src/HOL/Finite_Set.thy
author haftmann
Fri Jan 06 21:48:45 2012 +0100 (2012-01-06)
changeset 46146 6baea4fca6bd
parent 46033 6fc579c917b8
child 46898 1570b30ee040
permissions -rw-r--r--
incorporated various theorems from theory More_Set into corpus
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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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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` 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 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 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 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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lemma finite_Collect_bounded_ex [simp]:
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  assumes "finite {y. P y}"
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  shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
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   318
proof -
haftmann@41656
   319
  have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
haftmann@41656
   320
  with assms show ?thesis by simp
haftmann@41656
   321
qed
nipkow@29920
   322
haftmann@41656
   323
lemma finite_Plus:
haftmann@41656
   324
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
haftmann@41656
   325
  by (simp add: Plus_def)
nipkow@17022
   326
nipkow@31080
   327
lemma finite_PlusD: 
nipkow@31080
   328
  fixes A :: "'a set" and B :: "'b set"
nipkow@31080
   329
  assumes fin: "finite (A <+> B)"
nipkow@31080
   330
  shows "finite A" "finite B"
nipkow@31080
   331
proof -
nipkow@31080
   332
  have "Inl ` A \<subseteq> A <+> B" by auto
haftmann@41656
   333
  then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
haftmann@41656
   334
  then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
nipkow@31080
   335
next
nipkow@31080
   336
  have "Inr ` B \<subseteq> A <+> B" by auto
haftmann@41656
   337
  then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
haftmann@41656
   338
  then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
nipkow@31080
   339
qed
nipkow@31080
   340
haftmann@41656
   341
lemma finite_Plus_iff [simp]:
haftmann@41656
   342
  "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
haftmann@41656
   343
  by (auto intro: finite_PlusD finite_Plus)
nipkow@31080
   344
haftmann@41656
   345
lemma finite_Plus_UNIV_iff [simp]:
haftmann@41656
   346
  "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
haftmann@41656
   347
  by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
wenzelm@12396
   348
nipkow@40786
   349
lemma finite_SigmaI [simp, intro]:
haftmann@41656
   350
  "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
nipkow@40786
   351
  by (unfold Sigma_def) blast
wenzelm@12396
   352
haftmann@41656
   353
lemma finite_cartesian_product:
haftmann@41656
   354
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
nipkow@15402
   355
  by (rule finite_SigmaI)
nipkow@15402
   356
wenzelm@12396
   357
lemma finite_Prod_UNIV:
haftmann@41656
   358
  "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
haftmann@41656
   359
  by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
wenzelm@12396
   360
paulson@15409
   361
lemma finite_cartesian_productD1:
haftmann@42207
   362
  assumes "finite (A \<times> B)" and "B \<noteq> {}"
haftmann@42207
   363
  shows "finite A"
haftmann@42207
   364
proof -
haftmann@42207
   365
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
haftmann@42207
   366
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   367
  then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
haftmann@42207
   368
  with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
haftmann@42207
   369
    by (simp add: image_compose)
haftmann@42207
   370
  then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
haftmann@42207
   371
  then show ?thesis
haftmann@42207
   372
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   373
qed
paulson@15409
   374
paulson@15409
   375
lemma finite_cartesian_productD2:
haftmann@42207
   376
  assumes "finite (A \<times> B)" and "A \<noteq> {}"
haftmann@42207
   377
  shows "finite B"
haftmann@42207
   378
proof -
haftmann@42207
   379
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
haftmann@42207
   380
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   381
  then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
haftmann@42207
   382
  with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
haftmann@42207
   383
    by (simp add: image_compose)
haftmann@42207
   384
  then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
haftmann@42207
   385
  then show ?thesis
haftmann@42207
   386
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   387
qed
paulson@15409
   388
haftmann@41656
   389
lemma finite_Pow_iff [iff]:
haftmann@41656
   390
  "finite (Pow A) \<longleftrightarrow> finite A"
wenzelm@12396
   391
proof
wenzelm@12396
   392
  assume "finite (Pow A)"
haftmann@41656
   393
  then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
haftmann@41656
   394
  then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
wenzelm@12396
   395
next
wenzelm@12396
   396
  assume "finite A"
haftmann@41656
   397
  then show "finite (Pow A)"
huffman@35216
   398
    by induct (simp_all add: Pow_insert)
wenzelm@12396
   399
qed
wenzelm@12396
   400
haftmann@41656
   401
corollary finite_Collect_subsets [simp, intro]:
haftmann@41656
   402
  "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
haftmann@41656
   403
  by (simp add: Pow_def [symmetric])
nipkow@29918
   404
nipkow@15392
   405
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
haftmann@41656
   406
  by (blast intro: finite_subset [OF subset_Pow_Union])
nipkow@15392
   407
nipkow@15392
   408
haftmann@41656
   409
subsubsection {* Further induction rules on finite sets *}
haftmann@41656
   410
haftmann@41656
   411
lemma finite_ne_induct [case_names singleton insert, consumes 2]:
haftmann@41656
   412
  assumes "finite F" and "F \<noteq> {}"
haftmann@41656
   413
  assumes "\<And>x. P {x}"
haftmann@41656
   414
    and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
haftmann@41656
   415
  shows "P F"
haftmann@41656
   416
using assms proof induct
haftmann@41656
   417
  case empty then show ?case by simp
haftmann@41656
   418
next
haftmann@41656
   419
  case (insert x F) then show ?case by cases auto
haftmann@41656
   420
qed
haftmann@41656
   421
haftmann@41656
   422
lemma finite_subset_induct [consumes 2, case_names empty insert]:
haftmann@41656
   423
  assumes "finite F" and "F \<subseteq> A"
haftmann@41656
   424
  assumes empty: "P {}"
haftmann@41656
   425
    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
   426
  shows "P F"
haftmann@41656
   427
using `finite F` `F \<subseteq> A` proof induct
haftmann@41656
   428
  show "P {}" by fact
nipkow@31441
   429
next
haftmann@41656
   430
  fix x F
haftmann@41656
   431
  assume "finite F" and "x \<notin> F" and
haftmann@41656
   432
    P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
haftmann@41656
   433
  show "P (insert x F)"
haftmann@41656
   434
  proof (rule insert)
haftmann@41656
   435
    from i show "x \<in> A" by blast
haftmann@41656
   436
    from i have "F \<subseteq> A" by blast
haftmann@41656
   437
    with P show "P F" .
haftmann@41656
   438
    show "finite F" by fact
haftmann@41656
   439
    show "x \<notin> F" by fact
haftmann@41656
   440
  qed
haftmann@41656
   441
qed
haftmann@41656
   442
haftmann@41656
   443
lemma finite_empty_induct:
haftmann@41656
   444
  assumes "finite A"
haftmann@41656
   445
  assumes "P A"
haftmann@41656
   446
    and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
haftmann@41656
   447
  shows "P {}"
haftmann@41656
   448
proof -
haftmann@41656
   449
  have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
haftmann@41656
   450
  proof -
haftmann@41656
   451
    fix B :: "'a set"
haftmann@41656
   452
    assume "B \<subseteq> A"
haftmann@41656
   453
    with `finite A` have "finite B" by (rule rev_finite_subset)
haftmann@41656
   454
    from this `B \<subseteq> A` show "P (A - B)"
haftmann@41656
   455
    proof induct
haftmann@41656
   456
      case empty
haftmann@41656
   457
      from `P A` show ?case by simp
haftmann@41656
   458
    next
haftmann@41656
   459
      case (insert b B)
haftmann@41656
   460
      have "P (A - B - {b})"
haftmann@41656
   461
      proof (rule remove)
haftmann@41656
   462
        from `finite A` show "finite (A - B)" by induct auto
haftmann@41656
   463
        from insert show "b \<in> A - B" by simp
haftmann@41656
   464
        from insert show "P (A - B)" by simp
haftmann@41656
   465
      qed
haftmann@41656
   466
      also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
haftmann@41656
   467
      finally show ?case .
haftmann@41656
   468
    qed
haftmann@41656
   469
  qed
haftmann@41656
   470
  then have "P (A - A)" by blast
haftmann@41656
   471
  then show ?thesis by simp
nipkow@31441
   472
qed
nipkow@31441
   473
nipkow@31441
   474
haftmann@26441
   475
subsection {* Class @{text finite}  *}
haftmann@26041
   476
haftmann@29797
   477
class finite =
haftmann@26041
   478
  assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
huffman@27430
   479
begin
huffman@27430
   480
huffman@27430
   481
lemma finite [simp]: "finite (A \<Colon> 'a set)"
haftmann@26441
   482
  by (rule subset_UNIV finite_UNIV finite_subset)+
haftmann@26041
   483
haftmann@43866
   484
lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
bulwahn@40922
   485
  by simp
bulwahn@40922
   486
huffman@27430
   487
end
huffman@27430
   488
haftmann@37678
   489
instance prod :: (finite, finite) finite proof
haftmann@35719
   490
qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
haftmann@26146
   491
haftmann@26041
   492
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
nipkow@39302
   493
  by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
haftmann@26041
   494
haftmann@26146
   495
instance "fun" :: (finite, finite) finite
haftmann@26146
   496
proof
haftmann@26041
   497
  show "finite (UNIV :: ('a => 'b) set)"
haftmann@26041
   498
  proof (rule finite_imageD)
haftmann@26041
   499
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
berghofe@26792
   500
    have "range ?graph \<subseteq> Pow UNIV" by simp
berghofe@26792
   501
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
berghofe@26792
   502
      by (simp only: finite_Pow_iff finite)
berghofe@26792
   503
    ultimately show "finite (range ?graph)"
berghofe@26792
   504
      by (rule finite_subset)
haftmann@26041
   505
    show "inj ?graph" by (rule inj_graph)
haftmann@26041
   506
  qed
haftmann@26041
   507
qed
haftmann@26041
   508
haftmann@44831
   509
instance bool :: finite proof
haftmann@44831
   510
qed (simp add: UNIV_bool)
haftmann@44831
   511
haftmann@45962
   512
instance set :: (finite) finite
haftmann@45962
   513
  by default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
haftmann@45962
   514
haftmann@44831
   515
instance unit :: finite proof
haftmann@44831
   516
qed (simp add: UNIV_unit)
haftmann@44831
   517
haftmann@37678
   518
instance sum :: (finite, finite) finite proof
haftmann@35719
   519
qed (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
haftmann@27981
   520
haftmann@44831
   521
lemma finite_option_UNIV [simp]:
haftmann@44831
   522
  "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
haftmann@44831
   523
  by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
haftmann@44831
   524
haftmann@44831
   525
instance option :: (finite) finite proof
haftmann@44831
   526
qed (simp add: UNIV_option_conv)
haftmann@44831
   527
haftmann@26041
   528
haftmann@35817
   529
subsection {* A basic fold functional for finite sets *}
nipkow@15392
   530
nipkow@15392
   531
text {* The intended behaviour is
wenzelm@31916
   532
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
nipkow@28853
   533
if @{text f} is ``left-commutative'':
nipkow@15392
   534
*}
nipkow@15392
   535
haftmann@42871
   536
locale comp_fun_commute =
nipkow@28853
   537
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@42871
   538
  assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
nipkow@28853
   539
begin
nipkow@28853
   540
haftmann@42809
   541
lemma fun_left_comm: "f x (f y z) = f y (f x z)"
haftmann@42871
   542
  using comp_fun_commute by (simp add: fun_eq_iff)
nipkow@28853
   543
nipkow@28853
   544
end
nipkow@28853
   545
nipkow@28853
   546
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
nipkow@28853
   547
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
nipkow@28853
   548
  emptyI [intro]: "fold_graph f z {} z" |
nipkow@28853
   549
  insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
nipkow@28853
   550
      \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
nipkow@28853
   551
nipkow@28853
   552
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
nipkow@28853
   553
nipkow@28853
   554
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
haftmann@37767
   555
  "fold f z A = (THE y. fold_graph f z A y)"
nipkow@15392
   556
paulson@15498
   557
text{*A tempting alternative for the definiens is
nipkow@28853
   558
@{term "if finite A then THE y. fold_graph f z A y else e"}.
paulson@15498
   559
It allows the removal of finiteness assumptions from the theorems
nipkow@28853
   560
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
nipkow@28853
   561
The proofs become ugly. It is not worth the effort. (???) *}
nipkow@28853
   562
nipkow@28853
   563
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
haftmann@41656
   564
by (induct rule: finite_induct) auto
nipkow@28853
   565
nipkow@28853
   566
nipkow@28853
   567
subsubsection{*From @{const fold_graph} to @{term fold}*}
nipkow@15392
   568
haftmann@42871
   569
context comp_fun_commute
haftmann@26041
   570
begin
haftmann@26041
   571
huffman@36045
   572
lemma fold_graph_insertE_aux:
huffman@36045
   573
  "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
   574
proof (induct set: fold_graph)
huffman@36045
   575
  case (insertI x A y) show ?case
huffman@36045
   576
  proof (cases "x = a")
huffman@36045
   577
    assume "x = a" with insertI show ?case by auto
nipkow@28853
   578
  next
huffman@36045
   579
    assume "x \<noteq> a"
huffman@36045
   580
    then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
huffman@36045
   581
      using insertI by auto
haftmann@42875
   582
    have "f x y = f a (f x y')"
huffman@36045
   583
      unfolding y by (rule fun_left_comm)
haftmann@42875
   584
    moreover have "fold_graph f z (insert x A - {a}) (f x y')"
huffman@36045
   585
      using y' and `x \<noteq> a` and `x \<notin> A`
huffman@36045
   586
      by (simp add: insert_Diff_if fold_graph.insertI)
haftmann@42875
   587
    ultimately show ?case by fast
nipkow@15392
   588
  qed
huffman@36045
   589
qed simp
huffman@36045
   590
huffman@36045
   591
lemma fold_graph_insertE:
huffman@36045
   592
  assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
huffman@36045
   593
  obtains y where "v = f x y" and "fold_graph f z A y"
huffman@36045
   594
using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
nipkow@28853
   595
nipkow@28853
   596
lemma fold_graph_determ:
nipkow@28853
   597
  "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
huffman@36045
   598
proof (induct arbitrary: y set: fold_graph)
huffman@36045
   599
  case (insertI x A y v)
huffman@36045
   600
  from `fold_graph f z (insert x A) v` and `x \<notin> A`
huffman@36045
   601
  obtain y' where "v = f x y'" and "fold_graph f z A y'"
huffman@36045
   602
    by (rule fold_graph_insertE)
huffman@36045
   603
  from `fold_graph f z A y'` have "y' = y" by (rule insertI)
huffman@36045
   604
  with `v = f x y'` show "v = f x y" by simp
huffman@36045
   605
qed fast
nipkow@15392
   606
nipkow@28853
   607
lemma fold_equality:
nipkow@28853
   608
  "fold_graph f z A y \<Longrightarrow> fold f z A = y"
nipkow@28853
   609
by (unfold fold_def) (blast intro: fold_graph_determ)
nipkow@15392
   610
haftmann@42272
   611
lemma fold_graph_fold:
haftmann@42272
   612
  assumes "finite A"
haftmann@42272
   613
  shows "fold_graph f z A (fold f z A)"
haftmann@42272
   614
proof -
haftmann@42272
   615
  from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
haftmann@42272
   616
  moreover note fold_graph_determ
haftmann@42272
   617
  ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
haftmann@42272
   618
  then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
haftmann@42272
   619
  then show ?thesis by (unfold fold_def)
haftmann@42272
   620
qed
huffman@36045
   621
nipkow@15392
   622
text{* The base case for @{text fold}: *}
nipkow@15392
   623
nipkow@28853
   624
lemma (in -) fold_empty [simp]: "fold f z {} = z"
nipkow@28853
   625
by (unfold fold_def) blast
nipkow@28853
   626
nipkow@28853
   627
text{* The various recursion equations for @{const fold}: *}
nipkow@28853
   628
haftmann@26041
   629
lemma fold_insert [simp]:
haftmann@42875
   630
  assumes "finite A" and "x \<notin> A"
haftmann@42875
   631
  shows "fold f z (insert x A) = f x (fold f z A)"
haftmann@42875
   632
proof (rule fold_equality)
haftmann@42875
   633
  from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
haftmann@42875
   634
  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
   635
qed
nipkow@28853
   636
nipkow@28853
   637
lemma fold_fun_comm:
nipkow@28853
   638
  "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
nipkow@28853
   639
proof (induct rule: finite_induct)
nipkow@28853
   640
  case empty then show ?case by simp
nipkow@28853
   641
next
nipkow@28853
   642
  case (insert y A) then show ?case
nipkow@28853
   643
    by (simp add: fun_left_comm[of x])
nipkow@28853
   644
qed
nipkow@28853
   645
nipkow@28853
   646
lemma fold_insert2:
nipkow@28853
   647
  "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
huffman@35216
   648
by (simp add: fold_fun_comm)
nipkow@15392
   649
haftmann@26041
   650
lemma fold_rec:
haftmann@42875
   651
  assumes "finite A" and "x \<in> A"
haftmann@42875
   652
  shows "fold f z A = f x (fold f z (A - {x}))"
nipkow@28853
   653
proof -
nipkow@28853
   654
  have A: "A = insert x (A - {x})" using `x \<in> A` by blast
nipkow@28853
   655
  then have "fold f z A = fold f z (insert x (A - {x}))" by simp
nipkow@28853
   656
  also have "\<dots> = f x (fold f z (A - {x}))"
nipkow@28853
   657
    by (rule fold_insert) (simp add: `finite A`)+
nipkow@15535
   658
  finally show ?thesis .
nipkow@15535
   659
qed
nipkow@15535
   660
nipkow@28853
   661
lemma fold_insert_remove:
nipkow@28853
   662
  assumes "finite A"
nipkow@28853
   663
  shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
nipkow@28853
   664
proof -
nipkow@28853
   665
  from `finite A` have "finite (insert x A)" by auto
nipkow@28853
   666
  moreover have "x \<in> insert x A" by auto
nipkow@28853
   667
  ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
nipkow@28853
   668
    by (rule fold_rec)
nipkow@28853
   669
  then show ?thesis by simp
nipkow@28853
   670
qed
nipkow@28853
   671
haftmann@26041
   672
end
nipkow@15392
   673
nipkow@15480
   674
text{* A simplified version for idempotent functions: *}
nipkow@15480
   675
haftmann@42871
   676
locale comp_fun_idem = comp_fun_commute +
haftmann@42871
   677
  assumes comp_fun_idem: "f x o f x = f x"
haftmann@26041
   678
begin
haftmann@26041
   679
haftmann@42869
   680
lemma fun_left_idem: "f x (f x z) = f x z"
haftmann@42871
   681
  using comp_fun_idem by (simp add: fun_eq_iff)
nipkow@28853
   682
haftmann@26041
   683
lemma fold_insert_idem:
nipkow@28853
   684
  assumes fin: "finite A"
nipkow@28853
   685
  shows "fold f z (insert x A) = f x (fold f z A)"
nipkow@15480
   686
proof cases
nipkow@28853
   687
  assume "x \<in> A"
nipkow@28853
   688
  then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
nipkow@28853
   689
  then show ?thesis using assms by (simp add:fun_left_idem)
nipkow@15480
   690
next
nipkow@28853
   691
  assume "x \<notin> A" then show ?thesis using assms by simp
nipkow@15480
   692
qed
nipkow@15480
   693
nipkow@28853
   694
declare fold_insert[simp del] fold_insert_idem[simp]
nipkow@28853
   695
nipkow@28853
   696
lemma fold_insert_idem2:
nipkow@28853
   697
  "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
nipkow@28853
   698
by(simp add:fold_fun_comm)
nipkow@15484
   699
haftmann@26041
   700
end
haftmann@26041
   701
haftmann@35817
   702
haftmann@35817
   703
subsubsection {* Expressing set operations via @{const fold} *}
haftmann@35817
   704
haftmann@42871
   705
lemma (in comp_fun_commute) comp_comp_fun_commute:
haftmann@42871
   706
  "comp_fun_commute (f \<circ> g)"
haftmann@35817
   707
proof
haftmann@42871
   708
qed (simp_all add: comp_fun_commute)
haftmann@35817
   709
haftmann@42871
   710
lemma (in comp_fun_idem) comp_comp_fun_idem:
haftmann@42871
   711
  "comp_fun_idem (f \<circ> g)"
haftmann@42871
   712
  by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
haftmann@42871
   713
    (simp_all add: comp_fun_idem)
haftmann@35817
   714
haftmann@42871
   715
lemma comp_fun_idem_insert:
haftmann@42871
   716
  "comp_fun_idem insert"
haftmann@35817
   717
proof
haftmann@35817
   718
qed auto
haftmann@35817
   719
haftmann@42871
   720
lemma comp_fun_idem_remove:
haftmann@46146
   721
  "comp_fun_idem Set.remove"
haftmann@35817
   722
proof
haftmann@35817
   723
qed auto
nipkow@31992
   724
haftmann@42871
   725
lemma (in semilattice_inf) comp_fun_idem_inf:
haftmann@42871
   726
  "comp_fun_idem inf"
haftmann@35817
   727
proof
haftmann@35817
   728
qed (auto simp add: inf_left_commute)
haftmann@35817
   729
haftmann@42871
   730
lemma (in semilattice_sup) comp_fun_idem_sup:
haftmann@42871
   731
  "comp_fun_idem sup"
haftmann@35817
   732
proof
haftmann@35817
   733
qed (auto simp add: sup_left_commute)
nipkow@31992
   734
haftmann@35817
   735
lemma union_fold_insert:
haftmann@35817
   736
  assumes "finite A"
haftmann@35817
   737
  shows "A \<union> B = fold insert B A"
haftmann@35817
   738
proof -
haftmann@42871
   739
  interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
haftmann@35817
   740
  from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
haftmann@35817
   741
qed
nipkow@31992
   742
haftmann@35817
   743
lemma minus_fold_remove:
haftmann@35817
   744
  assumes "finite A"
haftmann@46146
   745
  shows "B - A = fold Set.remove B A"
haftmann@35817
   746
proof -
haftmann@46146
   747
  interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
haftmann@46146
   748
  from `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
haftmann@46146
   749
  then show ?thesis ..
haftmann@35817
   750
qed
haftmann@35817
   751
haftmann@35817
   752
context complete_lattice
nipkow@31992
   753
begin
nipkow@31992
   754
haftmann@35817
   755
lemma inf_Inf_fold_inf:
haftmann@35817
   756
  assumes "finite A"
haftmann@35817
   757
  shows "inf B (Inf A) = fold inf B A"
haftmann@35817
   758
proof -
haftmann@42871
   759
  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
haftmann@35817
   760
  from `finite A` show ?thesis by (induct A arbitrary: B)
noschinl@44919
   761
    (simp_all add: inf_commute fold_fun_comm)
haftmann@35817
   762
qed
nipkow@31992
   763
haftmann@35817
   764
lemma sup_Sup_fold_sup:
haftmann@35817
   765
  assumes "finite A"
haftmann@35817
   766
  shows "sup B (Sup A) = fold sup B A"
haftmann@35817
   767
proof -
haftmann@42871
   768
  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
haftmann@35817
   769
  from `finite A` show ?thesis by (induct A arbitrary: B)
noschinl@44919
   770
    (simp_all add: sup_commute fold_fun_comm)
nipkow@31992
   771
qed
nipkow@31992
   772
haftmann@35817
   773
lemma Inf_fold_inf:
haftmann@35817
   774
  assumes "finite A"
haftmann@35817
   775
  shows "Inf A = fold inf top A"
haftmann@35817
   776
  using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
haftmann@35817
   777
haftmann@35817
   778
lemma Sup_fold_sup:
haftmann@35817
   779
  assumes "finite A"
haftmann@35817
   780
  shows "Sup A = fold sup bot A"
haftmann@35817
   781
  using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
nipkow@31992
   782
haftmann@46146
   783
lemma inf_INF_fold_inf:
haftmann@35817
   784
  assumes "finite A"
haftmann@42873
   785
  shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold") 
haftmann@35817
   786
proof (rule sym)
haftmann@42871
   787
  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
haftmann@42871
   788
  interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
haftmann@42873
   789
  from `finite A` show "?fold = ?inf"
haftmann@42869
   790
    by (induct A arbitrary: B)
hoelzl@44928
   791
      (simp_all add: INF_def inf_left_commute)
haftmann@35817
   792
qed
nipkow@31992
   793
haftmann@46146
   794
lemma sup_SUP_fold_sup:
haftmann@35817
   795
  assumes "finite A"
haftmann@42873
   796
  shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold") 
haftmann@35817
   797
proof (rule sym)
haftmann@42871
   798
  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
haftmann@42871
   799
  interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
haftmann@42873
   800
  from `finite A` show "?fold = ?sup"
haftmann@42869
   801
    by (induct A arbitrary: B)
hoelzl@44928
   802
      (simp_all add: SUP_def sup_left_commute)
haftmann@35817
   803
qed
nipkow@31992
   804
haftmann@46146
   805
lemma INF_fold_inf:
haftmann@35817
   806
  assumes "finite A"
haftmann@42873
   807
  shows "INFI A f = fold (inf \<circ> f) top A"
haftmann@46146
   808
  using assms inf_INF_fold_inf [of A top] by simp
nipkow@31992
   809
haftmann@46146
   810
lemma SUP_fold_sup:
haftmann@35817
   811
  assumes "finite A"
haftmann@42873
   812
  shows "SUPR A f = fold (sup \<circ> f) bot A"
haftmann@46146
   813
  using assms sup_SUP_fold_sup [of A bot] by simp
nipkow@31992
   814
nipkow@31992
   815
end
nipkow@31992
   816
nipkow@31992
   817
haftmann@35817
   818
subsection {* The derived combinator @{text fold_image} *}
nipkow@28853
   819
nipkow@28853
   820
definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
haftmann@42875
   821
  where "fold_image f g = fold (\<lambda>x y. f (g x) y)"
nipkow@28853
   822
nipkow@28853
   823
lemma fold_image_empty[simp]: "fold_image f g z {} = z"
haftmann@42875
   824
  by (simp add:fold_image_def)
nipkow@15392
   825
haftmann@26041
   826
context ab_semigroup_mult
haftmann@26041
   827
begin
haftmann@26041
   828
nipkow@28853
   829
lemma fold_image_insert[simp]:
haftmann@42875
   830
  assumes "finite A" and "a \<notin> A"
haftmann@42875
   831
  shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
nipkow@28853
   832
proof -
haftmann@42875
   833
  interpret comp_fun_commute "%x y. (g x) * y" proof
haftmann@42809
   834
  qed (simp add: fun_eq_iff mult_ac)
haftmann@42809
   835
  show ?thesis using assms by (simp add: fold_image_def)
nipkow@28853
   836
qed
nipkow@28853
   837
nipkow@28853
   838
lemma fold_image_reindex:
haftmann@42875
   839
  assumes "finite A"
haftmann@42875
   840
  shows "inj_on h A \<Longrightarrow> fold_image times g z (h ` A) = fold_image times (g \<circ> h) z A"
haftmann@42875
   841
  using assms by induct auto
nipkow@28853
   842
nipkow@28853
   843
lemma fold_image_cong:
haftmann@42875
   844
  assumes "finite A" and g_h: "\<And>x. x\<in>A \<Longrightarrow> g x = h x"
haftmann@42875
   845
  shows "fold_image times g z A = fold_image times h z A"
haftmann@42875
   846
proof -
haftmann@42875
   847
  from `finite A`
haftmann@42875
   848
  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
   849
  proof (induct arbitrary: C)
haftmann@42875
   850
    case empty then show ?case by simp
haftmann@42875
   851
  next
haftmann@42875
   852
    case (insert x F) then show ?case apply -
haftmann@42875
   853
    apply (simp add: subset_insert_iff, clarify)
haftmann@42875
   854
    apply (subgoal_tac "finite C")
haftmann@42875
   855
      prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
haftmann@42875
   856
    apply (subgoal_tac "C = insert x (C - {x})")
haftmann@42875
   857
      prefer 2 apply blast
haftmann@42875
   858
    apply (erule ssubst)
haftmann@42875
   859
    apply (simp add: Ball_def del: insert_Diff_single)
haftmann@42875
   860
    done
haftmann@42875
   861
  qed
haftmann@42875
   862
  with g_h show ?thesis by simp
haftmann@42875
   863
qed
nipkow@15392
   864
haftmann@26041
   865
end
haftmann@26041
   866
haftmann@26041
   867
context comm_monoid_mult
haftmann@26041
   868
begin
haftmann@26041
   869
haftmann@35817
   870
lemma fold_image_1:
haftmann@35817
   871
  "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
haftmann@41656
   872
  apply (induct rule: finite_induct)
haftmann@35817
   873
  apply simp by auto
haftmann@35817
   874
nipkow@28853
   875
lemma fold_image_Un_Int:
haftmann@26041
   876
  "finite A ==> finite B ==>
nipkow@28853
   877
    fold_image times g 1 A * fold_image times g 1 B =
nipkow@28853
   878
    fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
haftmann@41656
   879
  apply (induct rule: finite_induct)
nipkow@28853
   880
by (induct set: finite) 
nipkow@28853
   881
   (auto simp add: mult_ac insert_absorb Int_insert_left)
haftmann@26041
   882
haftmann@35817
   883
lemma fold_image_Un_one:
haftmann@35817
   884
  assumes fS: "finite S" and fT: "finite T"
haftmann@35817
   885
  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
haftmann@35817
   886
  shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
haftmann@35817
   887
proof-
haftmann@35817
   888
  have "fold_image op * f 1 (S \<inter> T) = 1" 
haftmann@35817
   889
    apply (rule fold_image_1)
haftmann@35817
   890
    using fS fT I0 by auto 
haftmann@35817
   891
  with fold_image_Un_Int[OF fS fT] show ?thesis by simp
haftmann@35817
   892
qed
haftmann@35817
   893
haftmann@26041
   894
corollary fold_Un_disjoint:
haftmann@26041
   895
  "finite A ==> finite B ==> A Int B = {} ==>
nipkow@28853
   896
   fold_image times g 1 (A Un B) =
nipkow@28853
   897
   fold_image times g 1 A * fold_image times g 1 B"
nipkow@28853
   898
by (simp add: fold_image_Un_Int)
nipkow@28853
   899
nipkow@28853
   900
lemma fold_image_UN_disjoint:
haftmann@26041
   901
  "\<lbrakk> finite I; ALL i:I. finite (A i);
haftmann@26041
   902
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
nipkow@28853
   903
   \<Longrightarrow> fold_image times g 1 (UNION I A) =
nipkow@28853
   904
       fold_image times (%i. fold_image times g 1 (A i)) 1 I"
haftmann@41656
   905
apply (induct rule: finite_induct)
haftmann@41656
   906
apply simp
haftmann@41656
   907
apply atomize
nipkow@28853
   908
apply (subgoal_tac "ALL i:F. x \<noteq> i")
nipkow@28853
   909
 prefer 2 apply blast
nipkow@28853
   910
apply (subgoal_tac "A x Int UNION F A = {}")
nipkow@28853
   911
 prefer 2 apply blast
nipkow@28853
   912
apply (simp add: fold_Un_disjoint)
nipkow@28853
   913
done
nipkow@28853
   914
nipkow@28853
   915
lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@28853
   916
  fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
nipkow@28853
   917
  fold_image times (split g) 1 (SIGMA x:A. B x)"
nipkow@15392
   918
apply (subst Sigma_def)
nipkow@28853
   919
apply (subst fold_image_UN_disjoint, assumption, simp)
nipkow@15392
   920
 apply blast
nipkow@28853
   921
apply (erule fold_image_cong)
nipkow@28853
   922
apply (subst fold_image_UN_disjoint, simp, simp)
nipkow@15392
   923
 apply blast
paulson@15506
   924
apply simp
nipkow@15392
   925
done
nipkow@15392
   926
nipkow@28853
   927
lemma fold_image_distrib: "finite A \<Longrightarrow>
nipkow@28853
   928
   fold_image times (%x. g x * h x) 1 A =
nipkow@28853
   929
   fold_image times g 1 A *  fold_image times h 1 A"
nipkow@28853
   930
by (erule finite_induct) (simp_all add: mult_ac)
haftmann@26041
   931
chaieb@30260
   932
lemma fold_image_related: 
chaieb@30260
   933
  assumes Re: "R e e" 
chaieb@30260
   934
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
chaieb@30260
   935
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
chaieb@30260
   936
  shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
chaieb@30260
   937
  using fS by (rule finite_subset_induct) (insert assms, auto)
chaieb@30260
   938
chaieb@30260
   939
lemma  fold_image_eq_general:
chaieb@30260
   940
  assumes fS: "finite S"
chaieb@30260
   941
  and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 
chaieb@30260
   942
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
chaieb@30260
   943
  shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
chaieb@30260
   944
proof-
chaieb@30260
   945
  from h f12 have hS: "h ` S = S'" by auto
chaieb@30260
   946
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
chaieb@30260
   947
    from f12 h H  have "x = y" by auto }
chaieb@30260
   948
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
chaieb@30260
   949
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
chaieb@30260
   950
  from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
chaieb@30260
   951
  also have "\<dots> = fold_image (op *) (f2 o h) e S" 
chaieb@30260
   952
    using fold_image_reindex[OF fS hinj, of f2 e] .
chaieb@30260
   953
  also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
chaieb@30260
   954
    by blast
chaieb@30260
   955
  finally show ?thesis ..
chaieb@30260
   956
qed
chaieb@30260
   957
chaieb@30260
   958
lemma fold_image_eq_general_inverses:
chaieb@30260
   959
  assumes fS: "finite S" 
chaieb@30260
   960
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
chaieb@30260
   961
  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
   962
  shows "fold_image (op *) f e S = fold_image (op *) g e T"
chaieb@30260
   963
  (* metis solves it, but not yet available here *)
chaieb@30260
   964
  apply (rule fold_image_eq_general[OF fS, of T h g f e])
chaieb@30260
   965
  apply (rule ballI)
chaieb@30260
   966
  apply (frule kh)
chaieb@30260
   967
  apply (rule ex1I[])
chaieb@30260
   968
  apply blast
chaieb@30260
   969
  apply clarsimp
chaieb@30260
   970
  apply (drule hk) apply simp
chaieb@30260
   971
  apply (rule sym)
chaieb@30260
   972
  apply (erule conjunct1[OF conjunct2[OF hk]])
chaieb@30260
   973
  apply (rule ballI)
chaieb@30260
   974
  apply (drule  hk)
chaieb@30260
   975
  apply blast
chaieb@30260
   976
  done
chaieb@30260
   977
haftmann@26041
   978
end
haftmann@22917
   979
nipkow@25162
   980
haftmann@35817
   981
subsection {* A fold functional for non-empty sets *}
nipkow@15392
   982
nipkow@15392
   983
text{* Does not require start value. *}
wenzelm@12396
   984
berghofe@23736
   985
inductive
berghofe@22262
   986
  fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
berghofe@22262
   987
  for f :: "'a => 'a => 'a"
berghofe@22262
   988
where
paulson@15506
   989
  fold1Set_insertI [intro]:
nipkow@28853
   990
   "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
wenzelm@12396
   991
haftmann@35416
   992
definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
berghofe@22262
   993
  "fold1 f A == THE x. fold1Set f A x"
paulson@15506
   994
paulson@15506
   995
lemma fold1Set_nonempty:
haftmann@22917
   996
  "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
nipkow@28853
   997
by(erule fold1Set.cases, simp_all)
nipkow@15392
   998
berghofe@23736
   999
inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
berghofe@23736
  1000
berghofe@23736
  1001
inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
berghofe@22262
  1002
berghofe@22262
  1003
berghofe@22262
  1004
lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
huffman@35216
  1005
by (blast elim: fold_graph.cases)
nipkow@15392
  1006
haftmann@22917
  1007
lemma fold1_singleton [simp]: "fold1 f {a} = a"
nipkow@28853
  1008
by (unfold fold1_def) blast
wenzelm@12396
  1009
paulson@15508
  1010
lemma finite_nonempty_imp_fold1Set:
berghofe@22262
  1011
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
paulson@15508
  1012
apply (induct A rule: finite_induct)
nipkow@28853
  1013
apply (auto dest: finite_imp_fold_graph [of _ f])
paulson@15508
  1014
done
paulson@15506
  1015
nipkow@28853
  1016
text{*First, some lemmas about @{const fold_graph}.*}
nipkow@15392
  1017
haftmann@26041
  1018
context ab_semigroup_mult
haftmann@26041
  1019
begin
haftmann@26041
  1020
haftmann@42871
  1021
lemma comp_fun_commute: "comp_fun_commute (op *)" proof
haftmann@42809
  1022
qed (simp add: fun_eq_iff mult_ac)
nipkow@28853
  1023
nipkow@28853
  1024
lemma fold_graph_insert_swap:
nipkow@28853
  1025
assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
nipkow@28853
  1026
shows "fold_graph times z (insert b A) (z * y)"
nipkow@28853
  1027
proof -
haftmann@42871
  1028
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
nipkow@28853
  1029
from assms show ?thesis
nipkow@28853
  1030
proof (induct rule: fold_graph.induct)
huffman@36045
  1031
  case emptyI show ?case by (subst mult_commute [of z b], fast)
paulson@15508
  1032
next
berghofe@22262
  1033
  case (insertI x A y)
nipkow@28853
  1034
    have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
paulson@15521
  1035
      using insertI by force  --{*how does @{term id} get unfolded?*}
haftmann@26041
  1036
    thus ?case by (simp add: insert_commute mult_ac)
paulson@15508
  1037
qed
nipkow@28853
  1038
qed
nipkow@28853
  1039
nipkow@28853
  1040
lemma fold_graph_permute_diff:
nipkow@28853
  1041
assumes fold: "fold_graph times b A x"
nipkow@28853
  1042
shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
paulson@15508
  1043
using fold
nipkow@28853
  1044
proof (induct rule: fold_graph.induct)
paulson@15508
  1045
  case emptyI thus ?case by simp
paulson@15508
  1046
next
berghofe@22262
  1047
  case (insertI x A y)
paulson@15521
  1048
  have "a = x \<or> a \<in> A" using insertI by simp
paulson@15521
  1049
  thus ?case
paulson@15521
  1050
  proof
paulson@15521
  1051
    assume "a = x"
paulson@15521
  1052
    with insertI show ?thesis
nipkow@28853
  1053
      by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
paulson@15521
  1054
  next
paulson@15521
  1055
    assume ainA: "a \<in> A"
nipkow@28853
  1056
    hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
nipkow@28853
  1057
      using insertI by force
paulson@15521
  1058
    moreover
paulson@15521
  1059
    have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
paulson@15521
  1060
      using ainA insertI by blast
nipkow@28853
  1061
    ultimately show ?thesis by simp
paulson@15508
  1062
  qed
paulson@15508
  1063
qed
paulson@15508
  1064
haftmann@26041
  1065
lemma fold1_eq_fold:
nipkow@28853
  1066
assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
nipkow@28853
  1067
proof -
haftmann@42871
  1068
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
nipkow@28853
  1069
  from assms show ?thesis
nipkow@28853
  1070
apply (simp add: fold1_def fold_def)
paulson@15508
  1071
apply (rule the_equality)
nipkow@28853
  1072
apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
paulson@15508
  1073
apply (rule sym, clarify)
paulson@15508
  1074
apply (case_tac "Aa=A")
huffman@35216
  1075
 apply (best intro: fold_graph_determ)
nipkow@28853
  1076
apply (subgoal_tac "fold_graph times a A x")
huffman@35216
  1077
 apply (best intro: fold_graph_determ)
nipkow@28853
  1078
apply (subgoal_tac "insert aa (Aa - {a}) = A")
nipkow@28853
  1079
 prefer 2 apply (blast elim: equalityE)
nipkow@28853
  1080
apply (auto dest: fold_graph_permute_diff [where a=a])
paulson@15508
  1081
done
nipkow@28853
  1082
qed
paulson@15508
  1083
paulson@15521
  1084
lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
paulson@15521
  1085
apply safe
nipkow@28853
  1086
 apply simp
nipkow@28853
  1087
 apply (drule_tac x=x in spec)
nipkow@28853
  1088
 apply (drule_tac x="A-{x}" in spec, auto)
paulson@15508
  1089
done
paulson@15508
  1090
haftmann@26041
  1091
lemma fold1_insert:
paulson@15521
  1092
  assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
haftmann@26041
  1093
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1094
proof -
haftmann@42871
  1095
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
nipkow@28853
  1096
  from nonempty obtain a A' where "A = insert a A' & a ~: A'"
paulson@15521
  1097
    by (auto simp add: nonempty_iff)
paulson@15521
  1098
  with A show ?thesis
nipkow@28853
  1099
    by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
paulson@15521
  1100
qed
paulson@15521
  1101
haftmann@26041
  1102
end
haftmann@26041
  1103
haftmann@26041
  1104
context ab_semigroup_idem_mult
haftmann@26041
  1105
begin
haftmann@26041
  1106
haftmann@42871
  1107
lemma comp_fun_idem: "comp_fun_idem (op *)" proof
haftmann@42869
  1108
qed (simp_all add: fun_eq_iff mult_left_commute)
haftmann@35817
  1109
haftmann@26041
  1110
lemma fold1_insert_idem [simp]:
paulson@15521
  1111
  assumes nonempty: "A \<noteq> {}" and A: "finite A" 
haftmann@26041
  1112
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1113
proof -
haftmann@42871
  1114
  interpret comp_fun_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@42871
  1115
    by (rule comp_fun_idem)
nipkow@28853
  1116
  from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
paulson@15521
  1117
    by (auto simp add: nonempty_iff)
paulson@15521
  1118
  show ?thesis
paulson@15521
  1119
  proof cases
wenzelm@41550
  1120
    assume a: "a = x"
wenzelm@41550
  1121
    show ?thesis
paulson@15521
  1122
    proof cases
paulson@15521
  1123
      assume "A' = {}"
wenzelm@41550
  1124
      with A' a show ?thesis by simp
paulson@15521
  1125
    next
paulson@15521
  1126
      assume "A' \<noteq> {}"
wenzelm@41550
  1127
      with A A' a show ?thesis
huffman@35216
  1128
        by (simp add: fold1_insert mult_assoc [symmetric])
paulson@15521
  1129
    qed
paulson@15521
  1130
  next
paulson@15521
  1131
    assume "a \<noteq> x"
wenzelm@41550
  1132
    with A A' show ?thesis
huffman@35216
  1133
      by (simp add: insert_commute fold1_eq_fold)
paulson@15521
  1134
  qed
paulson@15521
  1135
qed
paulson@15506
  1136
haftmann@26041
  1137
lemma hom_fold1_commute:
haftmann@26041
  1138
assumes hom: "!!x y. h (x * y) = h x * h y"
haftmann@26041
  1139
and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
haftmann@22917
  1140
using N proof (induct rule: finite_ne_induct)
haftmann@22917
  1141
  case singleton thus ?case by simp
haftmann@22917
  1142
next
haftmann@22917
  1143
  case (insert n N)
haftmann@26041
  1144
  then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
haftmann@26041
  1145
  also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
haftmann@26041
  1146
  also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
haftmann@26041
  1147
  also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
haftmann@22917
  1148
    using insert by(simp)
haftmann@22917
  1149
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@22917
  1150
  finally show ?case .
haftmann@22917
  1151
qed
haftmann@22917
  1152
haftmann@32679
  1153
lemma fold1_eq_fold_idem:
haftmann@32679
  1154
  assumes "finite A"
haftmann@32679
  1155
  shows "fold1 times (insert a A) = fold times a A"
haftmann@32679
  1156
proof (cases "a \<in> A")
haftmann@32679
  1157
  case False
haftmann@32679
  1158
  with assms show ?thesis by (simp add: fold1_eq_fold)
haftmann@32679
  1159
next
haftmann@42871
  1160
  interpret comp_fun_idem times by (fact comp_fun_idem)
haftmann@32679
  1161
  case True then obtain b B
haftmann@32679
  1162
    where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
haftmann@32679
  1163
  with assms have "finite B" by auto
haftmann@32679
  1164
  then have "fold times a (insert a B) = fold times (a * a) B"
haftmann@32679
  1165
    using `a \<notin> B` by (rule fold_insert2)
haftmann@32679
  1166
  then show ?thesis
haftmann@32679
  1167
    using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
haftmann@32679
  1168
qed
haftmann@32679
  1169
haftmann@26041
  1170
end
haftmann@26041
  1171
paulson@15506
  1172
paulson@15508
  1173
text{* Now the recursion rules for definitions: *}
paulson@15508
  1174
haftmann@22917
  1175
lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
huffman@35216
  1176
by simp
paulson@15508
  1177
haftmann@26041
  1178
lemma (in ab_semigroup_mult) fold1_insert_def:
haftmann@26041
  1179
  "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1180
by (simp add:fold1_insert)
haftmann@26041
  1181
haftmann@26041
  1182
lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
haftmann@26041
  1183
  "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1184
by simp
paulson@15508
  1185
paulson@15508
  1186
subsubsection{* Determinacy for @{term fold1Set} *}
paulson@15508
  1187
nipkow@28853
  1188
(*Not actually used!!*)
nipkow@28853
  1189
(*
haftmann@26041
  1190
context ab_semigroup_mult
haftmann@26041
  1191
begin
haftmann@26041
  1192
nipkow@28853
  1193
lemma fold_graph_permute:
nipkow@28853
  1194
  "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
nipkow@28853
  1195
   ==> fold_graph times id a (insert b A) x"
haftmann@26041
  1196
apply (cases "a=b") 
nipkow@28853
  1197
apply (auto dest: fold_graph_permute_diff) 
paulson@15506
  1198
done
nipkow@15376
  1199
haftmann@26041
  1200
lemma fold1Set_determ:
haftmann@26041
  1201
  "fold1Set times A x ==> fold1Set times A y ==> y = x"
paulson@15506
  1202
proof (clarify elim!: fold1Set.cases)
paulson@15506
  1203
  fix A x B y a b
nipkow@28853
  1204
  assume Ax: "fold_graph times id a A x"
nipkow@28853
  1205
  assume By: "fold_graph times id b B y"
paulson@15506
  1206
  assume anotA:  "a \<notin> A"
paulson@15506
  1207
  assume bnotB:  "b \<notin> B"
paulson@15506
  1208
  assume eq: "insert a A = insert b B"
paulson@15506
  1209
  show "y=x"
paulson@15506
  1210
  proof cases
paulson@15506
  1211
    assume same: "a=b"
paulson@15506
  1212
    hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
nipkow@28853
  1213
    thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
nipkow@15392
  1214
  next
paulson@15506
  1215
    assume diff: "a\<noteq>b"
paulson@15506
  1216
    let ?D = "B - {a}"
paulson@15506
  1217
    have B: "B = insert a ?D" and A: "A = insert b ?D"
paulson@15506
  1218
     and aB: "a \<in> B" and bA: "b \<in> A"
paulson@15506
  1219
      using eq anotA bnotB diff by (blast elim!:equalityE)+
paulson@15506
  1220
    with aB bnotB By
nipkow@28853
  1221
    have "fold_graph times id a (insert b ?D) y" 
nipkow@28853
  1222
      by (auto intro: fold_graph_permute simp add: insert_absorb)
paulson@15506
  1223
    moreover
nipkow@28853
  1224
    have "fold_graph times id a (insert b ?D) x"
paulson@15506
  1225
      by (simp add: A [symmetric] Ax) 
nipkow@28853
  1226
    ultimately show ?thesis by (blast intro: fold_graph_determ) 
nipkow@15392
  1227
  qed
wenzelm@12396
  1228
qed
wenzelm@12396
  1229
haftmann@26041
  1230
lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
paulson@15506
  1231
  by (unfold fold1_def) (blast intro: fold1Set_determ)
paulson@15506
  1232
haftmann@26041
  1233
end
nipkow@28853
  1234
*)
haftmann@26041
  1235
paulson@15506
  1236
declare
nipkow@28853
  1237
  empty_fold_graphE [rule del]  fold_graph.intros [rule del]
paulson@15506
  1238
  empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
ballarin@19931
  1239
  -- {* No more proofs involve these relations. *}
nipkow@15376
  1240
haftmann@26041
  1241
subsubsection {* Lemmas about @{text fold1} *}
haftmann@26041
  1242
haftmann@26041
  1243
context ab_semigroup_mult
haftmann@22917
  1244
begin
haftmann@22917
  1245
haftmann@26041
  1246
lemma fold1_Un:
nipkow@15484
  1247
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  1248
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
haftmann@26041
  1249
       fold1 times (A Un B) = fold1 times A * fold1 times B"
haftmann@26041
  1250
using A by (induct rule: finite_ne_induct)
haftmann@26041
  1251
  (simp_all add: fold1_insert mult_assoc)
haftmann@26041
  1252
haftmann@26041
  1253
lemma fold1_in:
haftmann@26041
  1254
  assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
haftmann@26041
  1255
  shows "fold1 times A \<in> A"
nipkow@15484
  1256
using A
nipkow@15484
  1257
proof (induct rule:finite_ne_induct)
paulson@15506
  1258
  case singleton thus ?case by simp
nipkow@15484
  1259
next
nipkow@15484
  1260
  case insert thus ?case using elem by (force simp add:fold1_insert)
nipkow@15484
  1261
qed
nipkow@15484
  1262
haftmann@26041
  1263
end
haftmann@26041
  1264
haftmann@26041
  1265
lemma (in ab_semigroup_idem_mult) fold1_Un2:
nipkow@15497
  1266
assumes A: "finite A" "A \<noteq> {}"
haftmann@26041
  1267
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
haftmann@26041
  1268
       fold1 times (A Un B) = fold1 times A * fold1 times B"
nipkow@15497
  1269
using A
haftmann@26041
  1270
proof(induct rule:finite_ne_induct)
nipkow@15497
  1271
  case singleton thus ?case by simp
nipkow@15484
  1272
next
haftmann@26041
  1273
  case insert thus ?case by (simp add: mult_assoc)
nipkow@18423
  1274
qed
nipkow@18423
  1275
nipkow@18423
  1276
haftmann@35817
  1277
subsection {* Locales as mini-packages for fold operations *}
haftmann@34007
  1278
haftmann@35817
  1279
subsubsection {* The natural case *}
haftmann@35719
  1280
haftmann@35719
  1281
locale folding =
haftmann@35719
  1282
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@35719
  1283
  fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@42871
  1284
  assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
haftmann@35722
  1285
  assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
haftmann@35719
  1286
begin
haftmann@35719
  1287
haftmann@35719
  1288
lemma empty [simp]:
haftmann@35719
  1289
  "F {} = id"
nipkow@39302
  1290
  by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1291
haftmann@35719
  1292
lemma insert [simp]:
haftmann@35719
  1293
  assumes "finite A" and "x \<notin> A"
haftmann@35719
  1294
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35719
  1295
proof -
haftmann@42871
  1296
  interpret comp_fun_commute f proof
haftmann@42871
  1297
  qed (insert comp_fun_commute, simp add: fun_eq_iff)
haftmann@35719
  1298
  from fold_insert2 assms
haftmann@35722
  1299
  have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
nipkow@39302
  1300
  with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1301
qed
haftmann@35719
  1302
haftmann@35719
  1303
lemma remove:
haftmann@35719
  1304
  assumes "finite A" and "x \<in> A"
haftmann@35719
  1305
  shows "F A = F (A - {x}) \<circ> f x"
haftmann@35719
  1306
proof -
haftmann@35719
  1307
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35719
  1308
    by (auto dest: mk_disjoint_insert)
haftmann@35719
  1309
  moreover from `finite A` this have "finite B" by simp
haftmann@35719
  1310
  ultimately show ?thesis by simp
haftmann@35719
  1311
qed
haftmann@35719
  1312
haftmann@35719
  1313
lemma insert_remove:
haftmann@35719
  1314
  assumes "finite A"
haftmann@35719
  1315
  shows "F (insert x A) = F (A - {x}) \<circ> f x"
haftmann@35722
  1316
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35719
  1317
haftmann@35817
  1318
lemma commute_left_comp:
haftmann@35817
  1319
  "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
haftmann@42871
  1320
  by (simp add: o_assoc comp_fun_commute)
haftmann@35817
  1321
haftmann@42871
  1322
lemma comp_fun_commute':
haftmann@35719
  1323
  assumes "finite A"
haftmann@35719
  1324
  shows "f x \<circ> F A = F A \<circ> f x"
haftmann@35817
  1325
  using assms by (induct A)
haftmann@42871
  1326
    (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: o_assoc [symmetric] comp_fun_commute)
haftmann@35817
  1327
haftmann@35817
  1328
lemma commute_left_comp':
haftmann@35817
  1329
  assumes "finite A"
haftmann@35817
  1330
  shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
haftmann@42871
  1331
  using assms by (simp add: o_assoc comp_fun_commute')
haftmann@35817
  1332
haftmann@42871
  1333
lemma comp_fun_commute'':
haftmann@35817
  1334
  assumes "finite A" and "finite B"
haftmann@35817
  1335
  shows "F B \<circ> F A = F A \<circ> F B"
haftmann@35817
  1336
  using assms by (induct A)
haftmann@42871
  1337
    (simp_all add: o_assoc, simp add: o_assoc [symmetric] comp_fun_commute')
haftmann@35719
  1338
haftmann@35817
  1339
lemma commute_left_comp'':
haftmann@35817
  1340
  assumes "finite A" and "finite B"
haftmann@35817
  1341
  shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
haftmann@42871
  1342
  using assms by (simp add: o_assoc comp_fun_commute'')
haftmann@35817
  1343
haftmann@42871
  1344
lemmas comp_fun_commutes = o_assoc [symmetric] comp_fun_commute commute_left_comp
haftmann@42871
  1345
  comp_fun_commute' commute_left_comp' comp_fun_commute'' commute_left_comp''
haftmann@35817
  1346
haftmann@35817
  1347
lemma union_inter:
haftmann@35817
  1348
  assumes "finite A" and "finite B"
haftmann@35817
  1349
  shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
haftmann@35817
  1350
  using assms by (induct A)
haftmann@42871
  1351
    (simp_all del: o_apply add: insert_absorb Int_insert_left comp_fun_commutes,
haftmann@35817
  1352
      simp add: o_assoc)
haftmann@35719
  1353
haftmann@35719
  1354
lemma union:
haftmann@35719
  1355
  assumes "finite A" and "finite B"
haftmann@35719
  1356
  and "A \<inter> B = {}"
haftmann@35719
  1357
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1358
proof -
haftmann@35817
  1359
  from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
haftmann@35817
  1360
  with `A \<inter> B = {}` show ?thesis by simp
haftmann@35719
  1361
qed
haftmann@35719
  1362
haftmann@34007
  1363
end
haftmann@35719
  1364
haftmann@35817
  1365
haftmann@35817
  1366
subsubsection {* The natural case with idempotency *}
haftmann@35817
  1367
haftmann@35719
  1368
locale folding_idem = folding +
haftmann@35719
  1369
  assumes idem_comp: "f x \<circ> f x = f x"
haftmann@35719
  1370
begin
haftmann@35719
  1371
haftmann@35817
  1372
lemma idem_left_comp:
haftmann@35817
  1373
  "f x \<circ> (f x \<circ> g) = f x \<circ> g"
haftmann@35817
  1374
  by (simp add: o_assoc idem_comp)
haftmann@35817
  1375
haftmann@35817
  1376
lemma in_comp_idem:
haftmann@35817
  1377
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1378
  shows "F A \<circ> f x = F A"
haftmann@35817
  1379
using assms by (induct A)
haftmann@42871
  1380
  (auto simp add: comp_fun_commutes idem_comp, simp add: commute_left_comp' [symmetric] comp_fun_commute')
haftmann@35719
  1381
haftmann@35817
  1382
lemma subset_comp_idem:
haftmann@35817
  1383
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1384
  shows "F A \<circ> F B = F A"
haftmann@35817
  1385
proof -
haftmann@35817
  1386
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1387
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1388
    (simp_all add: o_assoc in_comp_idem `finite A`)
haftmann@35817
  1389
qed
haftmann@35719
  1390
haftmann@35817
  1391
declare insert [simp del]
haftmann@35719
  1392
haftmann@35719
  1393
lemma insert_idem [simp]:
haftmann@35719
  1394
  assumes "finite A"
haftmann@35719
  1395
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35817
  1396
  using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
haftmann@35719
  1397
haftmann@35719
  1398
lemma union_idem:
haftmann@35719
  1399
  assumes "finite A" and "finite B"
haftmann@35719
  1400
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1401
proof -
haftmann@35817
  1402
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1403
  then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
haftmann@35817
  1404
  with assms show ?thesis by (simp add: union_inter)
haftmann@35719
  1405
qed
haftmann@35719
  1406
haftmann@35719
  1407
end
haftmann@35719
  1408
haftmann@35817
  1409
haftmann@35817
  1410
subsubsection {* The image case with fixed function *}
haftmann@35817
  1411
haftmann@35796
  1412
no_notation times (infixl "*" 70)
haftmann@35796
  1413
no_notation Groups.one ("1")
haftmann@35722
  1414
haftmann@35722
  1415
locale folding_image_simple = comm_monoid +
haftmann@35722
  1416
  fixes g :: "('b \<Rightarrow> 'a)"
haftmann@35722
  1417
  fixes F :: "'b set \<Rightarrow> 'a"
haftmann@35817
  1418
  assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
haftmann@35722
  1419
begin
haftmann@35722
  1420
haftmann@35722
  1421
lemma empty [simp]:
haftmann@35722
  1422
  "F {} = 1"
haftmann@35817
  1423
  by (simp add: eq_fold_g)
haftmann@35722
  1424
haftmann@35722
  1425
lemma insert [simp]:
haftmann@35722
  1426
  assumes "finite A" and "x \<notin> A"
haftmann@35722
  1427
  shows "F (insert x A) = g x * F A"
haftmann@35722
  1428
proof -
haftmann@42871
  1429
  interpret comp_fun_commute "%x y. (g x) * y" proof
haftmann@42809
  1430
  qed (simp add: ac_simps fun_eq_iff)
haftmann@35722
  1431
  with assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
haftmann@35722
  1432
    by (simp add: fold_image_def)
haftmann@35817
  1433
  with `finite A` show ?thesis by (simp add: eq_fold_g)
haftmann@35722
  1434
qed
haftmann@35722
  1435
haftmann@35722
  1436
lemma remove:
haftmann@35722
  1437
  assumes "finite A" and "x \<in> A"
haftmann@35722
  1438
  shows "F A = g x * F (A - {x})"
haftmann@35722
  1439
proof -
haftmann@35722
  1440
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35722
  1441
    by (auto dest: mk_disjoint_insert)
haftmann@35722
  1442
  moreover from `finite A` this have "finite B" by simp
haftmann@35722
  1443
  ultimately show ?thesis by simp
haftmann@35722
  1444
qed
haftmann@35722
  1445
haftmann@35722
  1446
lemma insert_remove:
haftmann@35722
  1447
  assumes "finite A"
haftmann@35722
  1448
  shows "F (insert x A) = g x * F (A - {x})"
haftmann@35722
  1449
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35722
  1450
haftmann@35817
  1451
lemma neutral:
haftmann@35817
  1452
  assumes "finite A" and "\<forall>x\<in>A. g x = 1"
haftmann@35817
  1453
  shows "F A = 1"
haftmann@35817
  1454
  using assms by (induct A) simp_all
haftmann@35817
  1455
haftmann@35722
  1456
lemma union_inter:
haftmann@35722
  1457
  assumes "finite A" and "finite B"
haftmann@35817
  1458
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35722
  1459
using assms proof (induct A)
haftmann@35722
  1460
  case empty then show ?case by simp
haftmann@35722
  1461
next
haftmann@35722
  1462
  case (insert x A) then show ?case
haftmann@35722
  1463
    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
haftmann@35722
  1464
qed
haftmann@35722
  1465
haftmann@35817
  1466
corollary union_inter_neutral:
haftmann@35817
  1467
  assumes "finite A" and "finite B"
haftmann@35817
  1468
  and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
haftmann@35817
  1469
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1470
  using assms by (simp add: union_inter [symmetric] neutral)
haftmann@35817
  1471
haftmann@35722
  1472
corollary union_disjoint:
haftmann@35722
  1473
  assumes "finite A" and "finite B"
haftmann@35722
  1474
  assumes "A \<inter> B = {}"
haftmann@35722
  1475
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1476
  using assms by (simp add: union_inter_neutral)
haftmann@35722
  1477
haftmann@35719
  1478
end
haftmann@35722
  1479
haftmann@35817
  1480
haftmann@35817
  1481
subsubsection {* The image case with flexible function *}
haftmann@35817
  1482
haftmann@35722
  1483
locale folding_image = comm_monoid +
haftmann@35722
  1484
  fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@35722
  1485
  assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
haftmann@35722
  1486
haftmann@35722
  1487
sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
haftmann@35722
  1488
qed (fact eq_fold)
haftmann@35722
  1489
haftmann@35722
  1490
context folding_image
haftmann@35722
  1491
begin
haftmann@35722
  1492
haftmann@35817
  1493
lemma reindex: (* FIXME polymorhism *)
haftmann@35722
  1494
  assumes "finite A" and "inj_on h A"
haftmann@35722
  1495
  shows "F g (h ` A) = F (g \<circ> h) A"
haftmann@35722
  1496
  using assms by (induct A) auto
haftmann@35722
  1497
haftmann@35722
  1498
lemma cong:
haftmann@35722
  1499
  assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
haftmann@35722
  1500
  shows "F g A = F h A"
haftmann@35722
  1501
proof -
haftmann@35722
  1502
  from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
haftmann@35722
  1503
  apply - apply (erule finite_induct) apply simp
haftmann@35722
  1504
  apply (simp add: subset_insert_iff, clarify)
haftmann@35722
  1505
  apply (subgoal_tac "finite C")
haftmann@35722
  1506
  prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
haftmann@35722
  1507
  apply (subgoal_tac "C = insert x (C - {x})")
haftmann@35722
  1508
  prefer 2 apply blast
haftmann@35722
  1509
  apply (erule ssubst)
haftmann@35722
  1510
  apply (drule spec)
haftmann@35722
  1511
  apply (erule (1) notE impE)
haftmann@35722
  1512
  apply (simp add: Ball_def del: insert_Diff_single)
haftmann@35722
  1513
  done
haftmann@35722
  1514
  with assms show ?thesis by simp
haftmann@35722
  1515
qed
haftmann@35722
  1516
haftmann@35722
  1517
lemma UNION_disjoint:
haftmann@35722
  1518
  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
haftmann@35722
  1519
  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
haftmann@35722
  1520
  shows "F g (UNION I A) = F (F g \<circ> A) I"
haftmann@35722
  1521
apply (insert assms)
haftmann@41656
  1522
apply (induct rule: finite_induct)
haftmann@41656
  1523
apply simp
haftmann@41656
  1524
apply atomize
haftmann@35722
  1525
apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
haftmann@35722
  1526
 prefer 2 apply blast
haftmann@35722
  1527
apply (subgoal_tac "A x Int UNION Fa A = {}")
haftmann@35722
  1528
 prefer 2 apply blast
haftmann@35722
  1529
apply (simp add: union_disjoint)
haftmann@35722
  1530
done
haftmann@35722
  1531
haftmann@35722
  1532
lemma distrib:
haftmann@35722
  1533
  assumes "finite A"
haftmann@35722
  1534
  shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
haftmann@35722
  1535
  using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
haftmann@35722
  1536
haftmann@35722
  1537
lemma related: 
haftmann@35722
  1538
  assumes Re: "R 1 1" 
haftmann@35722
  1539
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
haftmann@35722
  1540
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
haftmann@35722
  1541
  shows "R (F h S) (F g S)"
haftmann@35722
  1542
  using fS by (rule finite_subset_induct) (insert assms, auto)
haftmann@35722
  1543
haftmann@35722
  1544
lemma eq_general:
haftmann@35722
  1545
  assumes fS: "finite S"
haftmann@35722
  1546
  and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 
haftmann@35722
  1547
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
haftmann@35722
  1548
  shows "F f1 S = F f2 S'"
haftmann@35722
  1549
proof-
haftmann@35722
  1550
  from h f12 have hS: "h ` S = S'" by blast
haftmann@35722
  1551
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
haftmann@35722
  1552
    from f12 h H  have "x = y" by auto }
haftmann@35722
  1553
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
haftmann@35722
  1554
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
haftmann@35722
  1555
  from hS have "F f2 S' = F f2 (h ` S)" by simp
haftmann@35722
  1556
  also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
haftmann@35722
  1557
  also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
haftmann@35722
  1558
    by blast
haftmann@35722
  1559
  finally show ?thesis ..
haftmann@35722
  1560
qed
haftmann@35722
  1561
haftmann@35722
  1562
lemma eq_general_inverses:
haftmann@35722
  1563
  assumes fS: "finite S" 
haftmann@35722
  1564
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
haftmann@35722
  1565
  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
  1566
  shows "F j S = F g T"
haftmann@35722
  1567
  (* metis solves it, but not yet available here *)
haftmann@35722
  1568
  apply (rule eq_general [OF fS, of T h g j])
haftmann@35722
  1569
  apply (rule ballI)
haftmann@35722
  1570
  apply (frule kh)
haftmann@35722
  1571
  apply (rule ex1I[])
haftmann@35722
  1572
  apply blast
haftmann@35722
  1573
  apply clarsimp
haftmann@35722
  1574
  apply (drule hk) apply simp
haftmann@35722
  1575
  apply (rule sym)
haftmann@35722
  1576
  apply (erule conjunct1[OF conjunct2[OF hk]])
haftmann@35722
  1577
  apply (rule ballI)
haftmann@35722
  1578
  apply (drule hk)
haftmann@35722
  1579
  apply blast
haftmann@35722
  1580
  done
haftmann@35722
  1581
haftmann@35722
  1582
end
haftmann@35722
  1583
haftmann@35817
  1584
haftmann@35817
  1585
subsubsection {* The image case with fixed function and idempotency *}
haftmann@35817
  1586
haftmann@35817
  1587
locale folding_image_simple_idem = folding_image_simple +
haftmann@35817
  1588
  assumes idem: "x * x = x"
haftmann@35817
  1589
haftmann@35817
  1590
sublocale folding_image_simple_idem < semilattice proof
haftmann@35817
  1591
qed (fact idem)
haftmann@35817
  1592
haftmann@35817
  1593
context folding_image_simple_idem
haftmann@35817
  1594
begin
haftmann@35817
  1595
haftmann@35817
  1596
lemma in_idem:
haftmann@35817
  1597
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1598
  shows "g x * F A = F A"
haftmann@35817
  1599
  using assms by (induct A) (auto simp add: left_commute)
haftmann@35817
  1600
haftmann@35817
  1601
lemma subset_idem:
haftmann@35817
  1602
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1603
  shows "F B * F A = F A"
haftmann@35817
  1604
proof -
haftmann@35817
  1605
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1606
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1607
    (auto simp add: assoc in_idem `finite A`)
haftmann@35817
  1608
qed
haftmann@35817
  1609
haftmann@35817
  1610
declare insert [simp del]
haftmann@35817
  1611
haftmann@35817
  1612
lemma insert_idem [simp]:
haftmann@35817
  1613
  assumes "finite A"
haftmann@35817
  1614
  shows "F (insert x A) = g x * F A"
haftmann@35817
  1615
  using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
haftmann@35817
  1616
haftmann@35817
  1617
lemma union_idem:
haftmann@35817
  1618
  assumes "finite A" and "finite B"
haftmann@35817
  1619
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1620
proof -
haftmann@35817
  1621
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1622
  then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
haftmann@35817
  1623
  with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1624
qed
haftmann@35817
  1625
haftmann@35817
  1626
end
haftmann@35817
  1627
haftmann@35817
  1628
haftmann@35817
  1629
subsubsection {* The image case with flexible function and idempotency *}
haftmann@35817
  1630
haftmann@35817
  1631
locale folding_image_idem = folding_image +
haftmann@35817
  1632
  assumes idem: "x * x = x"
haftmann@35817
  1633
haftmann@35817
  1634
sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
haftmann@35817
  1635
qed (fact idem)
haftmann@35817
  1636
haftmann@35817
  1637
haftmann@35817
  1638
subsubsection {* The neutral-less case *}
haftmann@35817
  1639
haftmann@35817
  1640
locale folding_one = abel_semigroup +
haftmann@35817
  1641
  fixes F :: "'a set \<Rightarrow> 'a"
haftmann@35817
  1642
  assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
haftmann@35817
  1643
begin
haftmann@35817
  1644
haftmann@35817
  1645
lemma singleton [simp]:
haftmann@35817
  1646
  "F {x} = x"
haftmann@35817
  1647
  by (simp add: eq_fold)
haftmann@35817
  1648
haftmann@35817
  1649
lemma eq_fold':
haftmann@35817
  1650
  assumes "finite A" and "x \<notin> A"
haftmann@35817
  1651
  shows "F (insert x A) = fold (op *) x A"
haftmann@35817
  1652
proof -
haftmann@35817
  1653
  interpret ab_semigroup_mult "op *" proof qed (simp_all add: ac_simps)
haftmann@35817
  1654
  with assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
haftmann@35817
  1655
qed
haftmann@35817
  1656
haftmann@35817
  1657
lemma insert [simp]:
haftmann@36637
  1658
  assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
haftmann@36637
  1659
  shows "F (insert x A) = x * F A"
haftmann@36637
  1660
proof -
haftmann@36637
  1661
  from `A \<noteq> {}` obtain b where "b \<in> A" by blast
haftmann@35817
  1662
  then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1663
  with `finite A` have "finite B" by simp
haftmann@35817
  1664
  interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
nipkow@39302
  1665
  qed (simp_all add: fun_eq_iff ac_simps)
haftmann@42871
  1666
  thm fold.comp_fun_commute' [of B b, simplified fun_eq_iff, simplified]
haftmann@42871
  1667
  from `finite B` fold.comp_fun_commute' [of B x]
haftmann@35817
  1668
    have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
nipkow@39302
  1669
  then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)
haftmann@35817
  1670
  from `finite B` * fold.insert [of B b]
haftmann@35817
  1671
    have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
nipkow@39302
  1672
  then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)
haftmann@35817
  1673
  from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
haftmann@35817
  1674
qed
haftmann@35817
  1675
haftmann@35817
  1676
lemma remove:
haftmann@35817
  1677
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1678
  shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1679
proof -
haftmann@35817
  1680
  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1681
  with assms show ?thesis by simp
haftmann@35817
  1682
qed
haftmann@35817
  1683
haftmann@35817
  1684
lemma insert_remove:
haftmann@35817
  1685
  assumes "finite A"
haftmann@35817
  1686
  shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1687
  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
haftmann@35817
  1688
haftmann@35817
  1689
lemma union_disjoint:
haftmann@35817
  1690
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
haftmann@35817
  1691
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1692
  using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
haftmann@35817
  1693
haftmann@35817
  1694
lemma union_inter:
haftmann@35817
  1695
  assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
haftmann@35817
  1696
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35817
  1697
proof -
haftmann@35817
  1698
  from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
haftmann@35817
  1699
  from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
haftmann@35817
  1700
    case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
haftmann@35817
  1701
  next
haftmann@35817
  1702
    case (insert x A) show ?case proof (cases "x \<in> B")
haftmann@35817
  1703
      case True then have "B \<noteq> {}" by auto
haftmann@35817
  1704
      with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
haftmann@35817
  1705
        (simp_all add: insert_absorb ac_simps union_disjoint)
haftmann@35817
  1706
    next
haftmann@35817
  1707
      case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
haftmann@35817
  1708
      moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
haftmann@35817
  1709
        by auto
haftmann@35817
  1710
      ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
haftmann@35817
  1711
    qed
haftmann@35817
  1712
  qed
haftmann@35817
  1713
qed
haftmann@35817
  1714
haftmann@35817
  1715
lemma closed:
haftmann@35817
  1716
  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
haftmann@35817
  1717
  shows "F A \<in> A"
haftmann@35817
  1718
using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
haftmann@35817
  1719
  case singleton then show ?case by simp
haftmann@35817
  1720
next
haftmann@35817
  1721
  case insert with elem show ?case by force
haftmann@35817
  1722
qed
haftmann@35817
  1723
haftmann@35817
  1724
end
haftmann@35817
  1725
haftmann@35817
  1726
haftmann@35817
  1727
subsubsection {* The neutral-less case with idempotency *}
haftmann@35817
  1728
haftmann@35817
  1729
locale folding_one_idem = folding_one +
haftmann@35817
  1730
  assumes idem: "x * x = x"
haftmann@35817
  1731
haftmann@35817
  1732
sublocale folding_one_idem < semilattice proof
haftmann@35817
  1733
qed (fact idem)
haftmann@35817
  1734
haftmann@35817
  1735
context folding_one_idem
haftmann@35817
  1736
begin
haftmann@35817
  1737
haftmann@35817
  1738
lemma in_idem:
haftmann@35817
  1739
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1740
  shows "x * F A = F A"
haftmann@35817
  1741
proof -
haftmann@35817
  1742
  from assms have "A \<noteq> {}" by auto
haftmann@35817
  1743
  with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
haftmann@35817
  1744
qed
haftmann@35817
  1745
haftmann@35817
  1746
lemma subset_idem:
haftmann@35817
  1747
  assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
haftmann@35817
  1748
  shows "F B * F A = F A"
haftmann@35817
  1749
proof -
haftmann@35817
  1750
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1751
  then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
haftmann@35817
  1752
    (simp_all add: assoc in_idem `finite A`)
haftmann@35817
  1753
qed
haftmann@35817
  1754
haftmann@35817
  1755
lemma eq_fold_idem':
haftmann@35817
  1756
  assumes "finite A"
haftmann@35817
  1757
  shows "F (insert a A) = fold (op *) a A"
haftmann@35817
  1758
proof -
haftmann@35817
  1759
  interpret ab_semigroup_idem_mult "op *" proof qed (simp_all add: ac_simps)
haftmann@35817
  1760
  with assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
haftmann@35817
  1761
qed
haftmann@35817
  1762
haftmann@35817
  1763
lemma insert_idem [simp]:
haftmann@36637
  1764
  assumes "finite A" and "A \<noteq> {}"
haftmann@36637
  1765
  shows "F (insert x A) = x * F A"
haftmann@35817
  1766
proof (cases "x \<in> A")
haftmann@36637
  1767
  case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
haftmann@35817
  1768
next
haftmann@36637
  1769
  case True
haftmann@36637
  1770
  from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
haftmann@35817
  1771
qed
haftmann@35817
  1772
  
haftmann@35817
  1773
lemma union_idem:
haftmann@35817
  1774
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
haftmann@35817
  1775
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1776
proof (cases "A \<inter> B = {}")
haftmann@35817
  1777
  case True with assms show ?thesis by (simp add: union_disjoint)
haftmann@35817
  1778
next
haftmann@35817
  1779
  case False
haftmann@35817
  1780
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1781
  with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
haftmann@35817
  1782
  with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1783
qed
haftmann@35817
  1784
haftmann@35817
  1785
lemma hom_commute:
haftmann@35817
  1786
  assumes hom: "\<And>x y. h (x * y) = h x * h y"
haftmann@35817
  1787
  and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
haftmann@35817
  1788
using N proof (induct rule: finite_ne_induct)
haftmann@35817
  1789
  case singleton thus ?case by simp
haftmann@35817
  1790
next
haftmann@35817
  1791
  case (insert n N)
haftmann@35817
  1792
  then have "h (F (insert n N)) = h (n * F N)" by simp
haftmann@35817
  1793
  also have "\<dots> = h n * h (F N)" by (rule hom)
haftmann@35817
  1794
  also have "h (F N) = F (h ` N)" by(rule insert)
haftmann@35817
  1795
  also have "h n * \<dots> = F (insert (h n) (h ` N))"
haftmann@35817
  1796
    using insert by(simp)
haftmann@35817
  1797
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@35817
  1798
  finally show ?case .
haftmann@35817
  1799
qed
haftmann@35817
  1800
haftmann@35817
  1801
end
haftmann@35817
  1802
haftmann@35796
  1803
notation times (infixl "*" 70)
haftmann@35796
  1804
notation Groups.one ("1")
haftmann@35722
  1805
haftmann@35722
  1806
haftmann@35722
  1807
subsection {* Finite cardinality *}
haftmann@35722
  1808
haftmann@35722
  1809
text {* This definition, although traditional, is ugly to work with:
haftmann@35722
  1810
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
haftmann@35722
  1811
But now that we have @{text fold_image} things are easy:
haftmann@35722
  1812
*}
haftmann@35722
  1813
haftmann@35722
  1814
definition card :: "'a set \<Rightarrow> nat" where
haftmann@35722
  1815
  "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
haftmann@35722
  1816
haftmann@37770
  1817
interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
haftmann@35722
  1818
qed (simp add: card_def)
haftmann@35722
  1819
haftmann@35722
  1820
lemma card_infinite [simp]:
haftmann@35722
  1821
  "\<not> finite A \<Longrightarrow> card A = 0"
haftmann@35722
  1822
  by (simp add: card_def)
haftmann@35722
  1823
haftmann@35722
  1824
lemma card_empty:
haftmann@35722
  1825
  "card {} = 0"
haftmann@35722
  1826
  by (fact card.empty)
haftmann@35722
  1827
haftmann@35722
  1828
lemma card_insert_disjoint:
haftmann@35722
  1829
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
haftmann@35722
  1830
  by simp
haftmann@35722
  1831
haftmann@35722
  1832
lemma card_insert_if:
haftmann@35722
  1833
  "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
haftmann@35722
  1834
  by auto (simp add: card.insert_remove card.remove)
haftmann@35722
  1835
haftmann@35722
  1836
lemma card_ge_0_finite:
haftmann@35722
  1837
  "card A > 0 \<Longrightarrow> finite A"
haftmann@35722
  1838
  by (rule ccontr) simp
haftmann@35722
  1839
blanchet@35828
  1840
lemma card_0_eq [simp, no_atp]:
haftmann@35722
  1841
  "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
haftmann@35722
  1842
  by (auto dest: mk_disjoint_insert)
haftmann@35722
  1843
haftmann@35722
  1844
lemma finite_UNIV_card_ge_0:
haftmann@35722
  1845
  "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
haftmann@35722
  1846
  by (rule ccontr) simp
haftmann@35722
  1847
haftmann@35722
  1848
lemma card_eq_0_iff:
haftmann@35722
  1849
  "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
haftmann@35722
  1850
  by auto
haftmann@35722
  1851
haftmann@35722
  1852
lemma card_gt_0_iff:
haftmann@35722
  1853
  "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
haftmann@35722
  1854
  by (simp add: neq0_conv [symmetric] card_eq_0_iff) 
haftmann@35722
  1855
haftmann@35722
  1856
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
haftmann@35722
  1857
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
haftmann@35722
  1858
apply(simp del:insert_Diff_single)
haftmann@35722
  1859
done
haftmann@35722
  1860
haftmann@35722
  1861
lemma card_Diff_singleton:
haftmann@35722
  1862
  "finite A ==> x: A ==> card (A - {x}) = card A - 1"
haftmann@35722
  1863
by (simp add: card_Suc_Diff1 [symmetric])
haftmann@35722
  1864
haftmann@35722
  1865
lemma card_Diff_singleton_if:
bulwahn@45166
  1866
  "finite A ==> card (A - {x}) = (if x : A then card A - 1 else card A)"
haftmann@35722
  1867
by (simp add: card_Diff_singleton)
haftmann@35722
  1868
haftmann@35722
  1869
lemma card_Diff_insert[simp]:
haftmann@35722
  1870
assumes "finite A" and "a:A" and "a ~: B"
haftmann@35722
  1871
shows "card(A - insert a B) = card(A - B) - 1"
haftmann@35722
  1872
proof -
haftmann@35722
  1873
  have "A - insert a B = (A - B) - {a}" using assms by blast
haftmann@35722
  1874
  then show ?thesis using assms by(simp add:card_Diff_singleton)
haftmann@35722
  1875
qed
haftmann@35722
  1876
haftmann@35722
  1877
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
haftmann@35722
  1878
by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  1879
haftmann@35722
  1880
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
haftmann@35722
  1881
by (simp add: card_insert_if)
haftmann@35722
  1882
nipkow@41987
  1883
lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
nipkow@41987
  1884
by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
nipkow@41987
  1885
nipkow@41988
  1886
lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
nipkow@41987
  1887
using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
nipkow@41987
  1888
haftmann@35722
  1889
lemma card_mono:
haftmann@35722
  1890
  assumes "finite B" and "A \<subseteq> B"
haftmann@35722
  1891
  shows "card A \<le> card B"
haftmann@35722
  1892
proof -
haftmann@35722
  1893
  from assms have "finite A" by (auto intro: finite_subset)
haftmann@35722
  1894
  then show ?thesis using assms proof (induct A arbitrary: B)
haftmann@35722
  1895
    case empty then show ?case by simp
haftmann@35722
  1896
  next
haftmann@35722
  1897
    case (insert x A)
haftmann@35722
  1898
    then have "x \<in> B" by simp
haftmann@35722
  1899
    from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
haftmann@35722
  1900
    with insert.hyps have "card A \<le> card (B - {x})" by auto
haftmann@35722
  1901
    with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
haftmann@35722
  1902
  qed
haftmann@35722
  1903
qed
haftmann@35722
  1904
haftmann@35722
  1905
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
haftmann@41656
  1906
apply (induct rule: finite_induct)
haftmann@41656
  1907
apply simp
haftmann@41656
  1908
apply clarify
haftmann@35722
  1909
apply (subgoal_tac "finite A & A - {x} <= F")
haftmann@35722
  1910
 prefer 2 apply (blast intro: finite_subset, atomize)
haftmann@35722
  1911
apply (drule_tac x = "A - {x}" in spec)
haftmann@35722
  1912
apply (simp add: card_Diff_singleton_if split add: split_if_asm)
haftmann@35722
  1913
apply (case_tac "card A", auto)
haftmann@35722
  1914
done
haftmann@35722
  1915
haftmann@35722
  1916
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
haftmann@35722
  1917
apply (simp add: psubset_eq linorder_not_le [symmetric])
haftmann@35722
  1918
apply (blast dest: card_seteq)
haftmann@35722
  1919
done
haftmann@35722
  1920
haftmann@35722
  1921
lemma card_Un_Int: "finite A ==> finite B
haftmann@35722
  1922
    ==> card A + card B = card (A Un B) + card (A Int B)"
haftmann@35817
  1923
  by (fact card.union_inter [symmetric])
haftmann@35722
  1924
haftmann@35722
  1925
lemma card_Un_disjoint: "finite A ==> finite B
haftmann@35722
  1926
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
haftmann@35722
  1927
  by (fact card.union_disjoint)
haftmann@35722
  1928
haftmann@35722
  1929
lemma card_Diff_subset:
haftmann@35722
  1930
  assumes "finite B" and "B \<subseteq> A"
haftmann@35722
  1931
  shows "card (A - B) = card A - card B"
haftmann@35722
  1932
proof (cases "finite A")
haftmann@35722
  1933
  case False with assms show ?thesis by simp
haftmann@35722
  1934
next
haftmann@35722
  1935
  case True with assms show ?thesis by (induct B arbitrary: A) simp_all
haftmann@35722
  1936
qed
haftmann@35722
  1937
haftmann@35722
  1938
lemma card_Diff_subset_Int:
haftmann@35722
  1939
  assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
haftmann@35722
  1940
proof -
haftmann@35722
  1941
  have "A - B = A - A \<inter> B" by auto
haftmann@35722
  1942
  thus ?thesis
haftmann@35722
  1943
    by (simp add: card_Diff_subset AB) 
haftmann@35722
  1944
qed
haftmann@35722
  1945
nipkow@40716
  1946
lemma diff_card_le_card_Diff:
nipkow@40716
  1947
assumes "finite B" shows "card A - card B \<le> card(A - B)"
nipkow@40716
  1948
proof-
nipkow@40716
  1949
  have "card A - card B \<le> card A - card (A \<inter> B)"
nipkow@40716
  1950
    using card_mono[OF assms Int_lower2, of A] by arith
nipkow@40716
  1951
  also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
nipkow@40716
  1952
  finally show ?thesis .
nipkow@40716
  1953
qed
nipkow@40716
  1954
haftmann@35722
  1955
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
haftmann@35722
  1956
apply (rule Suc_less_SucD)
haftmann@35722
  1957
apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  1958
done
haftmann@35722
  1959
haftmann@35722
  1960
lemma card_Diff2_less:
haftmann@35722
  1961
  "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
haftmann@35722
  1962
apply (case_tac "x = y")
haftmann@35722
  1963
 apply (simp add: card_Diff1_less del:card_Diff_insert)
haftmann@35722
  1964
apply (rule less_trans)
haftmann@35722
  1965
 prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
haftmann@35722
  1966
done
haftmann@35722
  1967
haftmann@35722
  1968
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
haftmann@35722
  1969
apply (case_tac "x : A")
haftmann@35722
  1970
 apply (simp_all add: card_Diff1_less less_imp_le)
haftmann@35722
  1971
done
haftmann@35722
  1972
haftmann@35722
  1973
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
haftmann@35722
  1974
by (erule psubsetI, blast)
haftmann@35722
  1975
haftmann@35722
  1976
lemma insert_partition:
haftmann@35722
  1977
  "\<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
  1978
  \<Longrightarrow> x \<inter> \<Union> F = {}"
haftmann@35722
  1979
by auto
haftmann@35722
  1980
haftmann@35722
  1981
lemma finite_psubset_induct[consumes 1, case_names psubset]:
urbanc@36079
  1982
  assumes fin: "finite A" 
urbanc@36079
  1983
  and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A" 
urbanc@36079
  1984
  shows "P A"
urbanc@36079
  1985
using fin
urbanc@36079
  1986
proof (induct A taking: card rule: measure_induct_rule)
haftmann@35722
  1987
  case (less A)
urbanc@36079
  1988
  have fin: "finite A" by fact
urbanc@36079
  1989
  have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
urbanc@36079
  1990
  { fix B 
urbanc@36079
  1991
    assume asm: "B \<subset> A"
urbanc@36079
  1992
    from asm have "card B < card A" using psubset_card_mono fin by blast
urbanc@36079
  1993
    moreover
urbanc@36079
  1994
    from asm have "B \<subseteq> A" by auto
urbanc@36079
  1995
    then have "finite B" using fin finite_subset by blast
urbanc@36079
  1996
    ultimately 
urbanc@36079
  1997
    have "P B" using ih by simp
urbanc@36079
  1998
  }
urbanc@36079
  1999
  with fin show "P A" using major by blast
haftmann@35722
  2000
qed
haftmann@35722
  2001
haftmann@35722
  2002
text{* main cardinality theorem *}
haftmann@35722
  2003
lemma card_partition [rule_format]:
haftmann@35722
  2004
  "finite C ==>
haftmann@35722
  2005
     finite (\<Union> C) -->
haftmann@35722
  2006
     (\<forall>c\<in>C. card c = k) -->
haftmann@35722
  2007
     (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
haftmann@35722
  2008
     k * card(C) = card (\<Union> C)"
haftmann@35722
  2009
apply (erule finite_induct, simp)
haftmann@35722
  2010
apply (simp add: card_Un_disjoint insert_partition 
haftmann@35722
  2011
       finite_subset [of _ "\<Union> (insert x F)"])
haftmann@35722
  2012
done
haftmann@35722
  2013
haftmann@35722
  2014
lemma card_eq_UNIV_imp_eq_UNIV:
haftmann@35722
  2015
  assumes fin: "finite (UNIV :: 'a set)"
haftmann@35722
  2016
  and card: "card A = card (UNIV :: 'a set)"
haftmann@35722
  2017
  shows "A = (UNIV :: 'a set)"
haftmann@35722
  2018
proof
haftmann@35722
  2019
  show "A \<subseteq> UNIV" by simp
haftmann@35722
  2020
  show "UNIV \<subseteq> A"
haftmann@35722
  2021
  proof
haftmann@35722
  2022
    fix x
haftmann@35722
  2023
    show "x \<in> A"
haftmann@35722
  2024
    proof (rule ccontr)
haftmann@35722
  2025
      assume "x \<notin> A"
haftmann@35722
  2026
      then have "A \<subset> UNIV" by auto
haftmann@35722
  2027
      with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
haftmann@35722
  2028
      with card show False by simp
haftmann@35722
  2029
    qed
haftmann@35722
  2030
  qed
haftmann@35722
  2031
qed
haftmann@35722
  2032
haftmann@35722
  2033
text{*The form of a finite set of given cardinality*}
haftmann@35722
  2034
haftmann@35722
  2035
lemma card_eq_SucD:
haftmann@35722
  2036
assumes "card A = Suc k"
haftmann@35722
  2037
shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
haftmann@35722
  2038
proof -
haftmann@35722
  2039
  have fin: "finite A" using assms by (auto intro: ccontr)
haftmann@35722
  2040
  moreover have "card A \<noteq> 0" using assms by auto
haftmann@35722
  2041
  ultimately obtain b where b: "b \<in> A" by auto
haftmann@35722
  2042
  show ?thesis
haftmann@35722
  2043
  proof (intro exI conjI)
haftmann@35722
  2044
    show "A = insert b (A-{b})" using b by blast
haftmann@35722
  2045
    show "b \<notin> A - {b}" by blast
haftmann@35722
  2046
    show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
nipkow@44890
  2047
      using assms b fin by(fastforce dest:mk_disjoint_insert)+
haftmann@35722
  2048
  qed
haftmann@35722
  2049
qed
haftmann@35722
  2050
haftmann@35722
  2051
lemma card_Suc_eq:
haftmann@35722
  2052
  "(card A = Suc k) =
haftmann@35722
  2053
   (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
haftmann@35722
  2054
apply(rule iffI)
haftmann@35722
  2055
 apply(erule card_eq_SucD)
haftmann@35722
  2056
apply(auto)
haftmann@35722
  2057
apply(subst card_insert)
haftmann@35722
  2058
 apply(auto intro:ccontr)
haftmann@35722
  2059
done
haftmann@35722
  2060
nipkow@44744
  2061
lemma card_le_Suc_iff: "finite A \<Longrightarrow>
nipkow@44744
  2062
  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
  2063
by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
nipkow@44744
  2064
  dest: subset_singletonD split: nat.splits if_splits)
nipkow@44744
  2065
haftmann@35722
  2066
lemma finite_fun_UNIVD2:
haftmann@35722
  2067
  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
haftmann@35722
  2068
  shows "finite (UNIV :: 'b set)"
haftmann@35722
  2069
proof -
haftmann@46146
  2070
  from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
haftmann@46146
  2071
    by (rule finite_imageI)
haftmann@46146
  2072
  moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
haftmann@46146
  2073
    by (rule UNIV_eq_I) auto
haftmann@35722
  2074
  ultimately show "finite (UNIV :: 'b set)" by simp
haftmann@35722
  2075
qed
haftmann@35722
  2076
haftmann@35722
  2077
lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
haftmann@35722
  2078
  unfolding UNIV_unit by simp
haftmann@35722
  2079
haftmann@35722
  2080
haftmann@35722
  2081
subsubsection {* Cardinality of image *}
haftmann@35722
  2082
haftmann@35722
  2083
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
haftmann@41656
  2084
apply (induct rule: finite_induct)
haftmann@35722
  2085
 apply simp
haftmann@35722
  2086
apply (simp add: le_SucI card_insert_if)
haftmann@35722
  2087
done
haftmann@35722
  2088
haftmann@35722
  2089
lemma card_image:
haftmann@35722
  2090
  assumes "inj_on f A"
haftmann@35722
  2091
  shows "card (f ` A) = card A"
haftmann@35722
  2092
proof (cases "finite A")
haftmann@35722
  2093
  case True then show ?thesis using assms by (induct A) simp_all
haftmann@35722
  2094
next
haftmann@35722
  2095
  case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
haftmann@35722
  2096
  with False show ?thesis by simp
haftmann@35722
  2097
qed
haftmann@35722
  2098
haftmann@35722
  2099
lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
haftmann@35722
  2100
by(auto simp: card_image bij_betw_def)
haftmann@35722
  2101
haftmann@35722
  2102
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
haftmann@35722
  2103
by (simp add: card_seteq card_image)
haftmann@35722
  2104
haftmann@35722
  2105
lemma eq_card_imp_inj_on:
haftmann@35722
  2106
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
haftmann@35722
  2107
apply (induct rule:finite_induct)
haftmann@35722
  2108
apply simp
haftmann@35722
  2109
apply(frule card_image_le[where f = f])
haftmann@35722
  2110
apply(simp add:card_insert_if split:if_splits)
haftmann@35722
  2111
done
haftmann@35722
  2112
haftmann@35722
  2113
lemma inj_on_iff_eq_card:
haftmann@35722
  2114
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
haftmann@35722
  2115
by(blast intro: card_image eq_card_imp_inj_on)
haftmann@35722
  2116
haftmann@35722
  2117
haftmann@35722
  2118
lemma card_inj_on_le:
haftmann@35722
  2119
  "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
haftmann@35722
  2120
apply (subgoal_tac "finite A") 
haftmann@35722
  2121
 apply (force intro: card_mono simp add: card_image [symmetric])
haftmann@35722
  2122
apply (blast intro: finite_imageD dest: finite_subset) 
haftmann@35722
  2123
done
haftmann@35722
  2124
haftmann@35722
  2125
lemma card_bij_eq:
haftmann@35722
  2126
  "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
haftmann@35722
  2127
     finite A; finite B |] ==> card A = card B"
haftmann@35722
  2128
by (auto intro: le_antisym card_inj_on_le)
haftmann@35722
  2129
hoelzl@40703
  2130
lemma bij_betw_finite:
hoelzl@40703
  2131
  assumes "bij_betw f A B"
hoelzl@40703
  2132
  shows "finite A \<longleftrightarrow> finite B"
hoelzl@40703
  2133
using assms unfolding bij_betw_def
hoelzl@40703
  2134
using finite_imageD[of f A] by auto
haftmann@35722
  2135
haftmann@41656
  2136
nipkow@37466
  2137
subsubsection {* Pigeonhole Principles *}
nipkow@37466
  2138
nipkow@40311
  2139
lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
nipkow@37466
  2140
by (auto dest: card_image less_irrefl_nat)
nipkow@37466
  2141
nipkow@37466
  2142
lemma pigeonhole_infinite:
nipkow@37466
  2143
assumes  "~ finite A" and "finite(f`A)"
nipkow@37466
  2144
shows "EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  2145
proof -
nipkow@37466
  2146
  have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  2147
  proof(induct "f`A" arbitrary: A rule: finite_induct)
nipkow@37466
  2148
    case empty thus ?case by simp
nipkow@37466
  2149
  next
nipkow@37466
  2150
    case (insert b F)
nipkow@37466
  2151
    show ?case
nipkow@37466
  2152
    proof cases
nipkow@37466
  2153
      assume "finite{a:A. f a = b}"
nipkow@37466
  2154
      hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
nipkow@37466
  2155
      also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
nipkow@37466
  2156
      finally have "~ finite({a:A. f a \<noteq> b})" .
nipkow@37466
  2157
      from insert(3)[OF _ this]
nipkow@37466
  2158
      show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
nipkow@37466
  2159
    next
nipkow@37466
  2160
      assume 1: "~finite{a:A. f a = b}"
nipkow@37466
  2161
      hence "{a \<in> A. f a = b} \<noteq> {}" by force
nipkow@37466
  2162
      thus ?thesis using 1 by blast
nipkow@37466
  2163
    qed
nipkow@37466
  2164
  qed
nipkow@37466
  2165
  from this[OF assms(2,1)] show ?thesis .
nipkow@37466
  2166
qed
nipkow@37466
  2167
nipkow@37466
  2168
lemma pigeonhole_infinite_rel:
nipkow@37466
  2169
assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
nipkow@37466
  2170
shows "EX b:B. ~finite{a:A. R a b}"
nipkow@37466
  2171
proof -
nipkow@37466
  2172
   let ?F = "%a. {b:B. R a b}"
nipkow@37466
  2173
   from finite_Pow_iff[THEN iffD2, OF `finite B`]
nipkow@37466
  2174
   have "finite(?F ` A)" by(blast intro: rev_finite_subset)
nipkow@37466
  2175
   from pigeonhole_infinite[where f = ?F, OF assms(1) this]
nipkow@37466
  2176
   obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
nipkow@37466
  2177
   obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
nipkow@37466
  2178
   { assume "finite{a:A. R a b0}"
nipkow@37466
  2179
     then have "finite {a\<in>A. ?F a = ?F a0}"
nipkow@37466
  2180
       using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
nipkow@37466
  2181
   }
nipkow@37466
  2182
   with 1 `b0 : B` show ?thesis by blast
nipkow@37466
  2183
qed
nipkow@37466
  2184
nipkow@37466
  2185
haftmann@35722
  2186
subsubsection {* Cardinality of sums *}
haftmann@35722
  2187
haftmann@35722
  2188
lemma card_Plus:
haftmann@35722
  2189
  assumes "finite A" and "finite B"
haftmann@35722
  2190
  shows "card (A <+> B) = card A + card B"
haftmann@35722
  2191
proof -
haftmann@35722
  2192
  have "Inl`A \<inter> Inr`B = {}" by fast
haftmann@35722
  2193
  with assms show ?thesis
haftmann@35722
  2194
    unfolding Plus_def
haftmann@35722
  2195
    by (simp add: card_Un_disjoint card_image)
haftmann@35722
  2196
qed
haftmann@35722
  2197
haftmann@35722
  2198
lemma card_Plus_conv_if:
haftmann@35722
  2199
  "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
haftmann@35722
  2200
  by (auto simp add: card_Plus)
haftmann@35722
  2201
haftmann@35722
  2202
haftmann@35722
  2203
subsubsection {* Cardinality of the Powerset *}
haftmann@35722
  2204
haftmann@35722
  2205
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
haftmann@41656
  2206
apply (induct rule: finite_induct)
haftmann@35722
  2207
 apply (simp_all add: Pow_insert)
haftmann@35722
  2208
apply (subst card_Un_disjoint, blast)
nipkow@40786
  2209
  apply (blast, blast)
haftmann@35722
  2210
apply (subgoal_tac "inj_on (insert x) (Pow F)")
haftmann@35722
  2211
 apply (simp add: card_image Pow_insert)
haftmann@35722
  2212
apply (unfold inj_on_def)
haftmann@35722
  2213
apply (blast elim!: equalityE)
haftmann@35722
  2214
done
haftmann@35722
  2215
nipkow@41987
  2216
text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
haftmann@35722
  2217
haftmann@35722
  2218
lemma dvd_partition:
haftmann@35722
  2219
  "finite (Union C) ==>
haftmann@35722
  2220
    ALL c : C. k dvd card c ==>
haftmann@35722
  2221
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
haftmann@35722
  2222
  k dvd card (Union C)"
haftmann@41656
  2223
apply (frule finite_UnionD)
haftmann@41656
  2224
apply (rotate_tac -1)
haftmann@41656
  2225
apply (induct rule: finite_induct)
haftmann@41656
  2226
apply simp_all
haftmann@41656
  2227
apply clarify
haftmann@35722
  2228
apply (subst card_Un_disjoint)
haftmann@35722
  2229
   apply (auto simp add: disjoint_eq_subset_Compl)
haftmann@35722
  2230
done
haftmann@35722
  2231
haftmann@35722
  2232
haftmann@35722
  2233
subsubsection {* Relating injectivity and surjectivity *}
haftmann@35722
  2234
haftmann@41656
  2235
lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
haftmann@35722
  2236
apply(rule eq_card_imp_inj_on, assumption)
haftmann@35722
  2237
apply(frule finite_imageI)
haftmann@35722
  2238
apply(drule (1) card_seteq)
haftmann@35722
  2239
 apply(erule card_image_le)
haftmann@35722
  2240
apply simp
haftmann@35722
  2241
done
haftmann@35722
  2242
haftmann@35722
  2243
lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2244
shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
hoelzl@40702
  2245
by (blast intro: finite_surj_inj subset_UNIV)
haftmann@35722
  2246
haftmann@35722
  2247
lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2248
shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
nipkow@44890
  2249
by(fastforce simp:surj_def dest!: endo_inj_surj)
haftmann@35722
  2250
haftmann@35722
  2251
corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
haftmann@35722
  2252
proof
haftmann@35722
  2253
  assume "finite(UNIV::nat set)"
haftmann@35722
  2254
  with finite_UNIV_inj_surj[of Suc]
haftmann@35722
  2255
  show False by simp (blast dest: Suc_neq_Zero surjD)
haftmann@35722
  2256
qed
haftmann@35722
  2257
blanchet@35828
  2258
(* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
blanchet@35828
  2259
lemma infinite_UNIV_char_0[no_atp]:
haftmann@35722
  2260
  "\<not> finite (UNIV::'a::semiring_char_0 set)"
haftmann@35722
  2261
proof
haftmann@35722
  2262
  assume "finite (UNIV::'a set)"
haftmann@35722
  2263
  with subset_UNIV have "finite (range of_nat::'a set)"
haftmann@35722
  2264
    by (rule finite_subset)
haftmann@35722
  2265
  moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
haftmann@35722
  2266
    by (simp add: inj_on_def)
haftmann@35722
  2267
  ultimately have "finite (UNIV::nat set)"
haftmann@35722
  2268
    by (rule finite_imageD)
haftmann@35722
  2269
  then show "False"
haftmann@35722
  2270
    by simp
haftmann@35722
  2271
qed
haftmann@35722
  2272
haftmann@46033
  2273
hide_const (open) Finite_Set.fold
haftmann@46033
  2274
haftmann@35722
  2275
end