src/HOL/Finite_Set.thy
author hoelzl
Thu Jan 31 11:31:27 2013 +0100 (2013-01-31)
changeset 50999 3de230ed0547
parent 49806 acb6fa98e310
child 51290 c48477e76de5
permissions -rw-r--r--
introduce order topology
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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
haftmann@41656
   359
lemma finite_cartesian_product:
haftmann@41656
   360
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
nipkow@15402
   361
  by (rule finite_SigmaI)
nipkow@15402
   362
wenzelm@12396
   363
lemma finite_Prod_UNIV:
haftmann@41656
   364
  "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
haftmann@41656
   365
  by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
wenzelm@12396
   366
paulson@15409
   367
lemma finite_cartesian_productD1:
haftmann@42207
   368
  assumes "finite (A \<times> B)" and "B \<noteq> {}"
haftmann@42207
   369
  shows "finite A"
haftmann@42207
   370
proof -
haftmann@42207
   371
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
haftmann@42207
   372
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   373
  then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
haftmann@42207
   374
  with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
haftmann@42207
   375
    by (simp add: image_compose)
haftmann@42207
   376
  then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
haftmann@42207
   377
  then show ?thesis
haftmann@42207
   378
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   379
qed
paulson@15409
   380
paulson@15409
   381
lemma finite_cartesian_productD2:
haftmann@42207
   382
  assumes "finite (A \<times> B)" and "A \<noteq> {}"
haftmann@42207
   383
  shows "finite B"
haftmann@42207
   384
proof -
haftmann@42207
   385
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
haftmann@42207
   386
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   387
  then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
haftmann@42207
   388
  with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
haftmann@42207
   389
    by (simp add: image_compose)
haftmann@42207
   390
  then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
haftmann@42207
   391
  then show ?thesis
haftmann@42207
   392
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   393
qed
paulson@15409
   394
Andreas@48175
   395
lemma finite_prod: 
Andreas@48175
   396
  "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
Andreas@48175
   397
by(auto simp add: UNIV_Times_UNIV[symmetric] simp del: UNIV_Times_UNIV 
Andreas@48175
   398
   dest: finite_cartesian_productD1 finite_cartesian_productD2)
Andreas@48175
   399
haftmann@41656
   400
lemma finite_Pow_iff [iff]:
haftmann@41656
   401
  "finite (Pow A) \<longleftrightarrow> finite A"
wenzelm@12396
   402
proof
wenzelm@12396
   403
  assume "finite (Pow A)"
haftmann@41656
   404
  then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
haftmann@41656
   405
  then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
wenzelm@12396
   406
next
wenzelm@12396
   407
  assume "finite A"
haftmann@41656
   408
  then show "finite (Pow A)"
huffman@35216
   409
    by induct (simp_all add: Pow_insert)
wenzelm@12396
   410
qed
wenzelm@12396
   411
haftmann@41656
   412
corollary finite_Collect_subsets [simp, intro]:
haftmann@41656
   413
  "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
haftmann@41656
   414
  by (simp add: Pow_def [symmetric])
nipkow@29918
   415
Andreas@48175
   416
lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
Andreas@48175
   417
by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
Andreas@48175
   418
nipkow@15392
   419
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
haftmann@41656
   420
  by (blast intro: finite_subset [OF subset_Pow_Union])
nipkow@15392
   421
nipkow@15392
   422
haftmann@41656
   423
subsubsection {* Further induction rules on finite sets *}
haftmann@41656
   424
haftmann@41656
   425
lemma finite_ne_induct [case_names singleton insert, consumes 2]:
haftmann@41656
   426
  assumes "finite F" and "F \<noteq> {}"
haftmann@41656
   427
  assumes "\<And>x. P {x}"
haftmann@41656
   428
    and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
haftmann@41656
   429
  shows "P F"
wenzelm@46898
   430
using assms
wenzelm@46898
   431
proof induct
haftmann@41656
   432
  case empty then show ?case by simp
haftmann@41656
   433
next
haftmann@41656
   434
  case (insert x F) then show ?case by cases auto
haftmann@41656
   435
qed
haftmann@41656
   436
haftmann@41656
   437
lemma finite_subset_induct [consumes 2, case_names empty insert]:
haftmann@41656
   438
  assumes "finite F" and "F \<subseteq> A"
haftmann@41656
   439
  assumes empty: "P {}"
haftmann@41656
   440
    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
   441
  shows "P F"
wenzelm@46898
   442
using `finite F` `F \<subseteq> A`
wenzelm@46898
   443
proof induct
haftmann@41656
   444
  show "P {}" by fact
nipkow@31441
   445
next
haftmann@41656
   446
  fix x F
haftmann@41656
   447
  assume "finite F" and "x \<notin> F" and
haftmann@41656
   448
    P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
haftmann@41656
   449
  show "P (insert x F)"
haftmann@41656
   450
  proof (rule insert)
haftmann@41656
   451
    from i show "x \<in> A" by blast
haftmann@41656
   452
    from i have "F \<subseteq> A" by blast
haftmann@41656
   453
    with P show "P F" .
haftmann@41656
   454
    show "finite F" by fact
haftmann@41656
   455
    show "x \<notin> F" by fact
haftmann@41656
   456
  qed
haftmann@41656
   457
qed
haftmann@41656
   458
haftmann@41656
   459
lemma finite_empty_induct:
haftmann@41656
   460
  assumes "finite A"
haftmann@41656
   461
  assumes "P A"
haftmann@41656
   462
    and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
haftmann@41656
   463
  shows "P {}"
haftmann@41656
   464
proof -
haftmann@41656
   465
  have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
haftmann@41656
   466
  proof -
haftmann@41656
   467
    fix B :: "'a set"
haftmann@41656
   468
    assume "B \<subseteq> A"
haftmann@41656
   469
    with `finite A` have "finite B" by (rule rev_finite_subset)
haftmann@41656
   470
    from this `B \<subseteq> A` show "P (A - B)"
haftmann@41656
   471
    proof induct
haftmann@41656
   472
      case empty
haftmann@41656
   473
      from `P A` show ?case by simp
haftmann@41656
   474
    next
haftmann@41656
   475
      case (insert b B)
haftmann@41656
   476
      have "P (A - B - {b})"
haftmann@41656
   477
      proof (rule remove)
haftmann@41656
   478
        from `finite A` show "finite (A - B)" by induct auto
haftmann@41656
   479
        from insert show "b \<in> A - B" by simp
haftmann@41656
   480
        from insert show "P (A - B)" by simp
haftmann@41656
   481
      qed
haftmann@41656
   482
      also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
haftmann@41656
   483
      finally show ?case .
haftmann@41656
   484
    qed
haftmann@41656
   485
  qed
haftmann@41656
   486
  then have "P (A - A)" by blast
haftmann@41656
   487
  then show ?thesis by simp
nipkow@31441
   488
qed
nipkow@31441
   489
nipkow@31441
   490
haftmann@26441
   491
subsection {* Class @{text finite}  *}
haftmann@26041
   492
haftmann@29797
   493
class finite =
haftmann@26041
   494
  assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
huffman@27430
   495
begin
huffman@27430
   496
huffman@27430
   497
lemma finite [simp]: "finite (A \<Colon> 'a set)"
haftmann@26441
   498
  by (rule subset_UNIV finite_UNIV finite_subset)+
haftmann@26041
   499
haftmann@43866
   500
lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
bulwahn@40922
   501
  by simp
bulwahn@40922
   502
huffman@27430
   503
end
huffman@27430
   504
wenzelm@46898
   505
instance prod :: (finite, finite) finite
wenzelm@46898
   506
  by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
haftmann@26146
   507
haftmann@26041
   508
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
nipkow@39302
   509
  by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
haftmann@26041
   510
haftmann@26146
   511
instance "fun" :: (finite, finite) finite
haftmann@26146
   512
proof
haftmann@26041
   513
  show "finite (UNIV :: ('a => 'b) set)"
haftmann@26041
   514
  proof (rule finite_imageD)
haftmann@26041
   515
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
berghofe@26792
   516
    have "range ?graph \<subseteq> Pow UNIV" by simp
berghofe@26792
   517
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
berghofe@26792
   518
      by (simp only: finite_Pow_iff finite)
berghofe@26792
   519
    ultimately show "finite (range ?graph)"
berghofe@26792
   520
      by (rule finite_subset)
haftmann@26041
   521
    show "inj ?graph" by (rule inj_graph)
haftmann@26041
   522
  qed
haftmann@26041
   523
qed
haftmann@26041
   524
wenzelm@46898
   525
instance bool :: finite
wenzelm@46898
   526
  by default (simp add: UNIV_bool)
haftmann@44831
   527
haftmann@45962
   528
instance set :: (finite) finite
haftmann@45962
   529
  by default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
haftmann@45962
   530
wenzelm@46898
   531
instance unit :: finite
wenzelm@46898
   532
  by default (simp add: UNIV_unit)
haftmann@44831
   533
wenzelm@46898
   534
instance sum :: (finite, finite) finite
wenzelm@46898
   535
  by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
haftmann@27981
   536
haftmann@44831
   537
lemma finite_option_UNIV [simp]:
haftmann@44831
   538
  "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
haftmann@44831
   539
  by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
haftmann@44831
   540
wenzelm@46898
   541
instance option :: (finite) finite
wenzelm@46898
   542
  by default (simp add: UNIV_option_conv)
haftmann@44831
   543
haftmann@26041
   544
haftmann@35817
   545
subsection {* A basic fold functional for finite sets *}
nipkow@15392
   546
nipkow@15392
   547
text {* The intended behaviour is
wenzelm@31916
   548
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
nipkow@28853
   549
if @{text f} is ``left-commutative'':
nipkow@15392
   550
*}
nipkow@15392
   551
haftmann@42871
   552
locale comp_fun_commute =
nipkow@28853
   553
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@42871
   554
  assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
nipkow@28853
   555
begin
nipkow@28853
   556
haftmann@42809
   557
lemma fun_left_comm: "f x (f y z) = f y (f x z)"
haftmann@42871
   558
  using comp_fun_commute by (simp add: fun_eq_iff)
nipkow@28853
   559
nipkow@28853
   560
end
nipkow@28853
   561
nipkow@28853
   562
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
nipkow@28853
   563
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
nipkow@28853
   564
  emptyI [intro]: "fold_graph f z {} z" |
nipkow@28853
   565
  insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
nipkow@28853
   566
      \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
nipkow@28853
   567
nipkow@28853
   568
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
nipkow@28853
   569
nipkow@28853
   570
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
haftmann@37767
   571
  "fold f z A = (THE y. fold_graph f z A y)"
nipkow@15392
   572
paulson@15498
   573
text{*A tempting alternative for the definiens is
nipkow@28853
   574
@{term "if finite A then THE y. fold_graph f z A y else e"}.
paulson@15498
   575
It allows the removal of finiteness assumptions from the theorems
nipkow@28853
   576
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
nipkow@28853
   577
The proofs become ugly. It is not worth the effort. (???) *}
nipkow@28853
   578
nipkow@28853
   579
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
haftmann@41656
   580
by (induct rule: finite_induct) auto
nipkow@28853
   581
nipkow@28853
   582
nipkow@28853
   583
subsubsection{*From @{const fold_graph} to @{term fold}*}
nipkow@15392
   584
haftmann@42871
   585
context comp_fun_commute
haftmann@26041
   586
begin
haftmann@26041
   587
huffman@36045
   588
lemma fold_graph_insertE_aux:
huffman@36045
   589
  "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
   590
proof (induct set: fold_graph)
huffman@36045
   591
  case (insertI x A y) show ?case
huffman@36045
   592
  proof (cases "x = a")
huffman@36045
   593
    assume "x = a" with insertI show ?case by auto
nipkow@28853
   594
  next
huffman@36045
   595
    assume "x \<noteq> a"
huffman@36045
   596
    then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
huffman@36045
   597
      using insertI by auto
haftmann@42875
   598
    have "f x y = f a (f x y')"
huffman@36045
   599
      unfolding y by (rule fun_left_comm)
haftmann@42875
   600
    moreover have "fold_graph f z (insert x A - {a}) (f x y')"
huffman@36045
   601
      using y' and `x \<noteq> a` and `x \<notin> A`
huffman@36045
   602
      by (simp add: insert_Diff_if fold_graph.insertI)
haftmann@42875
   603
    ultimately show ?case by fast
nipkow@15392
   604
  qed
huffman@36045
   605
qed simp
huffman@36045
   606
huffman@36045
   607
lemma fold_graph_insertE:
huffman@36045
   608
  assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
huffman@36045
   609
  obtains y where "v = f x y" and "fold_graph f z A y"
huffman@36045
   610
using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
nipkow@28853
   611
nipkow@28853
   612
lemma fold_graph_determ:
nipkow@28853
   613
  "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
huffman@36045
   614
proof (induct arbitrary: y set: fold_graph)
huffman@36045
   615
  case (insertI x A y v)
huffman@36045
   616
  from `fold_graph f z (insert x A) v` and `x \<notin> A`
huffman@36045
   617
  obtain y' where "v = f x y'" and "fold_graph f z A y'"
huffman@36045
   618
    by (rule fold_graph_insertE)
huffman@36045
   619
  from `fold_graph f z A y'` have "y' = y" by (rule insertI)
huffman@36045
   620
  with `v = f x y'` show "v = f x y" by simp
huffman@36045
   621
qed fast
nipkow@15392
   622
nipkow@28853
   623
lemma fold_equality:
nipkow@28853
   624
  "fold_graph f z A y \<Longrightarrow> fold f z A = y"
nipkow@28853
   625
by (unfold fold_def) (blast intro: fold_graph_determ)
nipkow@15392
   626
haftmann@42272
   627
lemma fold_graph_fold:
haftmann@42272
   628
  assumes "finite A"
haftmann@42272
   629
  shows "fold_graph f z A (fold f z A)"
haftmann@42272
   630
proof -
haftmann@42272
   631
  from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
haftmann@42272
   632
  moreover note fold_graph_determ
haftmann@42272
   633
  ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
haftmann@42272
   634
  then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
haftmann@42272
   635
  then show ?thesis by (unfold fold_def)
haftmann@42272
   636
qed
huffman@36045
   637
nipkow@15392
   638
text{* The base case for @{text fold}: *}
nipkow@15392
   639
nipkow@28853
   640
lemma (in -) fold_empty [simp]: "fold f z {} = z"
nipkow@28853
   641
by (unfold fold_def) blast
nipkow@28853
   642
nipkow@28853
   643
text{* The various recursion equations for @{const fold}: *}
nipkow@28853
   644
haftmann@26041
   645
lemma fold_insert [simp]:
haftmann@42875
   646
  assumes "finite A" and "x \<notin> A"
haftmann@42875
   647
  shows "fold f z (insert x A) = f x (fold f z A)"
haftmann@42875
   648
proof (rule fold_equality)
haftmann@42875
   649
  from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
haftmann@42875
   650
  with `x \<notin> A`show "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
haftmann@42875
   651
qed
nipkow@28853
   652
nipkow@28853
   653
lemma fold_fun_comm:
nipkow@28853
   654
  "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
nipkow@28853
   655
proof (induct rule: finite_induct)
nipkow@28853
   656
  case empty then show ?case by simp
nipkow@28853
   657
next
nipkow@28853
   658
  case (insert y A) then show ?case
nipkow@28853
   659
    by (simp add: fun_left_comm[of x])
nipkow@28853
   660
qed
nipkow@28853
   661
nipkow@28853
   662
lemma fold_insert2:
nipkow@28853
   663
  "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
huffman@35216
   664
by (simp add: fold_fun_comm)
nipkow@15392
   665
haftmann@26041
   666
lemma fold_rec:
haftmann@42875
   667
  assumes "finite A" and "x \<in> A"
haftmann@42875
   668
  shows "fold f z A = f x (fold f z (A - {x}))"
nipkow@28853
   669
proof -
nipkow@28853
   670
  have A: "A = insert x (A - {x})" using `x \<in> A` by blast
nipkow@28853
   671
  then have "fold f z A = fold f z (insert x (A - {x}))" by simp
nipkow@28853
   672
  also have "\<dots> = f x (fold f z (A - {x}))"
nipkow@28853
   673
    by (rule fold_insert) (simp add: `finite A`)+
nipkow@15535
   674
  finally show ?thesis .
nipkow@15535
   675
qed
nipkow@15535
   676
nipkow@28853
   677
lemma fold_insert_remove:
nipkow@28853
   678
  assumes "finite A"
nipkow@28853
   679
  shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
nipkow@28853
   680
proof -
nipkow@28853
   681
  from `finite A` have "finite (insert x A)" by auto
nipkow@28853
   682
  moreover have "x \<in> insert x A" by auto
nipkow@28853
   683
  ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
nipkow@28853
   684
    by (rule fold_rec)
nipkow@28853
   685
  then show ?thesis by simp
nipkow@28853
   686
qed
nipkow@28853
   687
kuncar@48619
   688
text{* Other properties of @{const fold}: *}
kuncar@48619
   689
kuncar@48619
   690
lemma fold_image:
kuncar@48619
   691
  assumes "finite A" and "inj_on g A"
kuncar@48619
   692
  shows "fold f x (g ` A) = fold (f \<circ> g) x A"
kuncar@48619
   693
using assms
kuncar@48619
   694
proof induction
kuncar@48619
   695
  case (insert a F)
kuncar@48619
   696
    interpret comp_fun_commute "\<lambda>x. f (g x)" by default (simp add: comp_fun_commute)
kuncar@48619
   697
    from insert show ?case by auto
kuncar@48619
   698
qed (simp)
kuncar@48619
   699
haftmann@26041
   700
end
nipkow@15392
   701
haftmann@49724
   702
lemma fold_cong:
haftmann@49724
   703
  assumes "comp_fun_commute f" "comp_fun_commute g"
haftmann@49724
   704
  assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
haftmann@49724
   705
    and "A = B" and "s = t"
haftmann@49724
   706
  shows "Finite_Set.fold f s A = Finite_Set.fold g t B"
haftmann@49724
   707
proof -
haftmann@49724
   708
  have "Finite_Set.fold f s A = Finite_Set.fold g s A"  
haftmann@49724
   709
  using `finite A` cong proof (induct A)
haftmann@49724
   710
    case empty then show ?case by simp
haftmann@49724
   711
  next
haftmann@49724
   712
    case (insert x A)
haftmann@49724
   713
    interpret f: comp_fun_commute f by (fact `comp_fun_commute f`)
haftmann@49724
   714
    interpret g: comp_fun_commute g by (fact `comp_fun_commute g`)
haftmann@49724
   715
    from insert show ?case by simp
haftmann@49724
   716
  qed
haftmann@49724
   717
  with assms show ?thesis by simp
haftmann@49724
   718
qed
haftmann@49724
   719
haftmann@49724
   720
nipkow@15480
   721
text{* A simplified version for idempotent functions: *}
nipkow@15480
   722
haftmann@42871
   723
locale comp_fun_idem = comp_fun_commute +
haftmann@42871
   724
  assumes comp_fun_idem: "f x o f x = f x"
haftmann@26041
   725
begin
haftmann@26041
   726
haftmann@42869
   727
lemma fun_left_idem: "f x (f x z) = f x z"
haftmann@42871
   728
  using comp_fun_idem by (simp add: fun_eq_iff)
nipkow@28853
   729
haftmann@26041
   730
lemma fold_insert_idem:
nipkow@28853
   731
  assumes fin: "finite A"
nipkow@28853
   732
  shows "fold f z (insert x A) = f x (fold f z A)"
nipkow@15480
   733
proof cases
nipkow@28853
   734
  assume "x \<in> A"
nipkow@28853
   735
  then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
nipkow@28853
   736
  then show ?thesis using assms by (simp add:fun_left_idem)
nipkow@15480
   737
next
nipkow@28853
   738
  assume "x \<notin> A" then show ?thesis using assms by simp
nipkow@15480
   739
qed
nipkow@15480
   740
nipkow@28853
   741
declare fold_insert[simp del] fold_insert_idem[simp]
nipkow@28853
   742
nipkow@28853
   743
lemma fold_insert_idem2:
nipkow@28853
   744
  "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
nipkow@28853
   745
by(simp add:fold_fun_comm)
nipkow@15484
   746
haftmann@26041
   747
end
haftmann@26041
   748
haftmann@35817
   749
haftmann@49723
   750
subsubsection {* Liftings to @{text comp_fun_commute} etc. *}
haftmann@35817
   751
haftmann@42871
   752
lemma (in comp_fun_commute) comp_comp_fun_commute:
haftmann@42871
   753
  "comp_fun_commute (f \<circ> g)"
haftmann@35817
   754
proof
haftmann@42871
   755
qed (simp_all add: comp_fun_commute)
haftmann@35817
   756
haftmann@42871
   757
lemma (in comp_fun_idem) comp_comp_fun_idem:
haftmann@42871
   758
  "comp_fun_idem (f \<circ> g)"
haftmann@42871
   759
  by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
haftmann@42871
   760
    (simp_all add: comp_fun_idem)
haftmann@35817
   761
haftmann@49723
   762
lemma (in comp_fun_commute) comp_fun_commute_funpow:
haftmann@49723
   763
  "comp_fun_commute (\<lambda>x. f x ^^ g x)"
haftmann@49723
   764
proof
haftmann@49723
   765
  fix y x
haftmann@49723
   766
  show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
haftmann@49723
   767
  proof (cases "x = y")
haftmann@49723
   768
    case True then show ?thesis by simp
haftmann@49723
   769
  next
haftmann@49723
   770
    case False show ?thesis
haftmann@49723
   771
    proof (induct "g x" arbitrary: g)
haftmann@49723
   772
      case 0 then show ?case by simp
haftmann@49723
   773
    next
haftmann@49723
   774
      case (Suc n g)
haftmann@49723
   775
      have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
haftmann@49723
   776
      proof (induct "g y" arbitrary: g)
haftmann@49723
   777
        case 0 then show ?case by simp
haftmann@49723
   778
      next
haftmann@49723
   779
        case (Suc n g)
haftmann@49723
   780
        def h \<equiv> "\<lambda>z. g z - 1"
haftmann@49723
   781
        with Suc have "n = h y" by simp
haftmann@49723
   782
        with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
haftmann@49723
   783
          by auto
haftmann@49723
   784
        from Suc h_def have "g y = Suc (h y)" by simp
haftmann@49739
   785
        then show ?case by (simp add: comp_assoc hyp)
haftmann@49723
   786
          (simp add: o_assoc comp_fun_commute)
haftmann@49723
   787
      qed
haftmann@49723
   788
      def h \<equiv> "\<lambda>z. if z = x then g x - 1 else g z"
haftmann@49723
   789
      with Suc have "n = h x" by simp
haftmann@49723
   790
      with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
haftmann@49723
   791
        by auto
haftmann@49723
   792
      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
   793
      from Suc h_def have "g x = Suc (h x)" by simp
haftmann@49723
   794
      then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
haftmann@49739
   795
        (simp add: comp_assoc hyp1)
haftmann@49723
   796
    qed
haftmann@49723
   797
  qed
haftmann@49723
   798
qed
haftmann@49723
   799
haftmann@49723
   800
haftmann@49723
   801
subsubsection {* Expressing set operations via @{const fold} *}
haftmann@49723
   802
haftmann@42871
   803
lemma comp_fun_idem_insert:
haftmann@42871
   804
  "comp_fun_idem insert"
haftmann@35817
   805
proof
haftmann@35817
   806
qed auto
haftmann@35817
   807
haftmann@42871
   808
lemma comp_fun_idem_remove:
haftmann@46146
   809
  "comp_fun_idem Set.remove"
haftmann@35817
   810
proof
haftmann@35817
   811
qed auto
nipkow@31992
   812
haftmann@42871
   813
lemma (in semilattice_inf) comp_fun_idem_inf:
haftmann@42871
   814
  "comp_fun_idem inf"
haftmann@35817
   815
proof
haftmann@35817
   816
qed (auto simp add: inf_left_commute)
haftmann@35817
   817
haftmann@42871
   818
lemma (in semilattice_sup) comp_fun_idem_sup:
haftmann@42871
   819
  "comp_fun_idem sup"
haftmann@35817
   820
proof
haftmann@35817
   821
qed (auto simp add: sup_left_commute)
nipkow@31992
   822
haftmann@35817
   823
lemma union_fold_insert:
haftmann@35817
   824
  assumes "finite A"
haftmann@35817
   825
  shows "A \<union> B = fold insert B A"
haftmann@35817
   826
proof -
haftmann@42871
   827
  interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
haftmann@35817
   828
  from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
haftmann@35817
   829
qed
nipkow@31992
   830
haftmann@35817
   831
lemma minus_fold_remove:
haftmann@35817
   832
  assumes "finite A"
haftmann@46146
   833
  shows "B - A = fold Set.remove B A"
haftmann@35817
   834
proof -
haftmann@46146
   835
  interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
haftmann@46146
   836
  from `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
haftmann@46146
   837
  then show ?thesis ..
haftmann@35817
   838
qed
haftmann@35817
   839
kuncar@48619
   840
lemma comp_fun_commute_filter_fold: "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
kuncar@48619
   841
proof - 
kuncar@48619
   842
  interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
kuncar@48619
   843
  show ?thesis by default (auto simp: fun_eq_iff)
kuncar@48619
   844
qed
kuncar@48619
   845
kuncar@49758
   846
lemma Set_filter_fold:
kuncar@48619
   847
  assumes "finite A"
kuncar@49758
   848
  shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
kuncar@48619
   849
using assms
kuncar@48619
   850
by (induct A) 
kuncar@49758
   851
  (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
kuncar@49758
   852
kuncar@49758
   853
lemma inter_Set_filter:     
kuncar@49758
   854
  assumes "finite B"
kuncar@49758
   855
  shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
kuncar@49758
   856
using assms 
kuncar@49758
   857
by (induct B) (auto simp: Set.filter_def)
kuncar@48619
   858
kuncar@48619
   859
lemma image_fold_insert:
kuncar@48619
   860
  assumes "finite A"
kuncar@48619
   861
  shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
kuncar@48619
   862
using assms
kuncar@48619
   863
proof -
kuncar@48619
   864
  interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by default auto
kuncar@48619
   865
  show ?thesis using assms by (induct A) auto
kuncar@48619
   866
qed
kuncar@48619
   867
kuncar@48619
   868
lemma Ball_fold:
kuncar@48619
   869
  assumes "finite A"
kuncar@48619
   870
  shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
kuncar@48619
   871
using assms
kuncar@48619
   872
proof -
kuncar@48619
   873
  interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by default auto
kuncar@48619
   874
  show ?thesis using assms by (induct A) auto
kuncar@48619
   875
qed
kuncar@48619
   876
kuncar@48619
   877
lemma Bex_fold:
kuncar@48619
   878
  assumes "finite A"
kuncar@48619
   879
  shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
kuncar@48619
   880
using assms
kuncar@48619
   881
proof -
kuncar@48619
   882
  interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by default auto
kuncar@48619
   883
  show ?thesis using assms by (induct A) auto
kuncar@48619
   884
qed
kuncar@48619
   885
kuncar@48619
   886
lemma comp_fun_commute_Pow_fold: 
kuncar@48619
   887
  "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)" 
kuncar@48619
   888
  by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
kuncar@48619
   889
kuncar@48619
   890
lemma Pow_fold:
kuncar@48619
   891
  assumes "finite A"
kuncar@48619
   892
  shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
kuncar@48619
   893
using assms
kuncar@48619
   894
proof -
kuncar@48619
   895
  interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
kuncar@48619
   896
  show ?thesis using assms by (induct A) (auto simp: Pow_insert)
kuncar@48619
   897
qed
kuncar@48619
   898
kuncar@48619
   899
lemma fold_union_pair:
kuncar@48619
   900
  assumes "finite B"
kuncar@48619
   901
  shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
kuncar@48619
   902
proof -
kuncar@48619
   903
  interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by default auto
kuncar@48619
   904
  show ?thesis using assms  by (induct B arbitrary: A) simp_all
kuncar@48619
   905
qed
kuncar@48619
   906
kuncar@48619
   907
lemma comp_fun_commute_product_fold: 
kuncar@48619
   908
  assumes "finite B"
kuncar@48619
   909
  shows "comp_fun_commute (\<lambda>x A. fold (\<lambda>y. Set.insert (x, y)) A B)" 
kuncar@48619
   910
by default (auto simp: fold_union_pair[symmetric] assms)
kuncar@48619
   911
kuncar@48619
   912
lemma product_fold:
kuncar@48619
   913
  assumes "finite A"
kuncar@48619
   914
  assumes "finite B"
kuncar@48619
   915
  shows "A \<times> B = fold (\<lambda>x A. fold (\<lambda>y. Set.insert (x, y)) A B) {} A"
kuncar@48619
   916
using assms unfolding Sigma_def 
kuncar@48619
   917
by (induct A) 
kuncar@48619
   918
  (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
kuncar@48619
   919
kuncar@48619
   920
haftmann@35817
   921
context complete_lattice
nipkow@31992
   922
begin
nipkow@31992
   923
haftmann@35817
   924
lemma inf_Inf_fold_inf:
haftmann@35817
   925
  assumes "finite A"
haftmann@35817
   926
  shows "inf B (Inf A) = fold inf B A"
haftmann@35817
   927
proof -
haftmann@42871
   928
  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
haftmann@35817
   929
  from `finite A` show ?thesis by (induct A arbitrary: B)
noschinl@44919
   930
    (simp_all add: inf_commute fold_fun_comm)
haftmann@35817
   931
qed
nipkow@31992
   932
haftmann@35817
   933
lemma sup_Sup_fold_sup:
haftmann@35817
   934
  assumes "finite A"
haftmann@35817
   935
  shows "sup B (Sup A) = fold sup B A"
haftmann@35817
   936
proof -
haftmann@42871
   937
  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
haftmann@35817
   938
  from `finite A` show ?thesis by (induct A arbitrary: B)
noschinl@44919
   939
    (simp_all add: sup_commute fold_fun_comm)
nipkow@31992
   940
qed
nipkow@31992
   941
haftmann@35817
   942
lemma Inf_fold_inf:
haftmann@35817
   943
  assumes "finite A"
haftmann@35817
   944
  shows "Inf A = fold inf top A"
haftmann@35817
   945
  using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
haftmann@35817
   946
haftmann@35817
   947
lemma Sup_fold_sup:
haftmann@35817
   948
  assumes "finite A"
haftmann@35817
   949
  shows "Sup A = fold sup bot A"
haftmann@35817
   950
  using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
nipkow@31992
   951
haftmann@46146
   952
lemma inf_INF_fold_inf:
haftmann@35817
   953
  assumes "finite A"
haftmann@42873
   954
  shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold") 
haftmann@35817
   955
proof (rule sym)
haftmann@42871
   956
  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
haftmann@42871
   957
  interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
haftmann@42873
   958
  from `finite A` show "?fold = ?inf"
haftmann@42869
   959
    by (induct A arbitrary: B)
hoelzl@44928
   960
      (simp_all add: INF_def inf_left_commute)
haftmann@35817
   961
qed
nipkow@31992
   962
haftmann@46146
   963
lemma sup_SUP_fold_sup:
haftmann@35817
   964
  assumes "finite A"
haftmann@42873
   965
  shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold") 
haftmann@35817
   966
proof (rule sym)
haftmann@42871
   967
  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
haftmann@42871
   968
  interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
haftmann@42873
   969
  from `finite A` show "?fold = ?sup"
haftmann@42869
   970
    by (induct A arbitrary: B)
hoelzl@44928
   971
      (simp_all add: SUP_def sup_left_commute)
haftmann@35817
   972
qed
nipkow@31992
   973
haftmann@46146
   974
lemma INF_fold_inf:
haftmann@35817
   975
  assumes "finite A"
haftmann@42873
   976
  shows "INFI A f = fold (inf \<circ> f) top A"
haftmann@46146
   977
  using assms inf_INF_fold_inf [of A top] by simp
nipkow@31992
   978
haftmann@46146
   979
lemma SUP_fold_sup:
haftmann@35817
   980
  assumes "finite A"
haftmann@42873
   981
  shows "SUPR A f = fold (sup \<circ> f) bot A"
haftmann@46146
   982
  using assms sup_SUP_fold_sup [of A bot] by simp
nipkow@31992
   983
nipkow@31992
   984
end
nipkow@31992
   985
nipkow@31992
   986
haftmann@35817
   987
subsection {* The derived combinator @{text fold_image} *}
nipkow@28853
   988
nipkow@28853
   989
definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
haftmann@42875
   990
  where "fold_image f g = fold (\<lambda>x y. f (g x) y)"
nipkow@28853
   991
nipkow@28853
   992
lemma fold_image_empty[simp]: "fold_image f g z {} = z"
haftmann@42875
   993
  by (simp add:fold_image_def)
nipkow@15392
   994
haftmann@26041
   995
context ab_semigroup_mult
haftmann@26041
   996
begin
haftmann@26041
   997
nipkow@28853
   998
lemma fold_image_insert[simp]:
haftmann@42875
   999
  assumes "finite A" and "a \<notin> A"
haftmann@42875
  1000
  shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
nipkow@28853
  1001
proof -
wenzelm@46898
  1002
  interpret comp_fun_commute "%x y. (g x) * y"
wenzelm@46898
  1003
    by default (simp add: fun_eq_iff mult_ac)
wenzelm@46898
  1004
  from assms show ?thesis by (simp add: fold_image_def)
nipkow@28853
  1005
qed
nipkow@28853
  1006
nipkow@28853
  1007
lemma fold_image_reindex:
haftmann@42875
  1008
  assumes "finite A"
haftmann@42875
  1009
  shows "inj_on h A \<Longrightarrow> fold_image times g z (h ` A) = fold_image times (g \<circ> h) z A"
haftmann@42875
  1010
  using assms by induct auto
nipkow@28853
  1011
nipkow@28853
  1012
lemma fold_image_cong:
haftmann@42875
  1013
  assumes "finite A" and g_h: "\<And>x. x\<in>A \<Longrightarrow> g x = h x"
haftmann@42875
  1014
  shows "fold_image times g z A = fold_image times h z A"
haftmann@42875
  1015
proof -
haftmann@42875
  1016
  from `finite A`
haftmann@42875
  1017
  have "\<And>C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C"
haftmann@42875
  1018
  proof (induct arbitrary: C)
haftmann@42875
  1019
    case empty then show ?case by simp
haftmann@42875
  1020
  next
haftmann@42875
  1021
    case (insert x F) then show ?case apply -
haftmann@42875
  1022
    apply (simp add: subset_insert_iff, clarify)
haftmann@42875
  1023
    apply (subgoal_tac "finite C")
wenzelm@48125
  1024
      prefer 2 apply (blast dest: finite_subset [rotated])
haftmann@42875
  1025
    apply (subgoal_tac "C = insert x (C - {x})")
haftmann@42875
  1026
      prefer 2 apply blast
haftmann@42875
  1027
    apply (erule ssubst)
haftmann@42875
  1028
    apply (simp add: Ball_def del: insert_Diff_single)
haftmann@42875
  1029
    done
haftmann@42875
  1030
  qed
haftmann@42875
  1031
  with g_h show ?thesis by simp
haftmann@42875
  1032
qed
nipkow@15392
  1033
haftmann@26041
  1034
end
haftmann@26041
  1035
haftmann@26041
  1036
context comm_monoid_mult
haftmann@26041
  1037
begin
haftmann@26041
  1038
haftmann@35817
  1039
lemma fold_image_1:
haftmann@35817
  1040
  "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
haftmann@41656
  1041
  apply (induct rule: finite_induct)
haftmann@35817
  1042
  apply simp by auto
haftmann@35817
  1043
nipkow@28853
  1044
lemma fold_image_Un_Int:
haftmann@26041
  1045
  "finite A ==> finite B ==>
nipkow@28853
  1046
    fold_image times g 1 A * fold_image times g 1 B =
nipkow@28853
  1047
    fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
haftmann@41656
  1048
  apply (induct rule: finite_induct)
nipkow@28853
  1049
by (induct set: finite) 
nipkow@28853
  1050
   (auto simp add: mult_ac insert_absorb Int_insert_left)
haftmann@26041
  1051
haftmann@35817
  1052
lemma fold_image_Un_one:
haftmann@35817
  1053
  assumes fS: "finite S" and fT: "finite T"
haftmann@35817
  1054
  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
haftmann@35817
  1055
  shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
haftmann@35817
  1056
proof-
haftmann@35817
  1057
  have "fold_image op * f 1 (S \<inter> T) = 1" 
haftmann@35817
  1058
    apply (rule fold_image_1)
haftmann@35817
  1059
    using fS fT I0 by auto 
haftmann@35817
  1060
  with fold_image_Un_Int[OF fS fT] show ?thesis by simp
haftmann@35817
  1061
qed
haftmann@35817
  1062
haftmann@26041
  1063
corollary fold_Un_disjoint:
haftmann@26041
  1064
  "finite A ==> finite B ==> A Int B = {} ==>
nipkow@28853
  1065
   fold_image times g 1 (A Un B) =
nipkow@28853
  1066
   fold_image times g 1 A * fold_image times g 1 B"
nipkow@28853
  1067
by (simp add: fold_image_Un_Int)
nipkow@28853
  1068
nipkow@28853
  1069
lemma fold_image_UN_disjoint:
haftmann@26041
  1070
  "\<lbrakk> finite I; ALL i:I. finite (A i);
haftmann@26041
  1071
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
nipkow@28853
  1072
   \<Longrightarrow> fold_image times g 1 (UNION I A) =
nipkow@28853
  1073
       fold_image times (%i. fold_image times g 1 (A i)) 1 I"
haftmann@41656
  1074
apply (induct rule: finite_induct)
haftmann@41656
  1075
apply simp
haftmann@41656
  1076
apply atomize
nipkow@28853
  1077
apply (subgoal_tac "ALL i:F. x \<noteq> i")
nipkow@28853
  1078
 prefer 2 apply blast
nipkow@28853
  1079
apply (subgoal_tac "A x Int UNION F A = {}")
nipkow@28853
  1080
 prefer 2 apply blast
nipkow@28853
  1081
apply (simp add: fold_Un_disjoint)
nipkow@28853
  1082
done
nipkow@28853
  1083
nipkow@28853
  1084
lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@28853
  1085
  fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
nipkow@28853
  1086
  fold_image times (split g) 1 (SIGMA x:A. B x)"
nipkow@15392
  1087
apply (subst Sigma_def)
nipkow@28853
  1088
apply (subst fold_image_UN_disjoint, assumption, simp)
nipkow@15392
  1089
 apply blast
nipkow@28853
  1090
apply (erule fold_image_cong)
nipkow@28853
  1091
apply (subst fold_image_UN_disjoint, simp, simp)
nipkow@15392
  1092
 apply blast
paulson@15506
  1093
apply simp
nipkow@15392
  1094
done
nipkow@15392
  1095
nipkow@28853
  1096
lemma fold_image_distrib: "finite A \<Longrightarrow>
nipkow@28853
  1097
   fold_image times (%x. g x * h x) 1 A =
nipkow@28853
  1098
   fold_image times g 1 A *  fold_image times h 1 A"
nipkow@28853
  1099
by (erule finite_induct) (simp_all add: mult_ac)
haftmann@26041
  1100
chaieb@30260
  1101
lemma fold_image_related: 
chaieb@30260
  1102
  assumes Re: "R e e" 
chaieb@30260
  1103
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
chaieb@30260
  1104
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
chaieb@30260
  1105
  shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
chaieb@30260
  1106
  using fS by (rule finite_subset_induct) (insert assms, auto)
chaieb@30260
  1107
chaieb@30260
  1108
lemma  fold_image_eq_general:
chaieb@30260
  1109
  assumes fS: "finite S"
chaieb@30260
  1110
  and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 
chaieb@30260
  1111
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
chaieb@30260
  1112
  shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
chaieb@30260
  1113
proof-
chaieb@30260
  1114
  from h f12 have hS: "h ` S = S'" by auto
chaieb@30260
  1115
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
chaieb@30260
  1116
    from f12 h H  have "x = y" by auto }
chaieb@30260
  1117
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
chaieb@30260
  1118
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
chaieb@30260
  1119
  from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
chaieb@30260
  1120
  also have "\<dots> = fold_image (op *) (f2 o h) e S" 
chaieb@30260
  1121
    using fold_image_reindex[OF fS hinj, of f2 e] .
chaieb@30260
  1122
  also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
chaieb@30260
  1123
    by blast
chaieb@30260
  1124
  finally show ?thesis ..
chaieb@30260
  1125
qed
chaieb@30260
  1126
chaieb@30260
  1127
lemma fold_image_eq_general_inverses:
chaieb@30260
  1128
  assumes fS: "finite S" 
chaieb@30260
  1129
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
chaieb@30260
  1130
  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
chaieb@30260
  1131
  shows "fold_image (op *) f e S = fold_image (op *) g e T"
chaieb@30260
  1132
  (* metis solves it, but not yet available here *)
chaieb@30260
  1133
  apply (rule fold_image_eq_general[OF fS, of T h g f e])
chaieb@30260
  1134
  apply (rule ballI)
chaieb@30260
  1135
  apply (frule kh)
chaieb@30260
  1136
  apply (rule ex1I[])
chaieb@30260
  1137
  apply blast
chaieb@30260
  1138
  apply clarsimp
chaieb@30260
  1139
  apply (drule hk) apply simp
chaieb@30260
  1140
  apply (rule sym)
chaieb@30260
  1141
  apply (erule conjunct1[OF conjunct2[OF hk]])
chaieb@30260
  1142
  apply (rule ballI)
chaieb@30260
  1143
  apply (drule  hk)
chaieb@30260
  1144
  apply blast
chaieb@30260
  1145
  done
chaieb@30260
  1146
haftmann@26041
  1147
end
haftmann@22917
  1148
nipkow@25162
  1149
haftmann@35817
  1150
subsection {* A fold functional for non-empty sets *}
nipkow@15392
  1151
nipkow@15392
  1152
text{* Does not require start value. *}
wenzelm@12396
  1153
berghofe@23736
  1154
inductive
berghofe@22262
  1155
  fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
berghofe@22262
  1156
  for f :: "'a => 'a => 'a"
berghofe@22262
  1157
where
paulson@15506
  1158
  fold1Set_insertI [intro]:
nipkow@28853
  1159
   "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
wenzelm@12396
  1160
haftmann@35416
  1161
definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
berghofe@22262
  1162
  "fold1 f A == THE x. fold1Set f A x"
paulson@15506
  1163
paulson@15506
  1164
lemma fold1Set_nonempty:
haftmann@22917
  1165
  "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
nipkow@28853
  1166
by(erule fold1Set.cases, simp_all)
nipkow@15392
  1167
berghofe@23736
  1168
inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
berghofe@23736
  1169
berghofe@23736
  1170
inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
berghofe@22262
  1171
berghofe@22262
  1172
berghofe@22262
  1173
lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
huffman@35216
  1174
by (blast elim: fold_graph.cases)
nipkow@15392
  1175
haftmann@22917
  1176
lemma fold1_singleton [simp]: "fold1 f {a} = a"
nipkow@28853
  1177
by (unfold fold1_def) blast
wenzelm@12396
  1178
paulson@15508
  1179
lemma finite_nonempty_imp_fold1Set:
berghofe@22262
  1180
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
paulson@15508
  1181
apply (induct A rule: finite_induct)
nipkow@28853
  1182
apply (auto dest: finite_imp_fold_graph [of _ f])
paulson@15508
  1183
done
paulson@15506
  1184
nipkow@28853
  1185
text{*First, some lemmas about @{const fold_graph}.*}
nipkow@15392
  1186
haftmann@26041
  1187
context ab_semigroup_mult
haftmann@26041
  1188
begin
haftmann@26041
  1189
wenzelm@46898
  1190
lemma comp_fun_commute: "comp_fun_commute (op *)"
wenzelm@46898
  1191
  by default (simp add: fun_eq_iff mult_ac)
nipkow@28853
  1192
nipkow@28853
  1193
lemma fold_graph_insert_swap:
nipkow@28853
  1194
assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
nipkow@28853
  1195
shows "fold_graph times z (insert b A) (z * y)"
nipkow@28853
  1196
proof -
haftmann@42871
  1197
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
nipkow@28853
  1198
from assms show ?thesis
nipkow@28853
  1199
proof (induct rule: fold_graph.induct)
huffman@36045
  1200
  case emptyI show ?case by (subst mult_commute [of z b], fast)
paulson@15508
  1201
next
berghofe@22262
  1202
  case (insertI x A y)
nipkow@28853
  1203
    have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
paulson@15521
  1204
      using insertI by force  --{*how does @{term id} get unfolded?*}
haftmann@26041
  1205
    thus ?case by (simp add: insert_commute mult_ac)
paulson@15508
  1206
qed
nipkow@28853
  1207
qed
nipkow@28853
  1208
nipkow@28853
  1209
lemma fold_graph_permute_diff:
nipkow@28853
  1210
assumes fold: "fold_graph times b A x"
nipkow@28853
  1211
shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
paulson@15508
  1212
using fold
nipkow@28853
  1213
proof (induct rule: fold_graph.induct)
paulson@15508
  1214
  case emptyI thus ?case by simp
paulson@15508
  1215
next
berghofe@22262
  1216
  case (insertI x A y)
paulson@15521
  1217
  have "a = x \<or> a \<in> A" using insertI by simp
paulson@15521
  1218
  thus ?case
paulson@15521
  1219
  proof
paulson@15521
  1220
    assume "a = x"
paulson@15521
  1221
    with insertI show ?thesis
nipkow@28853
  1222
      by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
paulson@15521
  1223
  next
paulson@15521
  1224
    assume ainA: "a \<in> A"
nipkow@28853
  1225
    hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
nipkow@28853
  1226
      using insertI by force
paulson@15521
  1227
    moreover
paulson@15521
  1228
    have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
paulson@15521
  1229
      using ainA insertI by blast
nipkow@28853
  1230
    ultimately show ?thesis by simp
paulson@15508
  1231
  qed
paulson@15508
  1232
qed
paulson@15508
  1233
haftmann@26041
  1234
lemma fold1_eq_fold:
nipkow@28853
  1235
assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
nipkow@28853
  1236
proof -
haftmann@42871
  1237
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
nipkow@28853
  1238
  from assms show ?thesis
nipkow@28853
  1239
apply (simp add: fold1_def fold_def)
paulson@15508
  1240
apply (rule the_equality)
nipkow@28853
  1241
apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
paulson@15508
  1242
apply (rule sym, clarify)
paulson@15508
  1243
apply (case_tac "Aa=A")
huffman@35216
  1244
 apply (best intro: fold_graph_determ)
nipkow@28853
  1245
apply (subgoal_tac "fold_graph times a A x")
huffman@35216
  1246
 apply (best intro: fold_graph_determ)
nipkow@28853
  1247
apply (subgoal_tac "insert aa (Aa - {a}) = A")
nipkow@28853
  1248
 prefer 2 apply (blast elim: equalityE)
nipkow@28853
  1249
apply (auto dest: fold_graph_permute_diff [where a=a])
paulson@15508
  1250
done
nipkow@28853
  1251
qed
paulson@15508
  1252
paulson@15521
  1253
lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
paulson@15521
  1254
apply safe
nipkow@28853
  1255
 apply simp
nipkow@28853
  1256
 apply (drule_tac x=x in spec)
nipkow@28853
  1257
 apply (drule_tac x="A-{x}" in spec, auto)
paulson@15508
  1258
done
paulson@15508
  1259
haftmann@26041
  1260
lemma fold1_insert:
paulson@15521
  1261
  assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
haftmann@26041
  1262
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1263
proof -
haftmann@42871
  1264
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
nipkow@28853
  1265
  from nonempty obtain a A' where "A = insert a A' & a ~: A'"
paulson@15521
  1266
    by (auto simp add: nonempty_iff)
paulson@15521
  1267
  with A show ?thesis
nipkow@28853
  1268
    by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
paulson@15521
  1269
qed
paulson@15521
  1270
haftmann@26041
  1271
end
haftmann@26041
  1272
haftmann@26041
  1273
context ab_semigroup_idem_mult
haftmann@26041
  1274
begin
haftmann@26041
  1275
wenzelm@46898
  1276
lemma comp_fun_idem: "comp_fun_idem (op *)"
wenzelm@46898
  1277
  by default (simp_all add: fun_eq_iff mult_left_commute)
haftmann@35817
  1278
haftmann@26041
  1279
lemma fold1_insert_idem [simp]:
paulson@15521
  1280
  assumes nonempty: "A \<noteq> {}" and A: "finite A" 
haftmann@26041
  1281
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1282
proof -
haftmann@42871
  1283
  interpret comp_fun_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@42871
  1284
    by (rule comp_fun_idem)
nipkow@28853
  1285
  from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
paulson@15521
  1286
    by (auto simp add: nonempty_iff)
paulson@15521
  1287
  show ?thesis
paulson@15521
  1288
  proof cases
wenzelm@41550
  1289
    assume a: "a = x"
wenzelm@41550
  1290
    show ?thesis
paulson@15521
  1291
    proof cases
paulson@15521
  1292
      assume "A' = {}"
wenzelm@41550
  1293
      with A' a show ?thesis by simp
paulson@15521
  1294
    next
paulson@15521
  1295
      assume "A' \<noteq> {}"
wenzelm@41550
  1296
      with A A' a show ?thesis
huffman@35216
  1297
        by (simp add: fold1_insert mult_assoc [symmetric])
paulson@15521
  1298
    qed
paulson@15521
  1299
  next
paulson@15521
  1300
    assume "a \<noteq> x"
wenzelm@41550
  1301
    with A A' show ?thesis
huffman@35216
  1302
      by (simp add: insert_commute fold1_eq_fold)
paulson@15521
  1303
  qed
paulson@15521
  1304
qed
paulson@15506
  1305
haftmann@26041
  1306
lemma hom_fold1_commute:
haftmann@26041
  1307
assumes hom: "!!x y. h (x * y) = h x * h y"
haftmann@26041
  1308
and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
wenzelm@46898
  1309
using N
wenzelm@46898
  1310
proof (induct rule: finite_ne_induct)
haftmann@22917
  1311
  case singleton thus ?case by simp
haftmann@22917
  1312
next
haftmann@22917
  1313
  case (insert n N)
haftmann@26041
  1314
  then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
haftmann@26041
  1315
  also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
haftmann@26041
  1316
  also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
haftmann@26041
  1317
  also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
haftmann@22917
  1318
    using insert by(simp)
haftmann@22917
  1319
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@22917
  1320
  finally show ?case .
haftmann@22917
  1321
qed
haftmann@22917
  1322
haftmann@32679
  1323
lemma fold1_eq_fold_idem:
haftmann@32679
  1324
  assumes "finite A"
haftmann@32679
  1325
  shows "fold1 times (insert a A) = fold times a A"
haftmann@32679
  1326
proof (cases "a \<in> A")
haftmann@32679
  1327
  case False
haftmann@32679
  1328
  with assms show ?thesis by (simp add: fold1_eq_fold)
haftmann@32679
  1329
next
haftmann@42871
  1330
  interpret comp_fun_idem times by (fact comp_fun_idem)
haftmann@32679
  1331
  case True then obtain b B
haftmann@32679
  1332
    where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
haftmann@32679
  1333
  with assms have "finite B" by auto
haftmann@32679
  1334
  then have "fold times a (insert a B) = fold times (a * a) B"
haftmann@32679
  1335
    using `a \<notin> B` by (rule fold_insert2)
haftmann@32679
  1336
  then show ?thesis
haftmann@32679
  1337
    using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
haftmann@32679
  1338
qed
haftmann@32679
  1339
haftmann@26041
  1340
end
haftmann@26041
  1341
paulson@15506
  1342
paulson@15508
  1343
text{* Now the recursion rules for definitions: *}
paulson@15508
  1344
haftmann@22917
  1345
lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
huffman@35216
  1346
by simp
paulson@15508
  1347
haftmann@26041
  1348
lemma (in ab_semigroup_mult) fold1_insert_def:
haftmann@26041
  1349
  "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1350
by (simp add:fold1_insert)
haftmann@26041
  1351
haftmann@26041
  1352
lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
haftmann@26041
  1353
  "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1354
by simp
paulson@15508
  1355
paulson@15508
  1356
subsubsection{* Determinacy for @{term fold1Set} *}
paulson@15508
  1357
nipkow@28853
  1358
(*Not actually used!!*)
nipkow@28853
  1359
(*
haftmann@26041
  1360
context ab_semigroup_mult
haftmann@26041
  1361
begin
haftmann@26041
  1362
nipkow@28853
  1363
lemma fold_graph_permute:
nipkow@28853
  1364
  "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
nipkow@28853
  1365
   ==> fold_graph times id a (insert b A) x"
haftmann@26041
  1366
apply (cases "a=b") 
nipkow@28853
  1367
apply (auto dest: fold_graph_permute_diff) 
paulson@15506
  1368
done
nipkow@15376
  1369
haftmann@26041
  1370
lemma fold1Set_determ:
haftmann@26041
  1371
  "fold1Set times A x ==> fold1Set times A y ==> y = x"
paulson@15506
  1372
proof (clarify elim!: fold1Set.cases)
paulson@15506
  1373
  fix A x B y a b
nipkow@28853
  1374
  assume Ax: "fold_graph times id a A x"
nipkow@28853
  1375
  assume By: "fold_graph times id b B y"
paulson@15506
  1376
  assume anotA:  "a \<notin> A"
paulson@15506
  1377
  assume bnotB:  "b \<notin> B"
paulson@15506
  1378
  assume eq: "insert a A = insert b B"
paulson@15506
  1379
  show "y=x"
paulson@15506
  1380
  proof cases
paulson@15506
  1381
    assume same: "a=b"
paulson@15506
  1382
    hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
nipkow@28853
  1383
    thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
nipkow@15392
  1384
  next
paulson@15506
  1385
    assume diff: "a\<noteq>b"
paulson@15506
  1386
    let ?D = "B - {a}"
paulson@15506
  1387
    have B: "B = insert a ?D" and A: "A = insert b ?D"
paulson@15506
  1388
     and aB: "a \<in> B" and bA: "b \<in> A"
paulson@15506
  1389
      using eq anotA bnotB diff by (blast elim!:equalityE)+
paulson@15506
  1390
    with aB bnotB By
nipkow@28853
  1391
    have "fold_graph times id a (insert b ?D) y" 
nipkow@28853
  1392
      by (auto intro: fold_graph_permute simp add: insert_absorb)
paulson@15506
  1393
    moreover
nipkow@28853
  1394
    have "fold_graph times id a (insert b ?D) x"
paulson@15506
  1395
      by (simp add: A [symmetric] Ax) 
nipkow@28853
  1396
    ultimately show ?thesis by (blast intro: fold_graph_determ) 
nipkow@15392
  1397
  qed
wenzelm@12396
  1398
qed
wenzelm@12396
  1399
haftmann@26041
  1400
lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
paulson@15506
  1401
  by (unfold fold1_def) (blast intro: fold1Set_determ)
paulson@15506
  1402
haftmann@26041
  1403
end
nipkow@28853
  1404
*)
haftmann@26041
  1405
paulson@15506
  1406
declare
nipkow@28853
  1407
  empty_fold_graphE [rule del]  fold_graph.intros [rule del]
paulson@15506
  1408
  empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
ballarin@19931
  1409
  -- {* No more proofs involve these relations. *}
nipkow@15376
  1410
haftmann@26041
  1411
subsubsection {* Lemmas about @{text fold1} *}
haftmann@26041
  1412
haftmann@26041
  1413
context ab_semigroup_mult
haftmann@22917
  1414
begin
haftmann@22917
  1415
haftmann@26041
  1416
lemma fold1_Un:
nipkow@15484
  1417
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  1418
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
haftmann@26041
  1419
       fold1 times (A Un B) = fold1 times A * fold1 times B"
haftmann@26041
  1420
using A by (induct rule: finite_ne_induct)
haftmann@26041
  1421
  (simp_all add: fold1_insert mult_assoc)
haftmann@26041
  1422
haftmann@26041
  1423
lemma fold1_in:
haftmann@26041
  1424
  assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
haftmann@26041
  1425
  shows "fold1 times A \<in> A"
nipkow@15484
  1426
using A
nipkow@15484
  1427
proof (induct rule:finite_ne_induct)
paulson@15506
  1428
  case singleton thus ?case by simp
nipkow@15484
  1429
next
nipkow@15484
  1430
  case insert thus ?case using elem by (force simp add:fold1_insert)
nipkow@15484
  1431
qed
nipkow@15484
  1432
haftmann@26041
  1433
end
haftmann@26041
  1434
haftmann@26041
  1435
lemma (in ab_semigroup_idem_mult) fold1_Un2:
nipkow@15497
  1436
assumes A: "finite A" "A \<noteq> {}"
haftmann@26041
  1437
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
haftmann@26041
  1438
       fold1 times (A Un B) = fold1 times A * fold1 times B"
nipkow@15497
  1439
using A
haftmann@26041
  1440
proof(induct rule:finite_ne_induct)
nipkow@15497
  1441
  case singleton thus ?case by simp
nipkow@15484
  1442
next
haftmann@26041
  1443
  case insert thus ?case by (simp add: mult_assoc)
nipkow@18423
  1444
qed
nipkow@18423
  1445
nipkow@18423
  1446
haftmann@35817
  1447
subsection {* Locales as mini-packages for fold operations *}
haftmann@34007
  1448
haftmann@35817
  1449
subsubsection {* The natural case *}
haftmann@35719
  1450
haftmann@35719
  1451
locale folding =
haftmann@35719
  1452
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@35719
  1453
  fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@42871
  1454
  assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
haftmann@35722
  1455
  assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
haftmann@35719
  1456
begin
haftmann@35719
  1457
haftmann@35719
  1458
lemma empty [simp]:
haftmann@35719
  1459
  "F {} = id"
nipkow@39302
  1460
  by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1461
haftmann@35719
  1462
lemma insert [simp]:
haftmann@35719
  1463
  assumes "finite A" and "x \<notin> A"
haftmann@35719
  1464
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35719
  1465
proof -
wenzelm@46898
  1466
  interpret comp_fun_commute f
wenzelm@46898
  1467
    by default (insert comp_fun_commute, simp add: fun_eq_iff)
haftmann@35719
  1468
  from fold_insert2 assms
haftmann@35722
  1469
  have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
nipkow@39302
  1470
  with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1471
qed
haftmann@35719
  1472
haftmann@35719
  1473
lemma remove:
haftmann@35719
  1474
  assumes "finite A" and "x \<in> A"
haftmann@35719
  1475
  shows "F A = F (A - {x}) \<circ> f x"
haftmann@35719
  1476
proof -
haftmann@35719
  1477
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35719
  1478
    by (auto dest: mk_disjoint_insert)
haftmann@35719
  1479
  moreover from `finite A` this have "finite B" by simp
haftmann@35719
  1480
  ultimately show ?thesis by simp
haftmann@35719
  1481
qed
haftmann@35719
  1482
haftmann@35719
  1483
lemma insert_remove:
haftmann@35719
  1484
  assumes "finite A"
haftmann@35719
  1485
  shows "F (insert x A) = F (A - {x}) \<circ> f x"
haftmann@35722
  1486
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35719
  1487
haftmann@35817
  1488
lemma commute_left_comp:
haftmann@35817
  1489
  "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
haftmann@42871
  1490
  by (simp add: o_assoc comp_fun_commute)
haftmann@35817
  1491
haftmann@42871
  1492
lemma comp_fun_commute':
haftmann@35719
  1493
  assumes "finite A"
haftmann@35719
  1494
  shows "f x \<circ> F A = F A \<circ> f x"
haftmann@35817
  1495
  using assms by (induct A)
haftmann@49739
  1496
    (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: comp_assoc comp_fun_commute)
haftmann@35817
  1497
haftmann@35817
  1498
lemma commute_left_comp':
haftmann@35817
  1499
  assumes "finite A"
haftmann@35817
  1500
  shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
haftmann@42871
  1501
  using assms by (simp add: o_assoc comp_fun_commute')
haftmann@35817
  1502
haftmann@42871
  1503
lemma comp_fun_commute'':
haftmann@35817
  1504
  assumes "finite A" and "finite B"
haftmann@35817
  1505
  shows "F B \<circ> F A = F A \<circ> F B"
haftmann@35817
  1506
  using assms by (induct A)
haftmann@49739
  1507
    (simp_all add: o_assoc, simp add: comp_assoc comp_fun_commute')
haftmann@35719
  1508
haftmann@35817
  1509
lemma commute_left_comp'':
haftmann@35817
  1510
  assumes "finite A" and "finite B"
haftmann@35817
  1511
  shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
haftmann@42871
  1512
  using assms by (simp add: o_assoc comp_fun_commute'')
haftmann@35817
  1513
haftmann@49739
  1514
lemmas comp_fun_commutes = comp_assoc comp_fun_commute commute_left_comp
haftmann@42871
  1515
  comp_fun_commute' commute_left_comp' comp_fun_commute'' commute_left_comp''
haftmann@35817
  1516
haftmann@35817
  1517
lemma union_inter:
haftmann@35817
  1518
  assumes "finite A" and "finite B"
haftmann@35817
  1519
  shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
haftmann@35817
  1520
  using assms by (induct A)
haftmann@42871
  1521
    (simp_all del: o_apply add: insert_absorb Int_insert_left comp_fun_commutes,
haftmann@35817
  1522
      simp add: o_assoc)
haftmann@35719
  1523
haftmann@35719
  1524
lemma union:
haftmann@35719
  1525
  assumes "finite A" and "finite B"
haftmann@35719
  1526
  and "A \<inter> B = {}"
haftmann@35719
  1527
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1528
proof -
haftmann@35817
  1529
  from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
haftmann@35817
  1530
  with `A \<inter> B = {}` show ?thesis by simp
haftmann@35719
  1531
qed
haftmann@35719
  1532
haftmann@34007
  1533
end
haftmann@35719
  1534
haftmann@35817
  1535
haftmann@35817
  1536
subsubsection {* The natural case with idempotency *}
haftmann@35817
  1537
haftmann@35719
  1538
locale folding_idem = folding +
haftmann@35719
  1539
  assumes idem_comp: "f x \<circ> f x = f x"
haftmann@35719
  1540
begin
haftmann@35719
  1541
haftmann@35817
  1542
lemma idem_left_comp:
haftmann@35817
  1543
  "f x \<circ> (f x \<circ> g) = f x \<circ> g"
haftmann@35817
  1544
  by (simp add: o_assoc idem_comp)
haftmann@35817
  1545
haftmann@35817
  1546
lemma in_comp_idem:
haftmann@35817
  1547
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1548
  shows "F A \<circ> f x = F A"
haftmann@35817
  1549
using assms by (induct A)
haftmann@42871
  1550
  (auto simp add: comp_fun_commutes idem_comp, simp add: commute_left_comp' [symmetric] comp_fun_commute')
haftmann@35719
  1551
haftmann@35817
  1552
lemma subset_comp_idem:
haftmann@35817
  1553
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1554
  shows "F A \<circ> F B = F A"
haftmann@35817
  1555
proof -
haftmann@35817
  1556
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1557
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1558
    (simp_all add: o_assoc in_comp_idem `finite A`)
haftmann@35817
  1559
qed
haftmann@35719
  1560
haftmann@35817
  1561
declare insert [simp del]
haftmann@35719
  1562
haftmann@35719
  1563
lemma insert_idem [simp]:
haftmann@35719
  1564
  assumes "finite A"
haftmann@35719
  1565
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35817
  1566
  using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
haftmann@35719
  1567
haftmann@35719
  1568
lemma union_idem:
haftmann@35719
  1569
  assumes "finite A" and "finite B"
haftmann@35719
  1570
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1571
proof -
haftmann@35817
  1572
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1573
  then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
haftmann@35817
  1574
  with assms show ?thesis by (simp add: union_inter)
haftmann@35719
  1575
qed
haftmann@35719
  1576
haftmann@35719
  1577
end
haftmann@35719
  1578
haftmann@35817
  1579
haftmann@35817
  1580
subsubsection {* The image case with fixed function *}
haftmann@35817
  1581
haftmann@35796
  1582
no_notation times (infixl "*" 70)
haftmann@35796
  1583
no_notation Groups.one ("1")
haftmann@35722
  1584
haftmann@35722
  1585
locale folding_image_simple = comm_monoid +
haftmann@35722
  1586
  fixes g :: "('b \<Rightarrow> 'a)"
haftmann@35722
  1587
  fixes F :: "'b set \<Rightarrow> 'a"
haftmann@35817
  1588
  assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
haftmann@35722
  1589
begin
haftmann@35722
  1590
haftmann@35722
  1591
lemma empty [simp]:
haftmann@35722
  1592
  "F {} = 1"
haftmann@35817
  1593
  by (simp add: eq_fold_g)
haftmann@35722
  1594
haftmann@35722
  1595
lemma insert [simp]:
haftmann@35722
  1596
  assumes "finite A" and "x \<notin> A"
haftmann@35722
  1597
  shows "F (insert x A) = g x * F A"
haftmann@35722
  1598
proof -
wenzelm@46898
  1599
  interpret comp_fun_commute "%x y. (g x) * y"
wenzelm@46898
  1600
    by default (simp add: ac_simps fun_eq_iff)
wenzelm@46898
  1601
  from assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
haftmann@35722
  1602
    by (simp add: fold_image_def)
haftmann@35817
  1603
  with `finite A` show ?thesis by (simp add: eq_fold_g)
haftmann@35722
  1604
qed
haftmann@35722
  1605
haftmann@35722
  1606
lemma remove:
haftmann@35722
  1607
  assumes "finite A" and "x \<in> A"
haftmann@35722
  1608
  shows "F A = g x * F (A - {x})"
haftmann@35722
  1609
proof -
haftmann@35722
  1610
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35722
  1611
    by (auto dest: mk_disjoint_insert)
haftmann@35722
  1612
  moreover from `finite A` this have "finite B" by simp
haftmann@35722
  1613
  ultimately show ?thesis by simp
haftmann@35722
  1614
qed
haftmann@35722
  1615
haftmann@35722
  1616
lemma insert_remove:
haftmann@35722
  1617
  assumes "finite A"
haftmann@35722
  1618
  shows "F (insert x A) = g x * F (A - {x})"
haftmann@35722
  1619
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35722
  1620
haftmann@35817
  1621
lemma neutral:
haftmann@35817
  1622
  assumes "finite A" and "\<forall>x\<in>A. g x = 1"
haftmann@35817
  1623
  shows "F A = 1"
haftmann@35817
  1624
  using assms by (induct A) simp_all
haftmann@35817
  1625
haftmann@35722
  1626
lemma union_inter:
haftmann@35722
  1627
  assumes "finite A" and "finite B"
haftmann@35817
  1628
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35722
  1629
using assms proof (induct A)
haftmann@35722
  1630
  case empty then show ?case by simp
haftmann@35722
  1631
next
haftmann@35722
  1632
  case (insert x A) then show ?case
haftmann@35722
  1633
    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
haftmann@35722
  1634
qed
haftmann@35722
  1635
haftmann@35817
  1636
corollary union_inter_neutral:
haftmann@35817
  1637
  assumes "finite A" and "finite B"
haftmann@35817
  1638
  and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
haftmann@35817
  1639
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1640
  using assms by (simp add: union_inter [symmetric] neutral)
haftmann@35817
  1641
haftmann@35722
  1642
corollary union_disjoint:
haftmann@35722
  1643
  assumes "finite A" and "finite B"
haftmann@35722
  1644
  assumes "A \<inter> B = {}"
haftmann@35722
  1645
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1646
  using assms by (simp add: union_inter_neutral)
haftmann@35722
  1647
haftmann@35719
  1648
end
haftmann@35722
  1649
haftmann@35817
  1650
haftmann@35817
  1651
subsubsection {* The image case with flexible function *}
haftmann@35817
  1652
haftmann@35722
  1653
locale folding_image = comm_monoid +
haftmann@35722
  1654
  fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@35722
  1655
  assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
haftmann@35722
  1656
haftmann@35722
  1657
sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
haftmann@35722
  1658
qed (fact eq_fold)
haftmann@35722
  1659
haftmann@35722
  1660
context folding_image
haftmann@35722
  1661
begin
haftmann@35722
  1662
haftmann@35817
  1663
lemma reindex: (* FIXME polymorhism *)
haftmann@35722
  1664
  assumes "finite A" and "inj_on h A"
haftmann@35722
  1665
  shows "F g (h ` A) = F (g \<circ> h) A"
haftmann@35722
  1666
  using assms by (induct A) auto
haftmann@35722
  1667
haftmann@35722
  1668
lemma cong:
haftmann@35722
  1669
  assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
haftmann@35722
  1670
  shows "F g A = F h A"
haftmann@35722
  1671
proof -
haftmann@35722
  1672
  from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
haftmann@35722
  1673
  apply - apply (erule finite_induct) apply simp
haftmann@35722
  1674
  apply (simp add: subset_insert_iff, clarify)
haftmann@35722
  1675
  apply (subgoal_tac "finite C")
wenzelm@48125
  1676
  prefer 2 apply (blast dest: finite_subset [rotated])
haftmann@35722
  1677
  apply (subgoal_tac "C = insert x (C - {x})")
haftmann@35722
  1678
  prefer 2 apply blast
haftmann@35722
  1679
  apply (erule ssubst)
haftmann@35722
  1680
  apply (drule spec)
haftmann@35722
  1681
  apply (erule (1) notE impE)
haftmann@35722
  1682
  apply (simp add: Ball_def del: insert_Diff_single)
haftmann@35722
  1683
  done
haftmann@35722
  1684
  with assms show ?thesis by simp
haftmann@35722
  1685
qed
haftmann@35722
  1686
haftmann@35722
  1687
lemma UNION_disjoint:
haftmann@35722
  1688
  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
haftmann@35722
  1689
  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
haftmann@35722
  1690
  shows "F g (UNION I A) = F (F g \<circ> A) I"
haftmann@35722
  1691
apply (insert assms)
haftmann@41656
  1692
apply (induct rule: finite_induct)
haftmann@41656
  1693
apply simp
haftmann@41656
  1694
apply atomize
haftmann@35722
  1695
apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
haftmann@35722
  1696
 prefer 2 apply blast
haftmann@35722
  1697
apply (subgoal_tac "A x Int UNION Fa A = {}")
haftmann@35722
  1698
 prefer 2 apply blast
haftmann@35722
  1699
apply (simp add: union_disjoint)
haftmann@35722
  1700
done
haftmann@35722
  1701
haftmann@35722
  1702
lemma distrib:
haftmann@35722
  1703
  assumes "finite A"
haftmann@35722
  1704
  shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
haftmann@35722
  1705
  using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
haftmann@35722
  1706
haftmann@35722
  1707
lemma related: 
haftmann@35722
  1708
  assumes Re: "R 1 1" 
haftmann@35722
  1709
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
haftmann@35722
  1710
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
haftmann@35722
  1711
  shows "R (F h S) (F g S)"
haftmann@35722
  1712
  using fS by (rule finite_subset_induct) (insert assms, auto)
haftmann@35722
  1713
haftmann@35722
  1714
lemma eq_general:
haftmann@35722
  1715
  assumes fS: "finite S"
haftmann@35722
  1716
  and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 
haftmann@35722
  1717
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
haftmann@35722
  1718
  shows "F f1 S = F f2 S'"
haftmann@35722
  1719
proof-
haftmann@35722
  1720
  from h f12 have hS: "h ` S = S'" by blast
haftmann@35722
  1721
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
haftmann@35722
  1722
    from f12 h H  have "x = y" by auto }
haftmann@35722
  1723
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
haftmann@35722
  1724
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
haftmann@35722
  1725
  from hS have "F f2 S' = F f2 (h ` S)" by simp
haftmann@35722
  1726
  also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
haftmann@35722
  1727
  also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
haftmann@35722
  1728
    by blast
haftmann@35722
  1729
  finally show ?thesis ..
haftmann@35722
  1730
qed
haftmann@35722
  1731
haftmann@35722
  1732
lemma eq_general_inverses:
haftmann@35722
  1733
  assumes fS: "finite S" 
haftmann@35722
  1734
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
haftmann@35722
  1735
  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
haftmann@35722
  1736
  shows "F j S = F g T"
haftmann@35722
  1737
  (* metis solves it, but not yet available here *)
haftmann@35722
  1738
  apply (rule eq_general [OF fS, of T h g j])
haftmann@35722
  1739
  apply (rule ballI)
haftmann@35722
  1740
  apply (frule kh)
haftmann@35722
  1741
  apply (rule ex1I[])
haftmann@35722
  1742
  apply blast
haftmann@35722
  1743
  apply clarsimp
haftmann@35722
  1744
  apply (drule hk) apply simp
haftmann@35722
  1745
  apply (rule sym)
haftmann@35722
  1746
  apply (erule conjunct1[OF conjunct2[OF hk]])
haftmann@35722
  1747
  apply (rule ballI)
haftmann@35722
  1748
  apply (drule hk)
haftmann@35722
  1749
  apply blast
haftmann@35722
  1750
  done
haftmann@35722
  1751
haftmann@35722
  1752
end
haftmann@35722
  1753
haftmann@35817
  1754
haftmann@35817
  1755
subsubsection {* The image case with fixed function and idempotency *}
haftmann@35817
  1756
haftmann@35817
  1757
locale folding_image_simple_idem = folding_image_simple +
haftmann@35817
  1758
  assumes idem: "x * x = x"
haftmann@35817
  1759
wenzelm@49756
  1760
sublocale folding_image_simple_idem < semilattice: semilattice proof
haftmann@35817
  1761
qed (fact idem)
haftmann@35817
  1762
haftmann@35817
  1763
context folding_image_simple_idem
haftmann@35817
  1764
begin
haftmann@35817
  1765
haftmann@35817
  1766
lemma in_idem:
haftmann@35817
  1767
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1768
  shows "g x * F A = F A"
haftmann@35817
  1769
  using assms by (induct A) (auto simp add: left_commute)
haftmann@35817
  1770
haftmann@35817
  1771
lemma subset_idem:
haftmann@35817
  1772
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1773
  shows "F B * F A = F A"
haftmann@35817
  1774
proof -
haftmann@35817
  1775
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1776
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1777
    (auto simp add: assoc in_idem `finite A`)
haftmann@35817
  1778
qed
haftmann@35817
  1779
haftmann@35817
  1780
declare insert [simp del]
haftmann@35817
  1781
haftmann@35817
  1782
lemma insert_idem [simp]:
haftmann@35817
  1783
  assumes "finite A"
haftmann@35817
  1784
  shows "F (insert x A) = g x * F A"
haftmann@35817
  1785
  using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
haftmann@35817
  1786
haftmann@35817
  1787
lemma union_idem:
haftmann@35817
  1788
  assumes "finite A" and "finite B"
haftmann@35817
  1789
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1790
proof -
haftmann@35817
  1791
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1792
  then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
haftmann@35817
  1793
  with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1794
qed
haftmann@35817
  1795
haftmann@35817
  1796
end
haftmann@35817
  1797
haftmann@35817
  1798
haftmann@35817
  1799
subsubsection {* The image case with flexible function and idempotency *}
haftmann@35817
  1800
haftmann@35817
  1801
locale folding_image_idem = folding_image +
haftmann@35817
  1802
  assumes idem: "x * x = x"
haftmann@35817
  1803
haftmann@35817
  1804
sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
haftmann@35817
  1805
qed (fact idem)
haftmann@35817
  1806
haftmann@35817
  1807
haftmann@35817
  1808
subsubsection {* The neutral-less case *}
haftmann@35817
  1809
haftmann@35817
  1810
locale folding_one = abel_semigroup +
haftmann@35817
  1811
  fixes F :: "'a set \<Rightarrow> 'a"
haftmann@35817
  1812
  assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
haftmann@35817
  1813
begin
haftmann@35817
  1814
haftmann@35817
  1815
lemma singleton [simp]:
haftmann@35817
  1816
  "F {x} = x"
haftmann@35817
  1817
  by (simp add: eq_fold)
haftmann@35817
  1818
haftmann@35817
  1819
lemma eq_fold':
haftmann@35817
  1820
  assumes "finite A" and "x \<notin> A"
haftmann@35817
  1821
  shows "F (insert x A) = fold (op *) x A"
haftmann@35817
  1822
proof -
wenzelm@46898
  1823
  interpret ab_semigroup_mult "op *" by default (simp_all add: ac_simps)
wenzelm@46898
  1824
  from assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
haftmann@35817
  1825
qed
haftmann@35817
  1826
haftmann@35817
  1827
lemma insert [simp]:
haftmann@36637
  1828
  assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
haftmann@36637
  1829
  shows "F (insert x A) = x * F A"
haftmann@36637
  1830
proof -
haftmann@36637
  1831
  from `A \<noteq> {}` obtain b where "b \<in> A" by blast
haftmann@35817
  1832
  then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1833
  with `finite A` have "finite B" by simp
haftmann@35817
  1834
  interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
nipkow@39302
  1835
  qed (simp_all add: fun_eq_iff ac_simps)
haftmann@42871
  1836
  from `finite B` fold.comp_fun_commute' [of B x]
haftmann@35817
  1837
    have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
nipkow@39302
  1838
  then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)
haftmann@35817
  1839
  from `finite B` * fold.insert [of B b]
haftmann@35817
  1840
    have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
nipkow@39302
  1841
  then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)
haftmann@35817
  1842
  from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
haftmann@35817
  1843
qed
haftmann@35817
  1844
haftmann@35817
  1845
lemma remove:
haftmann@35817
  1846
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1847
  shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1848
proof -
haftmann@35817
  1849
  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1850
  with assms show ?thesis by simp
haftmann@35817
  1851
qed
haftmann@35817
  1852
haftmann@35817
  1853
lemma insert_remove:
haftmann@35817
  1854
  assumes "finite A"
haftmann@35817
  1855
  shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1856
  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
haftmann@35817
  1857
haftmann@35817
  1858
lemma union_disjoint:
haftmann@35817
  1859
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
haftmann@35817
  1860
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1861
  using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
haftmann@35817
  1862
haftmann@35817
  1863
lemma union_inter:
haftmann@35817
  1864
  assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
haftmann@35817
  1865
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35817
  1866
proof -
haftmann@35817
  1867
  from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
haftmann@35817
  1868
  from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
haftmann@35817
  1869
    case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
haftmann@35817
  1870
  next
haftmann@35817
  1871
    case (insert x A) show ?case proof (cases "x \<in> B")
haftmann@35817
  1872
      case True then have "B \<noteq> {}" by auto
haftmann@35817
  1873
      with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
haftmann@35817
  1874
        (simp_all add: insert_absorb ac_simps union_disjoint)
haftmann@35817
  1875
    next
haftmann@35817
  1876
      case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
haftmann@35817
  1877
      moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
haftmann@35817
  1878
        by auto
haftmann@35817
  1879
      ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
haftmann@35817
  1880
    qed
haftmann@35817
  1881
  qed
haftmann@35817
  1882
qed
haftmann@35817
  1883
haftmann@35817
  1884
lemma closed:
haftmann@35817
  1885
  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
haftmann@35817
  1886
  shows "F A \<in> A"
haftmann@35817
  1887
using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
haftmann@35817
  1888
  case singleton then show ?case by simp
haftmann@35817
  1889
next
haftmann@35817
  1890
  case insert with elem show ?case by force
haftmann@35817
  1891
qed
haftmann@35817
  1892
haftmann@35817
  1893
end
haftmann@35817
  1894
haftmann@35817
  1895
haftmann@35817
  1896
subsubsection {* The neutral-less case with idempotency *}
haftmann@35817
  1897
haftmann@35817
  1898
locale folding_one_idem = folding_one +
haftmann@35817
  1899
  assumes idem: "x * x = x"
haftmann@35817
  1900
wenzelm@49756
  1901
sublocale folding_one_idem < semilattice: semilattice proof
haftmann@35817
  1902
qed (fact idem)
haftmann@35817
  1903
haftmann@35817
  1904
context folding_one_idem
haftmann@35817
  1905
begin
haftmann@35817
  1906
haftmann@35817
  1907
lemma in_idem:
haftmann@35817
  1908
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1909
  shows "x * F A = F A"
haftmann@35817
  1910
proof -
haftmann@35817
  1911
  from assms have "A \<noteq> {}" by auto
haftmann@35817
  1912
  with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
haftmann@35817
  1913
qed
haftmann@35817
  1914
haftmann@35817
  1915
lemma subset_idem:
haftmann@35817
  1916
  assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
haftmann@35817
  1917
  shows "F B * F A = F A"
haftmann@35817
  1918
proof -
haftmann@35817
  1919
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1920
  then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
haftmann@35817
  1921
    (simp_all add: assoc in_idem `finite A`)
haftmann@35817
  1922
qed
haftmann@35817
  1923
haftmann@35817
  1924
lemma eq_fold_idem':
haftmann@35817
  1925
  assumes "finite A"
haftmann@35817
  1926
  shows "F (insert a A) = fold (op *) a A"
haftmann@35817
  1927
proof -
wenzelm@46898
  1928
  interpret ab_semigroup_idem_mult "op *" by default (simp_all add: ac_simps)
wenzelm@46898
  1929
  from assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
haftmann@35817
  1930
qed
haftmann@35817
  1931
haftmann@35817
  1932
lemma insert_idem [simp]:
haftmann@36637
  1933
  assumes "finite A" and "A \<noteq> {}"
haftmann@36637
  1934
  shows "F (insert x A) = x * F A"
haftmann@35817
  1935
proof (cases "x \<in> A")
haftmann@36637
  1936
  case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
haftmann@35817
  1937
next
haftmann@36637
  1938
  case True
haftmann@36637
  1939
  from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
haftmann@35817
  1940
qed
haftmann@35817
  1941
  
haftmann@35817
  1942
lemma union_idem:
haftmann@35817
  1943
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
haftmann@35817
  1944
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1945
proof (cases "A \<inter> B = {}")
haftmann@35817
  1946
  case True with assms show ?thesis by (simp add: union_disjoint)
haftmann@35817
  1947
next
haftmann@35817
  1948
  case False
haftmann@35817
  1949
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1950
  with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
haftmann@35817
  1951
  with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1952
qed
haftmann@35817
  1953
haftmann@35817
  1954
lemma hom_commute:
haftmann@35817
  1955
  assumes hom: "\<And>x y. h (x * y) = h x * h y"
haftmann@35817
  1956
  and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
haftmann@35817
  1957
using N proof (induct rule: finite_ne_induct)
haftmann@35817
  1958
  case singleton thus ?case by simp
haftmann@35817
  1959
next
haftmann@35817
  1960
  case (insert n N)
haftmann@35817
  1961
  then have "h (F (insert n N)) = h (n * F N)" by simp
haftmann@35817
  1962
  also have "\<dots> = h n * h (F N)" by (rule hom)
haftmann@35817
  1963
  also have "h (F N) = F (h ` N)" by(rule insert)
haftmann@35817
  1964
  also have "h n * \<dots> = F (insert (h n) (h ` N))"
haftmann@35817
  1965
    using insert by(simp)
haftmann@35817
  1966
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@35817
  1967
  finally show ?case .
haftmann@35817
  1968
qed
haftmann@35817
  1969
haftmann@35817
  1970
end
haftmann@35817
  1971
haftmann@35796
  1972
notation times (infixl "*" 70)
haftmann@35796
  1973
notation Groups.one ("1")
haftmann@35722
  1974
haftmann@35722
  1975
haftmann@35722
  1976
subsection {* Finite cardinality *}
haftmann@35722
  1977
haftmann@35722
  1978
text {* This definition, although traditional, is ugly to work with:
haftmann@35722
  1979
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
haftmann@35722
  1980
But now that we have @{text fold_image} things are easy:
haftmann@35722
  1981
*}
haftmann@35722
  1982
haftmann@35722
  1983
definition card :: "'a set \<Rightarrow> nat" where
haftmann@35722
  1984
  "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
haftmann@35722
  1985
haftmann@37770
  1986
interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
haftmann@35722
  1987
qed (simp add: card_def)
haftmann@35722
  1988
haftmann@35722
  1989
lemma card_infinite [simp]:
haftmann@35722
  1990
  "\<not> finite A \<Longrightarrow> card A = 0"
haftmann@35722
  1991
  by (simp add: card_def)
haftmann@35722
  1992
haftmann@35722
  1993
lemma card_empty:
haftmann@35722
  1994
  "card {} = 0"
haftmann@35722
  1995
  by (fact card.empty)
haftmann@35722
  1996
haftmann@35722
  1997
lemma card_insert_disjoint:
haftmann@35722
  1998
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
haftmann@35722
  1999
  by simp
haftmann@35722
  2000
haftmann@35722
  2001
lemma card_insert_if:
haftmann@35722
  2002
  "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
haftmann@35722
  2003
  by auto (simp add: card.insert_remove card.remove)
haftmann@35722
  2004
haftmann@35722
  2005
lemma card_ge_0_finite:
haftmann@35722
  2006
  "card A > 0 \<Longrightarrow> finite A"
haftmann@35722
  2007
  by (rule ccontr) simp
haftmann@35722
  2008
blanchet@35828
  2009
lemma card_0_eq [simp, no_atp]:
haftmann@35722
  2010
  "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
haftmann@35722
  2011
  by (auto dest: mk_disjoint_insert)
haftmann@35722
  2012
haftmann@35722
  2013
lemma finite_UNIV_card_ge_0:
haftmann@35722
  2014
  "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
haftmann@35722
  2015
  by (rule ccontr) simp
haftmann@35722
  2016
haftmann@35722
  2017
lemma card_eq_0_iff:
haftmann@35722
  2018
  "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
haftmann@35722
  2019
  by auto
haftmann@35722
  2020
haftmann@35722
  2021
lemma card_gt_0_iff:
haftmann@35722
  2022
  "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
haftmann@35722
  2023
  by (simp add: neq0_conv [symmetric] card_eq_0_iff) 
haftmann@35722
  2024
haftmann@35722
  2025
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
haftmann@35722
  2026
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
haftmann@35722
  2027
apply(simp del:insert_Diff_single)
haftmann@35722
  2028
done
haftmann@35722
  2029
haftmann@35722
  2030
lemma card_Diff_singleton:
haftmann@35722
  2031
  "finite A ==> x: A ==> card (A - {x}) = card A - 1"
haftmann@35722
  2032
by (simp add: card_Suc_Diff1 [symmetric])
haftmann@35722
  2033
haftmann@35722
  2034
lemma card_Diff_singleton_if:
bulwahn@45166
  2035
  "finite A ==> card (A - {x}) = (if x : A then card A - 1 else card A)"
haftmann@35722
  2036
by (simp add: card_Diff_singleton)
haftmann@35722
  2037
haftmann@35722
  2038
lemma card_Diff_insert[simp]:
haftmann@35722
  2039
assumes "finite A" and "a:A" and "a ~: B"
haftmann@35722
  2040
shows "card(A - insert a B) = card(A - B) - 1"
haftmann@35722
  2041
proof -
haftmann@35722
  2042
  have "A - insert a B = (A - B) - {a}" using assms by blast
haftmann@35722
  2043
  then show ?thesis using assms by(simp add:card_Diff_singleton)
haftmann@35722
  2044
qed
haftmann@35722
  2045
haftmann@35722
  2046
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
haftmann@35722
  2047
by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  2048
haftmann@35722
  2049
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
haftmann@35722
  2050
by (simp add: card_insert_if)
haftmann@35722
  2051
nipkow@41987
  2052
lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
nipkow@41987
  2053
by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
nipkow@41987
  2054
nipkow@41988
  2055
lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
nipkow@41987
  2056
using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
nipkow@41987
  2057
haftmann@35722
  2058
lemma card_mono:
haftmann@35722
  2059
  assumes "finite B" and "A \<subseteq> B"
haftmann@35722
  2060
  shows "card A \<le> card B"
haftmann@35722
  2061
proof -
haftmann@35722
  2062
  from assms have "finite A" by (auto intro: finite_subset)
haftmann@35722
  2063
  then show ?thesis using assms proof (induct A arbitrary: B)
haftmann@35722
  2064
    case empty then show ?case by simp
haftmann@35722
  2065
  next
haftmann@35722
  2066
    case (insert x A)
haftmann@35722
  2067
    then have "x \<in> B" by simp
haftmann@35722
  2068
    from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
haftmann@35722
  2069
    with insert.hyps have "card A \<le> card (B - {x})" by auto
haftmann@35722
  2070
    with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
haftmann@35722
  2071
  qed
haftmann@35722
  2072
qed
haftmann@35722
  2073
haftmann@35722
  2074
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
haftmann@41656
  2075
apply (induct rule: finite_induct)
haftmann@41656
  2076
apply simp
haftmann@41656
  2077
apply clarify
haftmann@35722
  2078
apply (subgoal_tac "finite A & A - {x} <= F")
haftmann@35722
  2079
 prefer 2 apply (blast intro: finite_subset, atomize)
haftmann@35722
  2080
apply (drule_tac x = "A - {x}" in spec)
haftmann@35722
  2081
apply (simp add: card_Diff_singleton_if split add: split_if_asm)
haftmann@35722
  2082
apply (case_tac "card A", auto)
haftmann@35722
  2083
done
haftmann@35722
  2084
haftmann@35722
  2085
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
haftmann@35722
  2086
apply (simp add: psubset_eq linorder_not_le [symmetric])
haftmann@35722
  2087
apply (blast dest: card_seteq)
haftmann@35722
  2088
done
haftmann@35722
  2089
haftmann@35722
  2090
lemma card_Un_Int: "finite A ==> finite B
haftmann@35722
  2091
    ==> card A + card B = card (A Un B) + card (A Int B)"
haftmann@35817
  2092
  by (fact card.union_inter [symmetric])
haftmann@35722
  2093
haftmann@35722
  2094
lemma card_Un_disjoint: "finite A ==> finite B
haftmann@35722
  2095
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
haftmann@35722
  2096
  by (fact card.union_disjoint)
haftmann@35722
  2097
haftmann@35722
  2098
lemma card_Diff_subset:
haftmann@35722
  2099
  assumes "finite B" and "B \<subseteq> A"
haftmann@35722
  2100
  shows "card (A - B) = card A - card B"
haftmann@35722
  2101
proof (cases "finite A")
haftmann@35722
  2102
  case False with assms show ?thesis by simp
haftmann@35722
  2103
next
haftmann@35722
  2104
  case True with assms show ?thesis by (induct B arbitrary: A) simp_all
haftmann@35722
  2105
qed
haftmann@35722
  2106
haftmann@35722
  2107
lemma card_Diff_subset_Int:
haftmann@35722
  2108
  assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
haftmann@35722
  2109
proof -
haftmann@35722
  2110
  have "A - B = A - A \<inter> B" by auto
haftmann@35722
  2111
  thus ?thesis
haftmann@35722
  2112
    by (simp add: card_Diff_subset AB) 
haftmann@35722
  2113
qed
haftmann@35722
  2114
nipkow@40716
  2115
lemma diff_card_le_card_Diff:
nipkow@40716
  2116
assumes "finite B" shows "card A - card B \<le> card(A - B)"
nipkow@40716
  2117
proof-
nipkow@40716
  2118
  have "card A - card B \<le> card A - card (A \<inter> B)"
nipkow@40716
  2119
    using card_mono[OF assms Int_lower2, of A] by arith
nipkow@40716
  2120
  also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
nipkow@40716
  2121
  finally show ?thesis .
nipkow@40716
  2122
qed
nipkow@40716
  2123
haftmann@35722
  2124
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
haftmann@35722
  2125
apply (rule Suc_less_SucD)
haftmann@35722
  2126
apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  2127
done
haftmann@35722
  2128
haftmann@35722
  2129
lemma card_Diff2_less:
haftmann@35722
  2130
  "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
haftmann@35722
  2131
apply (case_tac "x = y")
haftmann@35722
  2132
 apply (simp add: card_Diff1_less del:card_Diff_insert)
haftmann@35722
  2133
apply (rule less_trans)
haftmann@35722
  2134
 prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
haftmann@35722
  2135
done
haftmann@35722
  2136
haftmann@35722
  2137
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
haftmann@35722
  2138
apply (case_tac "x : A")
haftmann@35722
  2139
 apply (simp_all add: card_Diff1_less less_imp_le)
haftmann@35722
  2140
done
haftmann@35722
  2141
haftmann@35722
  2142
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
haftmann@35722
  2143
by (erule psubsetI, blast)
haftmann@35722
  2144
haftmann@35722
  2145
lemma insert_partition:
haftmann@35722
  2146
  "\<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
  2147
  \<Longrightarrow> x \<inter> \<Union> F = {}"
haftmann@35722
  2148
by auto
haftmann@35722
  2149
haftmann@35722
  2150
lemma finite_psubset_induct[consumes 1, case_names psubset]:
urbanc@36079
  2151
  assumes fin: "finite A" 
urbanc@36079
  2152
  and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A" 
urbanc@36079
  2153
  shows "P A"
urbanc@36079
  2154
using fin
urbanc@36079
  2155
proof (induct A taking: card rule: measure_induct_rule)
haftmann@35722
  2156
  case (less A)
urbanc@36079
  2157
  have fin: "finite A" by fact
urbanc@36079
  2158
  have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
urbanc@36079
  2159
  { fix B 
urbanc@36079
  2160
    assume asm: "B \<subset> A"
urbanc@36079
  2161
    from asm have "card B < card A" using psubset_card_mono fin by blast
urbanc@36079
  2162
    moreover
urbanc@36079
  2163
    from asm have "B \<subseteq> A" by auto
urbanc@36079
  2164
    then have "finite B" using fin finite_subset by blast
urbanc@36079
  2165
    ultimately 
urbanc@36079
  2166
    have "P B" using ih by simp
urbanc@36079
  2167
  }
urbanc@36079
  2168
  with fin show "P A" using major by blast
haftmann@35722
  2169
qed
haftmann@35722
  2170
haftmann@35722
  2171
text{* main cardinality theorem *}
haftmann@35722
  2172
lemma card_partition [rule_format]:
haftmann@35722
  2173
  "finite C ==>
haftmann@35722
  2174
     finite (\<Union> C) -->
haftmann@35722
  2175
     (\<forall>c\<in>C. card c = k) -->
haftmann@35722
  2176
     (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
haftmann@35722
  2177
     k * card(C) = card (\<Union> C)"
haftmann@35722
  2178
apply (erule finite_induct, simp)
haftmann@35722
  2179
apply (simp add: card_Un_disjoint insert_partition 
haftmann@35722
  2180
       finite_subset [of _ "\<Union> (insert x F)"])
haftmann@35722
  2181
done
haftmann@35722
  2182
haftmann@35722
  2183
lemma card_eq_UNIV_imp_eq_UNIV:
haftmann@35722
  2184
  assumes fin: "finite (UNIV :: 'a set)"
haftmann@35722
  2185
  and card: "card A = card (UNIV :: 'a set)"
haftmann@35722
  2186
  shows "A = (UNIV :: 'a set)"
haftmann@35722
  2187
proof
haftmann@35722
  2188
  show "A \<subseteq> UNIV" by simp
haftmann@35722
  2189
  show "UNIV \<subseteq> A"
haftmann@35722
  2190
  proof
haftmann@35722
  2191
    fix x
haftmann@35722
  2192
    show "x \<in> A"
haftmann@35722
  2193
    proof (rule ccontr)
haftmann@35722
  2194
      assume "x \<notin> A"
haftmann@35722
  2195
      then have "A \<subset> UNIV" by auto
haftmann@35722
  2196
      with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
haftmann@35722
  2197
      with card show False by simp
haftmann@35722
  2198
    qed
haftmann@35722
  2199
  qed
haftmann@35722
  2200
qed
haftmann@35722
  2201
haftmann@35722
  2202
text{*The form of a finite set of given cardinality*}
haftmann@35722
  2203
haftmann@35722
  2204
lemma card_eq_SucD:
haftmann@35722
  2205
assumes "card A = Suc k"
haftmann@35722
  2206
shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
haftmann@35722
  2207
proof -
haftmann@35722
  2208
  have fin: "finite A" using assms by (auto intro: ccontr)
haftmann@35722
  2209
  moreover have "card A \<noteq> 0" using assms by auto
haftmann@35722
  2210
  ultimately obtain b where b: "b \<in> A" by auto
haftmann@35722
  2211
  show ?thesis
haftmann@35722
  2212
  proof (intro exI conjI)
haftmann@35722
  2213
    show "A = insert b (A-{b})" using b by blast
haftmann@35722
  2214
    show "b \<notin> A - {b}" by blast
haftmann@35722
  2215
    show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
nipkow@44890
  2216
      using assms b fin by(fastforce dest:mk_disjoint_insert)+
haftmann@35722
  2217
  qed
haftmann@35722
  2218
qed
haftmann@35722
  2219
haftmann@35722
  2220
lemma card_Suc_eq:
haftmann@35722
  2221
  "(card A = Suc k) =
haftmann@35722
  2222
   (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
haftmann@35722
  2223
apply(rule iffI)
haftmann@35722
  2224
 apply(erule card_eq_SucD)
haftmann@35722
  2225
apply(auto)
haftmann@35722
  2226
apply(subst card_insert)
haftmann@35722
  2227
 apply(auto intro:ccontr)
haftmann@35722
  2228
done
haftmann@35722
  2229
nipkow@44744
  2230
lemma card_le_Suc_iff: "finite A \<Longrightarrow>
nipkow@44744
  2231
  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
  2232
by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
nipkow@44744
  2233
  dest: subset_singletonD split: nat.splits if_splits)
nipkow@44744
  2234
haftmann@35722
  2235
lemma finite_fun_UNIVD2:
haftmann@35722
  2236
  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
haftmann@35722
  2237
  shows "finite (UNIV :: 'b set)"
haftmann@35722
  2238
proof -
haftmann@46146
  2239
  from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
haftmann@46146
  2240
    by (rule finite_imageI)
haftmann@46146
  2241
  moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
haftmann@46146
  2242
    by (rule UNIV_eq_I) auto
haftmann@35722
  2243
  ultimately show "finite (UNIV :: 'b set)" by simp
haftmann@35722
  2244
qed
haftmann@35722
  2245
huffman@48063
  2246
lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
haftmann@35722
  2247
  unfolding UNIV_unit by simp
haftmann@35722
  2248
huffman@47210
  2249
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
huffman@47210
  2250
  unfolding UNIV_bool by simp
huffman@47210
  2251
haftmann@35722
  2252
haftmann@35722
  2253
subsubsection {* Cardinality of image *}
haftmann@35722
  2254
haftmann@35722
  2255
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
haftmann@41656
  2256
apply (induct rule: finite_induct)
haftmann@35722
  2257
 apply simp
haftmann@35722
  2258
apply (simp add: le_SucI card_insert_if)
haftmann@35722
  2259
done
haftmann@35722
  2260
haftmann@35722
  2261
lemma card_image:
haftmann@35722
  2262
  assumes "inj_on f A"
haftmann@35722
  2263
  shows "card (f ` A) = card A"
haftmann@35722
  2264
proof (cases "finite A")
haftmann@35722
  2265
  case True then show ?thesis using assms by (induct A) simp_all
haftmann@35722
  2266
next
haftmann@35722
  2267
  case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
haftmann@35722
  2268
  with False show ?thesis by simp
haftmann@35722
  2269
qed
haftmann@35722
  2270
haftmann@35722
  2271
lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
haftmann@35722
  2272
by(auto simp: card_image bij_betw_def)
haftmann@35722
  2273
haftmann@35722
  2274
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
haftmann@35722
  2275
by (simp add: card_seteq card_image)
haftmann@35722
  2276
haftmann@35722
  2277
lemma eq_card_imp_inj_on:
haftmann@35722
  2278
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
haftmann@35722
  2279
apply (induct rule:finite_induct)
haftmann@35722
  2280
apply simp
haftmann@35722
  2281
apply(frule card_image_le[where f = f])
haftmann@35722
  2282
apply(simp add:card_insert_if split:if_splits)
haftmann@35722
  2283
done
haftmann@35722
  2284
haftmann@35722
  2285
lemma inj_on_iff_eq_card:
haftmann@35722
  2286
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
haftmann@35722
  2287
by(blast intro: card_image eq_card_imp_inj_on)
haftmann@35722
  2288
haftmann@35722
  2289
haftmann@35722
  2290
lemma card_inj_on_le:
haftmann@35722
  2291
  "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
haftmann@35722
  2292
apply (subgoal_tac "finite A") 
haftmann@35722
  2293
 apply (force intro: card_mono simp add: card_image [symmetric])
haftmann@35722
  2294
apply (blast intro: finite_imageD dest: finite_subset) 
haftmann@35722
  2295
done
haftmann@35722
  2296
haftmann@35722
  2297
lemma card_bij_eq:
haftmann@35722
  2298
  "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
haftmann@35722
  2299
     finite A; finite B |] ==> card A = card B"
haftmann@35722
  2300
by (auto intro: le_antisym card_inj_on_le)
haftmann@35722
  2301
hoelzl@40703
  2302
lemma bij_betw_finite:
hoelzl@40703
  2303
  assumes "bij_betw f A B"
hoelzl@40703
  2304
  shows "finite A \<longleftrightarrow> finite B"
hoelzl@40703
  2305
using assms unfolding bij_betw_def
hoelzl@40703
  2306
using finite_imageD[of f A] by auto
haftmann@35722
  2307
haftmann@41656
  2308
nipkow@37466
  2309
subsubsection {* Pigeonhole Principles *}
nipkow@37466
  2310
nipkow@40311
  2311
lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
nipkow@37466
  2312
by (auto dest: card_image less_irrefl_nat)
nipkow@37466
  2313
nipkow@37466
  2314
lemma pigeonhole_infinite:
nipkow@37466
  2315
assumes  "~ finite A" and "finite(f`A)"
nipkow@37466
  2316
shows "EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  2317
proof -
nipkow@37466
  2318
  have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  2319
  proof(induct "f`A" arbitrary: A rule: finite_induct)
nipkow@37466
  2320
    case empty thus ?case by simp
nipkow@37466
  2321
  next
nipkow@37466
  2322
    case (insert b F)
nipkow@37466
  2323
    show ?case
nipkow@37466
  2324
    proof cases
nipkow@37466
  2325
      assume "finite{a:A. f a = b}"
nipkow@37466
  2326
      hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
nipkow@37466
  2327
      also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
nipkow@37466
  2328
      finally have "~ finite({a:A. f a \<noteq> b})" .
nipkow@37466
  2329
      from insert(3)[OF _ this]
nipkow@37466
  2330
      show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
nipkow@37466
  2331
    next
nipkow@37466
  2332
      assume 1: "~finite{a:A. f a = b}"
nipkow@37466
  2333
      hence "{a \<in> A. f a = b} \<noteq> {}" by force
nipkow@37466
  2334
      thus ?thesis using 1 by blast
nipkow@37466
  2335
    qed
nipkow@37466
  2336
  qed
nipkow@37466
  2337
  from this[OF assms(2,1)] show ?thesis .
nipkow@37466
  2338
qed
nipkow@37466
  2339
nipkow@37466
  2340
lemma pigeonhole_infinite_rel:
nipkow@37466
  2341
assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
nipkow@37466
  2342
shows "EX b:B. ~finite{a:A. R a b}"
nipkow@37466
  2343
proof -
nipkow@37466
  2344
   let ?F = "%a. {b:B. R a b}"
nipkow@37466
  2345
   from finite_Pow_iff[THEN iffD2, OF `finite B`]
nipkow@37466
  2346
   have "finite(?F ` A)" by(blast intro: rev_finite_subset)
nipkow@37466
  2347
   from pigeonhole_infinite[where f = ?F, OF assms(1) this]
nipkow@37466
  2348
   obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
nipkow@37466
  2349
   obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
nipkow@37466
  2350
   { assume "finite{a:A. R a b0}"
nipkow@37466
  2351
     then have "finite {a\<in>A. ?F a = ?F a0}"
nipkow@37466
  2352
       using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
nipkow@37466
  2353
   }
nipkow@37466
  2354
   with 1 `b0 : B` show ?thesis by blast
nipkow@37466
  2355
qed
nipkow@37466
  2356
nipkow@37466
  2357
haftmann@35722
  2358
subsubsection {* Cardinality of sums *}
haftmann@35722
  2359
haftmann@35722
  2360
lemma card_Plus:
haftmann@35722
  2361
  assumes "finite A" and "finite B"
haftmann@35722
  2362
  shows "card (A <+> B) = card A + card B"
haftmann@35722
  2363
proof -
haftmann@35722
  2364
  have "Inl`A \<inter> Inr`B = {}" by fast
haftmann@35722
  2365
  with assms show ?thesis
haftmann@35722
  2366
    unfolding Plus_def
haftmann@35722
  2367
    by (simp add: card_Un_disjoint card_image)
haftmann@35722
  2368
qed
haftmann@35722
  2369
haftmann@35722
  2370
lemma card_Plus_conv_if:
haftmann@35722
  2371
  "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
haftmann@35722
  2372
  by (auto simp add: card_Plus)
haftmann@35722
  2373
haftmann@35722
  2374
haftmann@35722
  2375
subsubsection {* Cardinality of the Powerset *}
haftmann@35722
  2376
huffman@47221
  2377
lemma card_Pow: "finite A ==> card (Pow A) = 2 ^ card A"
haftmann@41656
  2378
apply (induct rule: finite_induct)
haftmann@35722
  2379
 apply (simp_all add: Pow_insert)
haftmann@35722
  2380
apply (subst card_Un_disjoint, blast)
nipkow@40786
  2381
  apply (blast, blast)
haftmann@35722
  2382
apply (subgoal_tac "inj_on (insert x) (Pow F)")
huffman@47221
  2383
 apply (subst mult_2)
haftmann@35722
  2384
 apply (simp add: card_image Pow_insert)
haftmann@35722
  2385
apply (unfold inj_on_def)
haftmann@35722
  2386
apply (blast elim!: equalityE)
haftmann@35722
  2387
done
haftmann@35722
  2388
nipkow@41987
  2389
text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
haftmann@35722
  2390
haftmann@35722
  2391
lemma dvd_partition:
haftmann@35722
  2392
  "finite (Union C) ==>
haftmann@35722
  2393
    ALL c : C. k dvd card c ==>
haftmann@35722
  2394
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
haftmann@35722
  2395
  k dvd card (Union C)"
haftmann@41656
  2396
apply (frule finite_UnionD)
haftmann@41656
  2397
apply (rotate_tac -1)
haftmann@41656
  2398
apply (induct rule: finite_induct)
haftmann@41656
  2399
apply simp_all
haftmann@41656
  2400
apply clarify
haftmann@35722
  2401
apply (subst card_Un_disjoint)
haftmann@35722
  2402
   apply (auto simp add: disjoint_eq_subset_Compl)
haftmann@35722
  2403
done
haftmann@35722
  2404
haftmann@35722
  2405
haftmann@35722
  2406
subsubsection {* Relating injectivity and surjectivity *}
haftmann@35722
  2407
haftmann@41656
  2408
lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
haftmann@35722
  2409
apply(rule eq_card_imp_inj_on, assumption)
haftmann@35722
  2410
apply(frule finite_imageI)
haftmann@35722
  2411
apply(drule (1) card_seteq)
haftmann@35722
  2412
 apply(erule card_image_le)
haftmann@35722
  2413
apply simp
haftmann@35722
  2414
done
haftmann@35722
  2415
haftmann@35722
  2416
lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2417
shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
hoelzl@40702
  2418
by (blast intro: finite_surj_inj subset_UNIV)
haftmann@35722
  2419
haftmann@35722
  2420
lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2421
shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
nipkow@44890
  2422
by(fastforce simp:surj_def dest!: endo_inj_surj)
haftmann@35722
  2423
haftmann@35722
  2424
corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
haftmann@35722
  2425
proof
haftmann@35722
  2426
  assume "finite(UNIV::nat set)"
haftmann@35722
  2427
  with finite_UNIV_inj_surj[of Suc]
haftmann@35722
  2428
  show False by simp (blast dest: Suc_neq_Zero surjD)
haftmann@35722
  2429
qed
haftmann@35722
  2430
blanchet@35828
  2431
(* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
blanchet@35828
  2432
lemma infinite_UNIV_char_0[no_atp]:
haftmann@35722
  2433
  "\<not> finite (UNIV::'a::semiring_char_0 set)"
haftmann@35722
  2434
proof
haftmann@35722
  2435
  assume "finite (UNIV::'a set)"
haftmann@35722
  2436
  with subset_UNIV have "finite (range of_nat::'a set)"
haftmann@35722
  2437
    by (rule finite_subset)
haftmann@35722
  2438
  moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
haftmann@35722
  2439
    by (simp add: inj_on_def)
haftmann@35722
  2440
  ultimately have "finite (UNIV::nat set)"
haftmann@35722
  2441
    by (rule finite_imageD)
haftmann@35722
  2442
  then show "False"
haftmann@35722
  2443
    by simp
haftmann@35722
  2444
qed
haftmann@35722
  2445
kuncar@49758
  2446
hide_const (open) Finite_Set.fold
haftmann@46033
  2447
haftmann@35722
  2448
end
haftmann@49723
  2449