empty insert]:
haftmann@41656
   448
  assumes "finite F" and "F \<subseteq> A"
haftmann@41656
   449
  assumes empty: "P {}"
haftmann@41656
   450
    and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
haftmann@41656
   451
  shows "P F"
wenzelm@46898
   452
using `finite F` `F \<subseteq> A`
wenzelm@46898
   453
proof induct
haftmann@41656
   454
  show "P {}" by fact
nipkow@31441
   455
next
haftmann@41656
   456
  fix x F
haftmann@41656
   457
  assume "finite F" and "x \<notin> F" and
haftmann@41656
   458
    P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
haftmann@41656
   459
  show "P (insert x F)"
haftmann@41656
   460
  proof (rule insert)
haftmann@41656
   461
    from i show "x \<in> A" by blast
haftmann@41656
   462
    from i have "F \<subseteq> A" by blast
haftmann@41656
   463
    with P show "P F" .
haftmann@41656
   464
    show "finite F" by fact
haftmann@41656
   465
    show "x \<notin> F" by fact
haftmann@41656
   466
  qed
haftmann@41656
   467
qed
haftmann@41656
   468
haftmann@41656
   469
lemma finite_empty_induct:
haftmann@41656
   470
  assumes "finite A"
haftmann@41656
   471
  assumes "P A"
haftmann@41656
   472
    and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
haftmann@41656
   473
  shows "P {}"
haftmann@41656
   474
proof -
haftmann@41656
   475
  have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
haftmann@41656
   476
  proof -
haftmann@41656
   477
    fix B :: "'a set"
haftmann@41656
   478
    assume "B \<subseteq> A"
haftmann@41656
   479
    with `finite A` have "finite B" by (rule rev_finite_subset)
haftmann@41656
   480
    from this `B \<subseteq> A` show "P (A - B)"
haftmann@41656
   481
    proof induct
haftmann@41656
   482
      case empty
haftmann@41656
   483
      from `P A` show ?case by simp
haftmann@41656
   484
    next
haftmann@41656
   485
      case (insert b B)
haftmann@41656
   486
      have "P (A - B - {b})"
haftmann@41656
   487
      proof (rule remove)
haftmann@41656
   488
        from `finite A` show "finite (A - B)" by induct auto
haftmann@41656
   489
        from insert show "b \<in> A - B" by simp
haftmann@41656
   490
        from insert show "P (A - B)" by simp
haftmann@41656
   491
      qed
haftmann@41656
   492
      also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
haftmann@41656
   493
      finally show ?case .
haftmann@41656
   494
    qed
haftmann@41656
   495
  qed
haftmann@41656
   496
  then have "P (A - A)" by blast
haftmann@41656
   497
  then show ?thesis by simp
nipkow@31441
   498
qed
nipkow@31441
   499
nipkow@31441
   500
haftmann@26441
   501
subsection {* Class @{text finite}  *}
haftmann@26041
   502
haftmann@29797
   503
class finite =
haftmann@26041
   504
  assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
huffman@27430
   505
begin
huffman@27430
   506
huffman@27430
   507
lemma finite [simp]: "finite (A \<Colon> 'a set)"
haftmann@26441
   508
  by (rule subset_UNIV finite_UNIV finite_subset)+
haftmann@26041
   509
haftmann@43866
   510
lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
bulwahn@40922
   511
  by simp
bulwahn@40922
   512
huffman@27430
   513
end
huffman@27430
   514
wenzelm@46898
   515
instance prod :: (finite, finite) finite
wenzelm@46898
   516
  by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
haftmann@26146
   517
haftmann@26041
   518
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
nipkow@39302
   519
  by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
haftmann@26041
   520
haftmann@26146
   521
instance "fun" :: (finite, finite) finite
haftmann@26146
   522
proof
haftmann@26041
   523
  show "finite (UNIV :: ('a => 'b) set)"
haftmann@26041
   524
  proof (rule finite_imageD)
haftmann@26041
   525
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
berghofe@26792
   526
    have "range ?graph \<subseteq> Pow UNIV" by simp
berghofe@26792
   527
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
berghofe@26792
   528
      by (simp only: finite_Pow_iff finite)
berghofe@26792
   529
    ultimately show "finite (range ?graph)"
berghofe@26792
   530
      by (rule finite_subset)
haftmann@26041
   531
    show "inj ?graph" by (rule inj_graph)
haftmann@26041
   532
  qed
haftmann@26041
   533
qed
haftmann@26041
   534
wenzelm@46898
   535
instance bool :: finite
wenzelm@46898
   536
  by default (simp add: UNIV_bool)
haftmann@44831
   537
haftmann@45962
   538
instance set :: (finite) finite
haftmann@45962
   539
  by default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
haftmann@45962
   540
wenzelm@46898
   541
instance unit :: finite
wenzelm@46898
   542
  by default (simp add: UNIV_unit)
haftmann@44831
   543
wenzelm@46898
   544
instance sum :: (finite, finite) finite
wenzelm@46898
   545
  by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
haftmann@27981
   546
haftmann@44831
   547
lemma finite_option_UNIV [simp]:
haftmann@44831
   548
  "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
haftmann@44831
   549
  by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
haftmann@44831
   550
wenzelm@46898
   551
instance option :: (finite) finite
wenzelm@46898
   552
  by default (simp add: UNIV_option_conv)
haftmann@44831
   553
haftmann@26041
   554
haftmann@35817
   555
subsection {* A basic fold functional for finite sets *}
nipkow@15392
   556
nipkow@15392
   557
text {* The intended behaviour is
wenzelm@31916
   558
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
nipkow@28853
   559
if @{text f} is ``left-commutative'':
nipkow@15392
   560
*}
nipkow@15392
   561
haftmann@42871
   562
locale comp_fun_commute =
nipkow@28853
   563
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@42871
   564
  assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
nipkow@28853
   565
begin
nipkow@28853
   566
haftmann@51489
   567
lemma fun_left_comm: "f y (f x z) = f x (f y z)"
haftmann@42871
   568
  using comp_fun_commute by (simp add: fun_eq_iff)
nipkow@28853
   569
haftmann@51489
   570
lemma commute_left_comp:
haftmann@51489
   571
  "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
haftmann@51489
   572
  by (simp add: o_assoc comp_fun_commute)
haftmann@51489
   573
nipkow@28853
   574
end
nipkow@28853
   575
nipkow@28853
   576
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
nipkow@28853
   577
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
nipkow@28853
   578
  emptyI [intro]: "fold_graph f z {} z" |
nipkow@28853
   579
  insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
nipkow@28853
   580
      \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
nipkow@28853
   581
nipkow@28853
   582
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
nipkow@28853
   583
nipkow@28853
   584
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
haftmann@51489
   585
  "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"
nipkow@15392
   586
paulson@15498
   587
text{*A tempting alternative for the definiens is
nipkow@28853
   588
@{term "if finite A then THE y. fold_graph f z A y else e"}.
paulson@15498
   589
It allows the removal of finiteness assumptions from the theorems
nipkow@28853
   590
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
nipkow@28853
   591
The proofs become ugly. It is not worth the effort. (???) *}
nipkow@28853
   592
nipkow@28853
   593
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
haftmann@41656
   594
by (induct rule: finite_induct) auto
nipkow@28853
   595
nipkow@28853
   596
nipkow@28853
   597
subsubsection{*From @{const fold_graph} to @{term fold}*}
nipkow@15392
   598
haftmann@42871
   599
context comp_fun_commute
haftmann@26041
   600
begin
haftmann@26041
   601
haftmann@51489
   602
lemma fold_graph_finite:
haftmann@51489
   603
  assumes "fold_graph f z A y"
haftmann@51489
   604
  shows "finite A"
haftmann@51489
   605
  using assms by induct simp_all
haftmann@51489
   606
huffman@36045
   607
lemma fold_graph_insertE_aux:
huffman@36045
   608
  "fold_graph f z A y \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'"
huffman@36045
   609
proof (induct set: fold_graph)
huffman@36045
   610
  case (insertI x A y) show ?case
huffman@36045
   611
  proof (cases "x = a")
huffman@36045
   612
    assume "x = a" with insertI show ?case by auto
nipkow@28853
   613
  next
huffman@36045
   614
    assume "x \<noteq> a"
huffman@36045
   615
    then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
huffman@36045
   616
      using insertI by auto
haftmann@42875
   617
    have "f x y = f a (f x y')"
huffman@36045
   618
      unfolding y by (rule fun_left_comm)
haftmann@42875
   619
    moreover have "fold_graph f z (insert x A - {a}) (f x y')"
huffman@36045
   620
      using y' and `x \<noteq> a` and `x \<notin> A`
huffman@36045
   621
      by (simp add: insert_Diff_if fold_graph.insertI)
haftmann@42875
   622
    ultimately show ?case by fast
nipkow@15392
   623
  qed
huffman@36045
   624
qed simp
huffman@36045
   625
huffman@36045
   626
lemma fold_graph_insertE:
huffman@36045
   627
  assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
huffman@36045
   628
  obtains y where "v = f x y" and "fold_graph f z A y"
huffman@36045
   629
using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
nipkow@28853
   630
nipkow@28853
   631
lemma fold_graph_determ:
nipkow@28853
   632
  "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
huffman@36045
   633
proof (induct arbitrary: y set: fold_graph)
huffman@36045
   634
  case (insertI x A y v)
huffman@36045
   635
  from `fold_graph f z (insert x A) v` and `x \<notin> A`
huffman@36045
   636
  obtain y' where "v = f x y'" and "fold_graph f z A y'"
huffman@36045
   637
    by (rule fold_graph_insertE)
huffman@36045
   638
  from `fold_graph f z A y'` have "y' = y" by (rule insertI)
huffman@36045
   639
  with `v = f x y'` show "v = f x y" by simp
huffman@36045
   640
qed fast
nipkow@15392
   641
nipkow@28853
   642
lemma fold_equality:
nipkow@28853
   643
  "fold_graph f z A y \<Longrightarrow> fold f z A = y"
haftmann@51489
   644
  by (cases "finite A") (auto simp add: fold_def intro: fold_graph_determ dest: fold_graph_finite)
nipkow@15392
   645
haftmann@42272
   646
lemma fold_graph_fold:
haftmann@42272
   647
  assumes "finite A"
haftmann@42272
   648
  shows "fold_graph f z A (fold f z A)"
haftmann@42272
   649
proof -
haftmann@42272
   650
  from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
haftmann@42272
   651
  moreover note fold_graph_determ
haftmann@42272
   652
  ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
haftmann@42272
   653
  then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
haftmann@51489
   654
  with assms show ?thesis by (simp add: fold_def)
haftmann@42272
   655
qed
huffman@36045
   656
haftmann@51489
   657
text {* The base case for @{text fold}: *}
nipkow@15392
   658
haftmann@51489
   659
lemma (in -) fold_infinite [simp]:
haftmann@51489
   660
  assumes "\<not> finite A"
haftmann@51489
   661
  shows "fold f z A = z"
haftmann@51489
   662
  using assms by (auto simp add: fold_def)
haftmann@51489
   663
haftmann@51489
   664
lemma (in -) fold_empty [simp]:
haftmann@51489
   665
  "fold f z {} = z"
haftmann@51489
   666
  by (auto simp add: fold_def)
nipkow@28853
   667
nipkow@28853
   668
text{* The various recursion equations for @{const fold}: *}
nipkow@28853
   669
haftmann@26041
   670
lemma fold_insert [simp]:
haftmann@42875
   671
  assumes "finite A" and "x \<notin> A"
haftmann@42875
   672
  shows "fold f z (insert x A) = f x (fold f z A)"
haftmann@42875
   673
proof (rule fold_equality)
haftmann@51489
   674
  fix z
haftmann@42875
   675
  from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
haftmann@51489
   676
  with `x \<notin> A` have "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
haftmann@51489
   677
  then show "fold_graph f z (insert x A) (f x (fold f z A))" by simp
haftmann@42875
   678
qed
nipkow@28853
   679
haftmann@51489
   680
declare (in -) empty_fold_graphE [rule del] fold_graph.intros [rule del]
haftmann@51489
   681
  -- {* No more proofs involve these. *}
haftmann@51489
   682
haftmann@51489
   683
lemma fold_fun_left_comm:
nipkow@28853
   684
  "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
nipkow@28853
   685
proof (induct rule: finite_induct)
nipkow@28853
   686
  case empty then show ?case by simp
nipkow@28853
   687
next
nipkow@28853
   688
  case (insert y A) then show ?case
haftmann@51489
   689
    by (simp add: fun_left_comm [of x])
nipkow@28853
   690
qed
nipkow@28853
   691
nipkow@28853
   692
lemma fold_insert2:
haftmann@51489
   693
  "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A)  = fold f (f x z) A"
haftmann@51489
   694
  by (simp add: fold_fun_left_comm)
nipkow@15392
   695
haftmann@26041
   696
lemma fold_rec:
haftmann@42875
   697
  assumes "finite A" and "x \<in> A"
haftmann@42875
   698
  shows "fold f z A = f x (fold f z (A - {x}))"
nipkow@28853
   699
proof -
nipkow@28853
   700
  have A: "A = insert x (A - {x})" using `x \<in> A` by blast
nipkow@28853
   701
  then have "fold f z A = fold f z (insert x (A - {x}))" by simp
nipkow@28853
   702
  also have "\<dots> = f x (fold f z (A - {x}))"
nipkow@28853
   703
    by (rule fold_insert) (simp add: `finite A`)+
nipkow@15535
   704
  finally show ?thesis .
nipkow@15535
   705
qed
nipkow@15535
   706
nipkow@28853
   707
lemma fold_insert_remove:
nipkow@28853
   708
  assumes "finite A"
nipkow@28853
   709
  shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
nipkow@28853
   710
proof -
nipkow@28853
   711
  from `finite A` have "finite (insert x A)" by auto
nipkow@28853
   712
  moreover have "x \<in> insert x A" by auto
nipkow@28853
   713
  ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
nipkow@28853
   714
    by (rule fold_rec)
nipkow@28853
   715
  then show ?thesis by simp
nipkow@28853
   716
qed
nipkow@28853
   717
kuncar@48619
   718
text{* Other properties of @{const fold}: *}
kuncar@48619
   719
kuncar@48619
   720
lemma fold_image:
kuncar@48619
   721
  assumes "finite A" and "inj_on g A"
haftmann@51489
   722
  shows "fold f z (g ` A) = fold (f \<circ> g) z A"
kuncar@48619
   723
using assms
kuncar@48619
   724
proof induction
kuncar@48619
   725
  case (insert a F)
kuncar@48619
   726
    interpret comp_fun_commute "\<lambda>x. f (g x)" by default (simp add: comp_fun_commute)
kuncar@48619
   727
    from insert show ?case by auto
haftmann@51489
   728
qed simp
kuncar@48619
   729
haftmann@26041
   730
end
nipkow@15392
   731
haftmann@49724
   732
lemma fold_cong:
haftmann@49724
   733
  assumes "comp_fun_commute f" "comp_fun_commute g"
haftmann@49724
   734
  assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
haftmann@51489
   735
    and "s = t" and "A = B"
haftmann@51489
   736
  shows "fold f s A = fold g t B"
haftmann@49724
   737
proof -
haftmann@51489
   738
  have "fold f s A = fold g s A"  
haftmann@49724
   739
  using `finite A` cong proof (induct A)
haftmann@49724
   740
    case empty then show ?case by simp
haftmann@49724
   741
  next
haftmann@49724
   742
    case (insert x A)
haftmann@49724
   743
    interpret f: comp_fun_commute f by (fact `comp_fun_commute f`)
haftmann@49724
   744
    interpret g: comp_fun_commute g by (fact `comp_fun_commute g`)
haftmann@49724
   745
    from insert show ?case by simp
haftmann@49724
   746
  qed
haftmann@49724
   747
  with assms show ?thesis by simp
haftmann@49724
   748
qed
haftmann@49724
   749
haftmann@49724
   750
haftmann@51489
   751
text {* A simplified version for idempotent functions: *}
nipkow@15480
   752
haftmann@42871
   753
locale comp_fun_idem = comp_fun_commute +
haftmann@51489
   754
  assumes comp_fun_idem: "f x \<circ> f x = f x"
haftmann@26041
   755
begin
haftmann@26041
   756
haftmann@42869
   757
lemma fun_left_idem: "f x (f x z) = f x z"
haftmann@42871
   758
  using comp_fun_idem by (simp add: fun_eq_iff)
nipkow@28853
   759
haftmann@26041
   760
lemma fold_insert_idem:
nipkow@28853
   761
  assumes fin: "finite A"
haftmann@51489
   762
  shows "fold f z (insert x A)  = f x (fold f z A)"
nipkow@15480
   763
proof cases
nipkow@28853
   764
  assume "x \<in> A"
nipkow@28853
   765
  then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
haftmann@51489
   766
  then show ?thesis using assms by (simp add: comp_fun_idem fun_left_idem)
nipkow@15480
   767
next
nipkow@28853
   768
  assume "x \<notin> A" then show ?thesis using assms by simp
nipkow@15480
   769
qed
nipkow@15480
   770
haftmann@51489
   771
declare fold_insert [simp del] fold_insert_idem [simp]
nipkow@28853
   772
nipkow@28853
   773
lemma fold_insert_idem2:
nipkow@28853
   774
  "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
haftmann@51489
   775
  by (simp add: fold_fun_left_comm)
nipkow@15484
   776
haftmann@26041
   777
end
haftmann@26041
   778
haftmann@35817
   779
haftmann@49723
   780
subsubsection {* Liftings to @{text comp_fun_commute} etc. *}
haftmann@35817
   781
haftmann@42871
   782
lemma (in comp_fun_commute) comp_comp_fun_commute:
haftmann@42871
   783
  "comp_fun_commute (f \<circ> g)"
haftmann@35817
   784
proof
haftmann@42871
   785
qed (simp_all add: comp_fun_commute)
haftmann@35817
   786
haftmann@42871
   787
lemma (in comp_fun_idem) comp_comp_fun_idem:
haftmann@42871
   788
  "comp_fun_idem (f \<circ> g)"
haftmann@42871
   789
  by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
haftmann@42871
   790
    (simp_all add: comp_fun_idem)
haftmann@35817
   791
haftmann@49723
   792
lemma (in comp_fun_commute) comp_fun_commute_funpow:
haftmann@49723
   793
  "comp_fun_commute (\<lambda>x. f x ^^ g x)"
haftmann@49723
   794
proof
haftmann@49723
   795
  fix y x
haftmann@49723
   796
  show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
haftmann@49723
   797
  proof (cases "x = y")
haftmann@49723
   798
    case True then show ?thesis by simp
haftmann@49723
   799
  next
haftmann@49723
   800
    case False show ?thesis
haftmann@49723
   801
    proof (induct "g x" arbitrary: g)
haftmann@49723
   802
      case 0 then show ?case by simp
haftmann@49723
   803
    next
haftmann@49723
   804
      case (Suc n g)
haftmann@49723
   805
      have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
haftmann@49723
   806
      proof (induct "g y" arbitrary: g)
haftmann@49723
   807
        case 0 then show ?case by simp
haftmann@49723
   808
      next
haftmann@49723
   809
        case (Suc n g)
haftmann@49723
   810
        def h \<equiv> "\<lambda>z. g z - 1"
haftmann@49723
   811
        with Suc have "n = h y" by simp
haftmann@49723
   812
        with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
haftmann@49723
   813
          by auto
haftmann@49723
   814
        from Suc h_def have "g y = Suc (h y)" by simp
haftmann@49739
   815
        then show ?case by (simp add: comp_assoc hyp)
haftmann@49723
   816
          (simp add: o_assoc comp_fun_commute)
haftmann@49723
   817
      qed
haftmann@49723
   818
      def h \<equiv> "\<lambda>z. if z = x then g x - 1 else g z"
haftmann@49723
   819
      with Suc have "n = h x" by simp
haftmann@49723
   820
      with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
haftmann@49723
   821
        by auto
haftmann@49723
   822
      with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y" by simp
haftmann@49723
   823
      from Suc h_def have "g x = Suc (h x)" by simp
haftmann@49723
   824
      then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
haftmann@49739
   825
        (simp add: comp_assoc hyp1)
haftmann@49723
   826
    qed
haftmann@49723
   827
  qed
haftmann@49723
   828
qed
haftmann@49723
   829
haftmann@49723
   830
haftmann@49723
   831
subsubsection {* Expressing set operations via @{const fold} *}
haftmann@49723
   832
haftmann@51489
   833
lemma comp_fun_commute_const:
haftmann@51489
   834
  "comp_fun_commute (\<lambda>_. f)"
haftmann@51489
   835
proof
haftmann@51489
   836
qed rule
haftmann@51489
   837
haftmann@42871
   838
lemma comp_fun_idem_insert:
haftmann@42871
   839
  "comp_fun_idem insert"
haftmann@35817
   840
proof
haftmann@35817
   841
qed auto
haftmann@35817
   842
haftmann@42871
   843
lemma comp_fun_idem_remove:
haftmann@46146
   844
  "comp_fun_idem Set.remove"
haftmann@35817
   845
proof
haftmann@35817
   846
qed auto
nipkow@31992
   847
haftmann@42871
   848
lemma (in semilattice_inf) comp_fun_idem_inf:
haftmann@42871
   849
  "comp_fun_idem inf"
haftmann@35817
   850
proof
haftmann@35817
   851
qed (auto simp add: inf_left_commute)
haftmann@35817
   852
haftmann@42871
   853
lemma (in semilattice_sup) comp_fun_idem_sup:
haftmann@42871
   854
  "comp_fun_idem sup"
haftmann@35817
   855
proof
haftmann@35817
   856
qed (auto simp add: sup_left_commute)
nipkow@31992
   857
haftmann@35817
   858
lemma union_fold_insert:
haftmann@35817
   859
  assumes "finite A"
haftmann@35817
   860
  shows "A \<union> B = fold insert B A"
haftmann@35817
   861
proof -
haftmann@42871
   862
  interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
haftmann@35817
   863
  from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
haftmann@35817
   864
qed
nipkow@31992
   865
haftmann@35817
   866
lemma minus_fold_remove:
haftmann@35817
   867
  assumes "finite A"
haftmann@46146
   868
  shows "B - A = fold Set.remove B A"
haftmann@35817
   869
proof -
haftmann@46146
   870
  interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
haftmann@46146
   871
  from `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
haftmann@46146
   872
  then show ?thesis ..
haftmann@35817
   873
qed
haftmann@35817
   874
haftmann@51489
   875
lemma comp_fun_commute_filter_fold:
haftmann@51489
   876
  "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
kuncar@48619
   877
proof - 
kuncar@48619
   878
  interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
kuncar@48619
   879
  show ?thesis by default (auto simp: fun_eq_iff)
kuncar@48619
   880
qed
kuncar@48619
   881
kuncar@49758
   882
lemma Set_filter_fold:
kuncar@48619
   883
  assumes "finite A"
kuncar@49758
   884
  shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
kuncar@48619
   885
using assms
kuncar@48619
   886
by (induct A) 
kuncar@49758
   887
  (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
kuncar@49758
   888
kuncar@49758
   889
lemma inter_Set_filter:     
kuncar@49758
   890
  assumes "finite B"
kuncar@49758
   891
  shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
kuncar@49758
   892
using assms 
kuncar@49758
   893
by (induct B) (auto simp: Set.filter_def)
kuncar@48619
   894
kuncar@48619
   895
lemma image_fold_insert:
kuncar@48619
   896
  assumes "finite A"
kuncar@48619
   897
  shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
kuncar@48619
   898
using assms
kuncar@48619
   899
proof -
kuncar@48619
   900
  interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by default auto
kuncar@48619
   901
  show ?thesis using assms by (induct A) auto
kuncar@48619
   902
qed
kuncar@48619
   903
kuncar@48619
   904
lemma Ball_fold:
kuncar@48619
   905
  assumes "finite A"
kuncar@48619
   906
  shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
kuncar@48619
   907
using assms
kuncar@48619
   908
proof -
kuncar@48619
   909
  interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by default auto
kuncar@48619
   910
  show ?thesis using assms by (induct A) auto
kuncar@48619
   911
qed
kuncar@48619
   912
kuncar@48619
   913
lemma Bex_fold:
kuncar@48619
   914
  assumes "finite A"
kuncar@48619
   915
  shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
kuncar@48619
   916
using assms
kuncar@48619
   917
proof -
kuncar@48619
   918
  interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by default auto
kuncar@48619
   919
  show ?thesis using assms by (induct A) auto
kuncar@48619
   920
qed
kuncar@48619
   921
kuncar@48619
   922
lemma comp_fun_commute_Pow_fold: 
kuncar@48619
   923
  "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)" 
kuncar@48619
   924
  by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
kuncar@48619
   925
kuncar@48619
   926
lemma Pow_fold:
kuncar@48619
   927
  assumes "finite A"
kuncar@48619
   928
  shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
kuncar@48619
   929
using assms
kuncar@48619
   930
proof -
kuncar@48619
   931
  interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
kuncar@48619
   932
  show ?thesis using assms by (induct A) (auto simp: Pow_insert)
kuncar@48619
   933
qed
kuncar@48619
   934
kuncar@48619
   935
lemma fold_union_pair:
kuncar@48619
   936
  assumes "finite B"
kuncar@48619
   937
  shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
kuncar@48619
   938
proof -
kuncar@48619
   939
  interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by default auto
kuncar@48619
   940
  show ?thesis using assms  by (induct B arbitrary: A) simp_all
kuncar@48619
   941
qed
kuncar@48619
   942
kuncar@48619
   943
lemma comp_fun_commute_product_fold: 
kuncar@48619
   944
  assumes "finite B"
haftmann@51489
   945
  shows "comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)" 
kuncar@48619
   946
by default (auto simp: fold_union_pair[symmetric] assms)
kuncar@48619
   947
kuncar@48619
   948
lemma product_fold:
kuncar@48619
   949
  assumes "finite A"
kuncar@48619
   950
  assumes "finite B"
haftmann@51489
   951
  shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A"
kuncar@48619
   952
using assms unfolding Sigma_def 
kuncar@48619
   953
by (induct A) 
kuncar@48619
   954
  (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
kuncar@48619
   955
kuncar@48619
   956
haftmann@35817
   957
context complete_lattice
nipkow@31992
   958
begin
nipkow@31992
   959
haftmann@35817
   960
lemma inf_Inf_fold_inf:
haftmann@35817
   961
  assumes "finite A"
haftmann@51489
   962
  shows "inf (Inf A) B = fold inf B A"
haftmann@35817
   963
proof -
haftmann@42871
   964
  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
haftmann@51489
   965
  from `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
haftmann@51489
   966
    (simp_all add: inf_commute fun_eq_iff)
haftmann@35817
   967
qed
nipkow@31992
   968
haftmann@35817
   969
lemma sup_Sup_fold_sup:
haftmann@35817
   970
  assumes "finite A"
haftmann@51489
   971
  shows "sup (Sup A) B = fold sup B A"
haftmann@35817
   972
proof -
haftmann@42871
   973
  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
haftmann@51489
   974
  from `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
haftmann@51489
   975
    (simp_all add: sup_commute fun_eq_iff)
nipkow@31992
   976
qed
nipkow@31992
   977
haftmann@35817
   978
lemma Inf_fold_inf:
haftmann@35817
   979
  assumes "finite A"
haftmann@35817
   980
  shows "Inf A = fold inf top A"
haftmann@35817
   981
  using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
haftmann@35817
   982
haftmann@35817
   983
lemma Sup_fold_sup:
haftmann@35817
   984
  assumes "finite A"
haftmann@35817
   985
  shows "Sup A = fold sup bot A"
haftmann@35817
   986
  using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
nipkow@31992
   987
haftmann@46146
   988
lemma inf_INF_fold_inf:
haftmann@35817
   989
  assumes "finite A"
haftmann@42873
   990
  shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold") 
haftmann@35817
   991
proof (rule sym)
haftmann@42871
   992
  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
haftmann@42871
   993
  interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
haftmann@42873
   994
  from `finite A` show "?fold = ?inf"
haftmann@42869
   995
    by (induct A arbitrary: B)
hoelzl@44928
   996
      (simp_all add: INF_def inf_left_commute)
haftmann@35817
   997
qed
nipkow@31992
   998
haftmann@46146
   999
lemma sup_SUP_fold_sup:
haftmann@35817
  1000
  assumes "finite A"
haftmann@42873
  1001
  shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold") 
haftmann@35817
  1002
proof (rule sym)
haftmann@42871
  1003
  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
haftmann@42871
  1004
  interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
haftmann@42873
  1005
  from `finite A` show "?fold = ?sup"
haftmann@42869
  1006
    by (induct A arbitrary: B)
hoelzl@44928
  1007
      (simp_all add: SUP_def sup_left_commute)
haftmann@35817
  1008
qed
nipkow@31992
  1009
haftmann@46146
  1010
lemma INF_fold_inf:
haftmann@35817
  1011
  assumes "finite A"
haftmann@42873
  1012
  shows "INFI A f = fold (inf \<circ> f) top A"
haftmann@46146
  1013
  using assms inf_INF_fold_inf [of A top] by simp
nipkow@31992
  1014
haftmann@46146
  1015
lemma SUP_fold_sup:
haftmann@35817
  1016
  assumes "finite A"
haftmann@42873
  1017
  shows "SUPR A f = fold (sup \<circ> f) bot A"
haftmann@46146
  1018
  using assms sup_SUP_fold_sup [of A bot] by simp
nipkow@31992
  1019
nipkow@31992
  1020
end
nipkow@31992
  1021
nipkow@31992
  1022
haftmann@35817
  1023
subsection {* Locales as mini-packages for fold operations *}
haftmann@34007
  1024
haftmann@35817
  1025
subsubsection {* The natural case *}
haftmann@35719
  1026
haftmann@35719
  1027
locale folding =
haftmann@35719
  1028
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@51489
  1029
  fixes z :: "'b"
haftmann@42871
  1030
  assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
haftmann@35719
  1031
begin
haftmann@35719
  1032
haftmann@51489
  1033
definition F :: "'a set \<Rightarrow> 'b"
haftmann@51489
  1034
where
haftmann@51489
  1035
  eq_fold: "F A = fold f z A"
haftmann@51489
  1036
haftmann@35719
  1037
lemma empty [simp]:
haftmann@51489
  1038
  "F {} = z"
haftmann@51489
  1039
  by (simp add: eq_fold)
haftmann@35719
  1040
haftmann@51489
  1041
lemma infinite [simp]:
haftmann@51489
  1042
  "\<not> finite A \<Longrightarrow> F A = z"
haftmann@51489
  1043
  by (simp add: eq_fold)
haftmann@51489
  1044
 
haftmann@35719
  1045
lemma insert [simp]:
haftmann@35719
  1046
  assumes "finite A" and "x \<notin> A"
haftmann@51489
  1047
  shows "F (insert x A) = f x (F A)"
haftmann@35719
  1048
proof -
wenzelm@46898
  1049
  interpret comp_fun_commute f
wenzelm@46898
  1050
    by default (insert comp_fun_commute, simp add: fun_eq_iff)
haftmann@51489
  1051
  from fold_insert assms
haftmann@51489
  1052
  have "fold f z (insert x A) = f x (fold f z A)" by simp
nipkow@39302
  1053
  with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1054
qed
haftmann@51489
  1055
 
haftmann@35719
  1056
lemma remove:
haftmann@35719
  1057
  assumes "finite A" and "x \<in> A"
haftmann@51489
  1058
  shows "F A = f x (F (A - {x}))"
haftmann@35719
  1059
proof -
haftmann@35719
  1060
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35719
  1061
    by (auto dest: mk_disjoint_insert)
haftmann@35719
  1062
  moreover from `finite A` this have "finite B" by simp
haftmann@35719
  1063
  ultimately show ?thesis by simp
haftmann@35719
  1064
qed
haftmann@35719
  1065
haftmann@35719
  1066
lemma insert_remove:
haftmann@35719
  1067
  assumes "finite A"
haftmann@51489
  1068
  shows "F (insert x A) = f x (F (A - {x}))"
haftmann@35722
  1069
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35719
  1070
haftmann@34007
  1071
end
haftmann@35719
  1072
haftmann@35817
  1073
haftmann@51489
  1074
subsubsection {* With idempotency *}
haftmann@35817
  1075
haftmann@35719
  1076
locale folding_idem = folding +
haftmann@51489
  1077
  assumes comp_fun_idem: "f x \<circ> f x = f x"
haftmann@35719
  1078
begin
haftmann@35719
  1079
haftmann@35817
  1080
declare insert [simp del]
haftmann@35719
  1081
haftmann@35719
  1082
lemma insert_idem [simp]:
haftmann@35719
  1083
  assumes "finite A"
haftmann@51489
  1084
  shows "F (insert x A) = f x (F A)"
haftmann@35817
  1085
proof -
haftmann@51489
  1086
  interpret comp_fun_idem f
haftmann@51489
  1087
    by default (insert comp_fun_commute comp_fun_idem, simp add: fun_eq_iff)
haftmann@51489
  1088
  from fold_insert_idem assms
haftmann@51489
  1089
  have "fold f z (insert x A) = f x (fold f z A)" by simp
haftmann@51489
  1090
  with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1091
qed
haftmann@35719
  1092
haftmann@35719
  1093
end
haftmann@35719
  1094
haftmann@35817
  1095
haftmann@35722
  1096
subsection {* Finite cardinality *}
haftmann@35722
  1097
haftmann@51489
  1098
text {*
haftmann@51489
  1099
  The traditional definition
haftmann@51489
  1100
  @{prop "card A \<equiv> LEAST n. EX f. A = {f i | i. i < n}"}
haftmann@51489
  1101
  is ugly to work with.
haftmann@51489
  1102
  But now that we have @{const fold} things are easy:
haftmann@35722
  1103
*}
haftmann@35722
  1104
haftmann@35722
  1105
definition card :: "'a set \<Rightarrow> nat" where
haftmann@51489
  1106
  "card = folding.F (\<lambda>_. Suc) 0"
haftmann@35722
  1107
haftmann@51489
  1108
interpretation card!: folding "\<lambda>_. Suc" 0
haftmann@51489
  1109
where
haftmann@51546
  1110
  "folding.F (\<lambda>_. Suc) 0 = card"
haftmann@51489
  1111
proof -
haftmann@51489
  1112
  show "folding (\<lambda>_. Suc)" by default rule
haftmann@51489
  1113
  then interpret card!: folding "\<lambda>_. Suc" 0 .
haftmann@51546
  1114
  from card_def show "folding.F (\<lambda>_. Suc) 0 = card" by rule
haftmann@51489
  1115
qed
haftmann@35722
  1116
haftmann@51489
  1117
lemma card_infinite:
haftmann@35722
  1118
  "\<not> finite A \<Longrightarrow> card A = 0"
haftmann@51489
  1119
  by (fact card.infinite)
haftmann@35722
  1120
haftmann@35722
  1121
lemma card_empty:
haftmann@35722
  1122
  "card {} = 0"
haftmann@35722
  1123
  by (fact card.empty)
haftmann@35722
  1124
haftmann@35722
  1125
lemma card_insert_disjoint:
haftmann@51489
  1126
  "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
haftmann@51489
  1127
  by (fact card.insert)
haftmann@35722
  1128
haftmann@35722
  1129
lemma card_insert_if:
haftmann@51489
  1130
  "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
haftmann@35722
  1131
  by auto (simp add: card.insert_remove card.remove)
haftmann@35722
  1132
haftmann@35722
  1133
lemma card_ge_0_finite:
haftmann@35722
  1134
  "card A > 0 \<Longrightarrow> finite A"
haftmann@35722
  1135
  by (rule ccontr) simp
haftmann@35722
  1136
blanchet@35828
  1137
lemma card_0_eq [simp, no_atp]:
haftmann@35722
  1138
  "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
haftmann@35722
  1139
  by (auto dest: mk_disjoint_insert)
haftmann@35722
  1140
haftmann@35722
  1141
lemma finite_UNIV_card_ge_0:
haftmann@35722
  1142
  "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
haftmann@35722
  1143
  by (rule ccontr) simp
haftmann@35722
  1144
haftmann@35722
  1145
lemma card_eq_0_iff:
haftmann@35722
  1146
  "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
haftmann@35722
  1147
  by auto
haftmann@35722
  1148
haftmann@35722
  1149
lemma card_gt_0_iff:
haftmann@35722
  1150
  "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
haftmann@35722
  1151
  by (simp add: neq0_conv [symmetric] card_eq_0_iff) 
haftmann@35722
  1152
haftmann@51489
  1153
lemma card_Suc_Diff1:
haftmann@51489
  1154
  "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
haftmann@35722
  1155
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
haftmann@35722
  1156
apply(simp del:insert_Diff_single)
haftmann@35722
  1157
done
haftmann@35722
  1158
haftmann@35722
  1159
lemma card_Diff_singleton:
haftmann@51489
  1160
  "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"
haftmann@51489
  1161
  by (simp add: card_Suc_Diff1 [symmetric])
haftmann@35722
  1162
haftmann@35722
  1163
lemma card_Diff_singleton_if:
haftmann@51489
  1164
  "finite A \<Longrightarrow> card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
haftmann@51489
  1165
  by (simp add: card_Diff_singleton)
haftmann@35722
  1166
haftmann@35722
  1167
lemma card_Diff_insert[simp]:
haftmann@51489
  1168
  assumes "finite A" and "a \<in> A" and "a \<notin> B"
haftmann@51489
  1169
  shows "card (A - insert a B) = card (A - B) - 1"
haftmann@35722
  1170
proof -
haftmann@35722
  1171
  have "A - insert a B = (A - B) - {a}" using assms by blast
haftmann@51489
  1172
  then show ?thesis using assms by(simp add: card_Diff_singleton)
haftmann@35722
  1173
qed
haftmann@35722
  1174
haftmann@35722
  1175
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
haftmann@51489
  1176
  by (fact card.insert_remove)
haftmann@35722
  1177
haftmann@35722
  1178
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
haftmann@35722
  1179
by (simp add: card_insert_if)
haftmann@35722
  1180
nipkow@41987
  1181
lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
nipkow@41987
  1182
by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
nipkow@41987
  1183
nipkow@41988
  1184
lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
nipkow@41987
  1185
using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
nipkow@41987
  1186
haftmann@35722
  1187
lemma card_mono:
haftmann@35722
  1188
  assumes "finite B" and "A \<subseteq> B"
haftmann@35722
  1189
  shows "card A \<le> card B"
haftmann@35722
  1190
proof -
haftmann@35722
  1191
  from assms have "finite A" by (auto intro: finite_subset)
haftmann@35722
  1192
  then show ?thesis using assms proof (induct A arbitrary: B)
haftmann@35722
  1193
    case empty then show ?case by simp
haftmann@35722
  1194
  next
haftmann@35722
  1195
    case (insert x A)
haftmann@35722
  1196
    then have "x \<in> B" by simp
haftmann@35722
  1197
    from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
haftmann@35722
  1198
    with insert.hyps have "card A \<le> card (B - {x})" by auto
haftmann@35722
  1199
    with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
haftmann@35722
  1200
  qed
haftmann@35722
  1201
qed
haftmann@35722
  1202
haftmann@35722
  1203
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
haftmann@41656
  1204
apply (induct rule: finite_induct)
haftmann@41656
  1205
apply simp
haftmann@41656
  1206
apply clarify
haftmann@35722
  1207
apply (subgoal_tac "finite A & A - {x} <= F")
haftmann@35722
  1208
 prefer 2 apply (blast intro: finite_subset, atomize)
haftmann@35722
  1209
apply (drule_tac x = "A - {x}" in spec)
haftmann@35722
  1210
apply (simp add: card_Diff_singleton_if split add: split_if_asm)
haftmann@35722
  1211
apply (case_tac "card A", auto)
haftmann@35722
  1212
done
haftmann@35722
  1213
haftmann@35722
  1214
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
haftmann@35722
  1215
apply (simp add: psubset_eq linorder_not_le [symmetric])
haftmann@35722
  1216
apply (blast dest: card_seteq)
haftmann@35722
  1217
done
haftmann@35722
  1218
haftmann@51489
  1219
lemma card_Un_Int:
haftmann@51489
  1220
  assumes "finite A" and "finite B"
haftmann@51489
  1221
  shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
haftmann@51489
  1222
using assms proof (induct A)
haftmann@51489
  1223
  case empty then show ?case by simp
haftmann@51489
  1224
next
haftmann@51489
  1225
 case (insert x A) then show ?case
haftmann@51489
  1226
    by (auto simp add: insert_absorb Int_insert_left)
haftmann@51489
  1227
qed
haftmann@35722
  1228
haftmann@51489
  1229
lemma card_Un_disjoint:
haftmann@51489
  1230
  assumes "finite A" and "finite B"
haftmann@51489
  1231
  assumes "A \<inter> B = {}"
haftmann@51489
  1232
  shows "card (A \<union> B) = card A + card B"
haftmann@51489
  1233
using assms card_Un_Int [of A B] by simp
haftmann@35722
  1234
haftmann@35722
  1235
lemma card_Diff_subset:
haftmann@35722
  1236
  assumes "finite B" and "B \<subseteq> A"
haftmann@35722
  1237
  shows "card (A - B) = card A - card B"
haftmann@35722
  1238
proof (cases "finite A")
haftmann@35722
  1239
  case False with assms show ?thesis by simp
haftmann@35722
  1240
next
haftmann@35722
  1241
  case True with assms show ?thesis by (induct B arbitrary: A) simp_all
haftmann@35722
  1242
qed
haftmann@35722
  1243
haftmann@35722
  1244
lemma card_Diff_subset_Int:
haftmann@35722
  1245
  assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
haftmann@35722
  1246
proof -
haftmann@35722
  1247
  have "A - B = A - A \<inter> B" by auto
haftmann@35722
  1248
  thus ?thesis
haftmann@35722
  1249
    by (simp add: card_Diff_subset AB) 
haftmann@35722
  1250
qed
haftmann@35722
  1251
nipkow@40716
  1252
lemma diff_card_le_card_Diff:
nipkow@40716
  1253
assumes "finite B" shows "card A - card B \<le> card(A - B)"
nipkow@40716
  1254
proof-
nipkow@40716
  1255
  have "card A - card B \<le> card A - card (A \<inter> B)"
nipkow@40716
  1256
    using card_mono[OF assms Int_lower2, of A] by arith
nipkow@40716
  1257
  also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
nipkow@40716
  1258
  finally show ?thesis .
nipkow@40716
  1259
qed
nipkow@40716
  1260
haftmann@35722
  1261
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
haftmann@35722
  1262
apply (rule Suc_less_SucD)
haftmann@35722
  1263
apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  1264
done
haftmann@35722
  1265
haftmann@35722
  1266
lemma card_Diff2_less:
haftmann@35722
  1267
  "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
haftmann@35722
  1268
apply (case_tac "x = y")
haftmann@35722
  1269
 apply (simp add: card_Diff1_less del:card_Diff_insert)
haftmann@35722
  1270
apply (rule less_trans)
haftmann@35722
  1271
 prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
haftmann@35722
  1272
done
haftmann@35722
  1273
haftmann@35722
  1274
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
haftmann@35722
  1275
apply (case_tac "x : A")
haftmann@35722
  1276
 apply (simp_all add: card_Diff1_less less_imp_le)
haftmann@35722
  1277
done
haftmann@35722
  1278
haftmann@35722
  1279
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
haftmann@35722
  1280
by (erule psubsetI, blast)
haftmann@35722
  1281
haftmann@35722
  1282
lemma insert_partition:
haftmann@35722
  1283
  "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
haftmann@35722
  1284
  \<Longrightarrow> x \<inter> \<Union> F = {}"
haftmann@35722
  1285
by auto
haftmann@35722
  1286
haftmann@35722
  1287
lemma finite_psubset_induct[consumes 1, case_names psubset]:
urbanc@36079
  1288
  assumes fin: "finite A" 
urbanc@36079
  1289
  and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A" 
urbanc@36079
  1290
  shows "P A"
urbanc@36079
  1291
using fin
urbanc@36079
  1292
proof (induct A taking: card rule: measure_induct_rule)
haftmann@35722
  1293
  case (less A)
urbanc@36079
  1294
  have fin: "finite A" by fact
urbanc@36079
  1295
  have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
urbanc@36079
  1296
  { fix B 
urbanc@36079
  1297
    assume asm: "B \<subset> A"
urbanc@36079
  1298
    from asm have "card B < card A" using psubset_card_mono fin by blast
urbanc@36079
  1299
    moreover
urbanc@36079
  1300
    from asm have "B \<subseteq> A" by auto
urbanc@36079
  1301
    then have "finite B" using fin finite_subset by blast
urbanc@36079
  1302
    ultimately 
urbanc@36079
  1303
    have "P B" using ih by simp
urbanc@36079
  1304
  }
urbanc@36079
  1305
  with fin show "P A" using major by blast
haftmann@35722
  1306
qed
haftmann@35722
  1307
haftmann@35722
  1308
text{* main cardinality theorem *}
haftmann@35722
  1309
lemma card_partition [rule_format]:
haftmann@35722
  1310
  "finite C ==>
haftmann@35722
  1311
     finite (\<Union> C) -->
haftmann@35722
  1312
     (\<forall>c\<in>C. card c = k) -->
haftmann@35722
  1313
     (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
haftmann@35722
  1314
     k * card(C) = card (\<Union> C)"
haftmann@35722
  1315
apply (erule finite_induct, simp)
haftmann@35722
  1316
apply (simp add: card_Un_disjoint insert_partition 
haftmann@35722
  1317
       finite_subset [of _ "\<Union> (insert x F)"])
haftmann@35722
  1318
done
haftmann@35722
  1319
haftmann@35722
  1320
lemma card_eq_UNIV_imp_eq_UNIV:
haftmann@35722
  1321
  assumes fin: "finite (UNIV :: 'a set)"
haftmann@35722
  1322
  and card: "card A = card (UNIV :: 'a set)"
haftmann@35722
  1323
  shows "A = (UNIV :: 'a set)"
haftmann@35722
  1324
proof
haftmann@35722
  1325
  show "A \<subseteq> UNIV" by simp
haftmann@35722
  1326
  show "UNIV \<subseteq> A"
haftmann@35722
  1327
  proof
haftmann@35722
  1328
    fix x
haftmann@35722
  1329
    show "x \<in> A"
haftmann@35722
  1330
    proof (rule ccontr)
haftmann@35722
  1331
      assume "x \<notin> A"
haftmann@35722
  1332
      then have "A \<subset> UNIV" by auto
haftmann@35722
  1333
      with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
haftmann@35722
  1334
      with card show False by simp
haftmann@35722
  1335
    qed
haftmann@35722
  1336
  qed
haftmann@35722
  1337
qed
haftmann@35722
  1338
haftmann@35722
  1339
text{*The form of a finite set of given cardinality*}
haftmann@35722
  1340
haftmann@35722
  1341
lemma card_eq_SucD:
haftmann@35722
  1342
assumes "card A = Suc k"
haftmann@35722
  1343
shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
haftmann@35722
  1344
proof -
haftmann@35722
  1345
  have fin: "finite A" using assms by (auto intro: ccontr)
haftmann@35722
  1346
  moreover have "card A \<noteq> 0" using assms by auto
haftmann@35722
  1347
  ultimately obtain b where b: "b \<in> A" by auto
haftmann@35722
  1348
  show ?thesis
haftmann@35722
  1349
  proof (intro exI conjI)
haftmann@35722
  1350
    show "A = insert b (A-{b})" using b by blast
haftmann@35722
  1351
    show "b \<notin> A - {b}" by blast
haftmann@35722
  1352
    show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
nipkow@44890
  1353
      using assms b fin by(fastforce dest:mk_disjoint_insert)+
haftmann@35722
  1354
  qed
haftmann@35722
  1355
qed
haftmann@35722
  1356
haftmann@35722
  1357
lemma card_Suc_eq:
haftmann@35722
  1358
  "(card A = Suc k) =
haftmann@35722
  1359
   (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
haftmann@35722
  1360
apply(rule iffI)
haftmann@35722
  1361
 apply(erule card_eq_SucD)
haftmann@35722
  1362
apply(auto)
haftmann@51489
  1363
apply(subst card.insert)
haftmann@35722
  1364
 apply(auto intro:ccontr)
haftmann@35722
  1365
done
haftmann@35722
  1366
nipkow@44744
  1367
lemma card_le_Suc_iff: "finite A \<Longrightarrow>
nipkow@44744
  1368
  Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
nipkow@44890
  1369
by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
nipkow@44744
  1370
  dest: subset_singletonD split: nat.splits if_splits)
nipkow@44744
  1371
haftmann@35722
  1372
lemma finite_fun_UNIVD2:
haftmann@35722
  1373
  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
haftmann@35722
  1374
  shows "finite (UNIV :: 'b set)"
haftmann@35722
  1375
proof -
haftmann@46146
  1376
  from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
haftmann@46146
  1377
    by (rule finite_imageI)
haftmann@46146
  1378
  moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
haftmann@46146
  1379
    by (rule UNIV_eq_I) auto
haftmann@35722
  1380
  ultimately show "finite (UNIV :: 'b set)" by simp
haftmann@35722
  1381
qed
haftmann@35722
  1382
huffman@48063
  1383
lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
haftmann@35722
  1384
  unfolding UNIV_unit by simp
haftmann@35722
  1385
huffman@47210
  1386
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
huffman@47210
  1387
  unfolding UNIV_bool by simp
huffman@47210
  1388
haftmann@35722
  1389
haftmann@35722
  1390
subsubsection {* Cardinality of image *}
haftmann@35722
  1391
haftmann@35722
  1392
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
haftmann@41656
  1393
apply (induct rule: finite_induct)
haftmann@35722
  1394
 apply simp
haftmann@35722
  1395
apply (simp add: le_SucI card_insert_if)
haftmann@35722
  1396
done
haftmann@35722
  1397
haftmann@35722
  1398
lemma card_image:
haftmann@35722
  1399
  assumes "inj_on f A"
haftmann@35722
  1400
  shows "card (f ` A) = card A"
haftmann@35722
  1401
proof (cases "finite A")
haftmann@35722
  1402
  case True then show ?thesis using assms by (induct A) simp_all
haftmann@35722
  1403
next
haftmann@35722
  1404
  case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
haftmann@35722
  1405
  with False show ?thesis by simp
haftmann@35722
  1406
qed
haftmann@35722
  1407
haftmann@35722
  1408
lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
haftmann@35722
  1409
by(auto simp: card_image bij_betw_def)
haftmann@35722
  1410
haftmann@35722
  1411
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
haftmann@35722
  1412
by (simp add: card_seteq card_image)
haftmann@35722
  1413
haftmann@35722
  1414
lemma eq_card_imp_inj_on:
haftmann@35722
  1415
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
haftmann@35722
  1416
apply (induct rule:finite_induct)
haftmann@35722
  1417
apply simp
haftmann@35722
  1418
apply(frule card_image_le[where f = f])
haftmann@35722
  1419
apply(simp add:card_insert_if split:if_splits)
haftmann@35722
  1420
done
haftmann@35722
  1421
haftmann@35722
  1422
lemma inj_on_iff_eq_card:
haftmann@35722
  1423
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
haftmann@35722
  1424
by(blast intro: card_image eq_card_imp_inj_on)
haftmann@35722
  1425
haftmann@35722
  1426
haftmann@35722
  1427
lemma card_inj_on_le:
haftmann@35722
  1428
  "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
haftmann@35722
  1429
apply (subgoal_tac "finite A") 
haftmann@35722
  1430
 apply (force intro: card_mono simp add: card_image [symmetric])
haftmann@35722
  1431
apply (blast intro: finite_imageD dest: finite_subset) 
haftmann@35722
  1432
done
haftmann@35722
  1433
haftmann@35722
  1434
lemma card_bij_eq:
haftmann@35722
  1435
  "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
haftmann@35722
  1436
     finite A; finite B |] ==> card A = card B"
haftmann@35722
  1437
by (auto intro: le_antisym card_inj_on_le)
haftmann@35722
  1438
hoelzl@40703
  1439
lemma bij_betw_finite:
hoelzl@40703
  1440
  assumes "bij_betw f A B"
hoelzl@40703
  1441
  shows "finite A \<longleftrightarrow> finite B"
hoelzl@40703
  1442
using assms unfolding bij_betw_def
hoelzl@40703
  1443
using finite_imageD[of f A] by auto
haftmann@35722
  1444
haftmann@41656
  1445
nipkow@37466
  1446
subsubsection {* Pigeonhole Principles *}
nipkow@37466
  1447
nipkow@40311
  1448
lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
nipkow@37466
  1449
by (auto dest: card_image less_irrefl_nat)
nipkow@37466
  1450
nipkow@37466
  1451
lemma pigeonhole_infinite:
nipkow@37466
  1452
assumes  "~ finite A" and "finite(f`A)"
nipkow@37466
  1453
shows "EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  1454
proof -
nipkow@37466
  1455
  have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  1456
  proof(induct "f`A" arbitrary: A rule: finite_induct)
nipkow@37466
  1457
    case empty thus ?case by simp
nipkow@37466
  1458
  next
nipkow@37466
  1459
    case (insert b F)
nipkow@37466
  1460
    show ?case
nipkow@37466
  1461
    proof cases
nipkow@37466
  1462
      assume "finite{a:A. f a = b}"
nipkow@37466
  1463
      hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
nipkow@37466
  1464
      also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
nipkow@37466
  1465
      finally have "~ finite({a:A. f a \<noteq> b})" .
nipkow@37466
  1466
      from insert(3)[OF _ this]
nipkow@37466
  1467
      show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
nipkow@37466
  1468
    next
nipkow@37466
  1469
      assume 1: "~finite{a:A. f a = b}"
nipkow@37466
  1470
      hence "{a \<in> A. f a = b} \<noteq> {}" by force
nipkow@37466
  1471
      thus ?thesis using 1 by blast
nipkow@37466
  1472
    qed
nipkow@37466
  1473
  qed
nipkow@37466
  1474
  from this[OF assms(2,1)] show ?thesis .
nipkow@37466
  1475
qed
nipkow@37466
  1476
nipkow@37466
  1477
lemma pigeonhole_infinite_rel:
nipkow@37466
  1478
assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
nipkow@37466
  1479
shows "EX b:B. ~finite{a:A. R a b}"
nipkow@37466
  1480
proof -
nipkow@37466
  1481
   let ?F = "%a. {b:B. R a b}"
nipkow@37466
  1482
   from finite_Pow_iff[THEN iffD2, OF `finite B`]
nipkow@37466
  1483
   have "finite(?F ` A)" by(blast intro: rev_finite_subset)
nipkow@37466
  1484
   from pigeonhole_infinite[where f = ?F, OF assms(1) this]
nipkow@37466
  1485
   obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
nipkow@37466
  1486
   obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
nipkow@37466
  1487
   { assume "finite{a:A. R a b0}"
nipkow@37466
  1488
     then have "finite {a\<in>A. ?F a = ?F a0}"
nipkow@37466
  1489
       using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
nipkow@37466
  1490
   }
nipkow@37466
  1491
   with 1 `b0 : B` show ?thesis by blast
nipkow@37466
  1492
qed
nipkow@37466
  1493
nipkow@37466
  1494
haftmann@35722
  1495
subsubsection {* Cardinality of sums *}
haftmann@35722
  1496
haftmann@35722
  1497
lemma card_Plus:
haftmann@35722
  1498
  assumes "finite A" and "finite B"
haftmann@35722
  1499
  shows "card (A <+> B) = card A + card B"
haftmann@35722
  1500
proof -
haftmann@35722
  1501
  have "Inl`A \<inter> Inr`B = {}" by fast
haftmann@35722
  1502
  with assms show ?thesis
haftmann@35722
  1503
    unfolding Plus_def
haftmann@35722
  1504
    by (simp add: card_Un_disjoint card_image)
haftmann@35722
  1505
qed
haftmann@35722
  1506
haftmann@35722
  1507
lemma card_Plus_conv_if:
haftmann@35722
  1508
  "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
haftmann@35722
  1509
  by (auto simp add: card_Plus)
haftmann@35722
  1510
haftmann@35722
  1511
haftmann@35722
  1512
subsubsection {* Cardinality of the Powerset *}
haftmann@35722
  1513
huffman@47221
  1514
lemma card_Pow: "finite A ==> card (Pow A) = 2 ^ card A"
haftmann@41656
  1515
apply (induct rule: finite_induct)
haftmann@35722
  1516
 apply (simp_all add: Pow_insert)
haftmann@35722
  1517
apply (subst card_Un_disjoint, blast)
nipkow@40786
  1518
  apply (blast, blast)
haftmann@35722
  1519
apply (subgoal_tac "inj_on (insert x) (Pow F)")
huffman@47221
  1520
 apply (subst mult_2)
haftmann@35722
  1521
 apply (simp add: card_image Pow_insert)
haftmann@35722
  1522
apply (unfold inj_on_def)
haftmann@35722
  1523
apply (blast elim!: equalityE)
haftmann@35722
  1524
done
haftmann@35722
  1525
nipkow@41987
  1526
text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
haftmann@35722
  1527
haftmann@35722
  1528
lemma dvd_partition:
haftmann@35722
  1529
  "finite (Union C) ==>
haftmann@35722
  1530
    ALL c : C. k dvd card c ==>
haftmann@35722
  1531
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
haftmann@35722
  1532
  k dvd card (Union C)"
haftmann@41656
  1533
apply (frule finite_UnionD)
haftmann@41656
  1534
apply (rotate_tac -1)
haftmann@41656
  1535
apply (induct rule: finite_induct)
haftmann@41656
  1536
apply simp_all
haftmann@41656
  1537
apply clarify
haftmann@35722
  1538
apply (subst card_Un_disjoint)
haftmann@35722
  1539
   apply (auto simp add: disjoint_eq_subset_Compl)
haftmann@35722
  1540
done
haftmann@35722
  1541
haftmann@35722
  1542
haftmann@35722
  1543
subsubsection {* Relating injectivity and surjectivity *}
haftmann@35722
  1544
haftmann@41656
  1545
lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
haftmann@35722
  1546
apply(rule eq_card_imp_inj_on, assumption)
haftmann@35722
  1547
apply(frule finite_imageI)
haftmann@35722
  1548
apply(drule (1) card_seteq)
haftmann@35722
  1549
 apply(erule card_image_le)
haftmann@35722
  1550
apply simp
haftmann@35722
  1551
done
haftmann@35722
  1552
haftmann@35722
  1553
lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  1554
shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
hoelzl@40702
  1555
by (blast intro: finite_surj_inj subset_UNIV)
haftmann@35722
  1556
haftmann@35722
  1557
lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  1558
shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
nipkow@44890
  1559
by(fastforce simp:surj_def dest!: endo_inj_surj)
haftmann@35722
  1560
haftmann@51489
  1561
corollary infinite_UNIV_nat [iff]:
haftmann@51489
  1562
  "\<not> finite (UNIV :: nat set)"
haftmann@35722
  1563
proof
haftmann@51489
  1564
  assume "finite (UNIV :: nat set)"
haftmann@51489
  1565
  with finite_UNIV_inj_surj [of Suc]
haftmann@35722
  1566
  show False by simp (blast dest: Suc_neq_Zero surjD)
haftmann@35722
  1567
qed
haftmann@35722
  1568
blanchet@35828
  1569
(* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
haftmann@51489
  1570
lemma infinite_UNIV_char_0 [no_atp]:
haftmann@51489
  1571
  "\<not> finite (UNIV :: 'a::semiring_char_0 set)"
haftmann@35722
  1572
proof
haftmann@51489
  1573
  assume "finite (UNIV :: 'a set)"
haftmann@51489
  1574
  with subset_UNIV have "finite (range of_nat :: 'a set)"
haftmann@35722
  1575
    by (rule finite_subset)
haftmann@51489
  1576
  moreover have "inj (of_nat :: nat \<Rightarrow> 'a)"
haftmann@35722
  1577
    by (simp add: inj_on_def)
haftmann@51489
  1578
  ultimately have "finite (UNIV :: nat set)"
haftmann@35722
  1579
    by (rule finite_imageD)
haftmann@51489
  1580
  then show False
haftmann@35722
  1581
    by simp
haftmann@35722
  1582
qed
haftmann@35722
  1583
kuncar@49758
  1584
hide_const (open) Finite_Set.fold
haftmann@46033
  1585
haftmann@35722
  1586
end
haftmann@49723
  1587