src/HOL/Finite_Set.thy
author wenzelm
Tue Mar 13 13:31:26 2012 +0100 (2012-03-13)
changeset 46898 1570b30ee040
parent 46146 6baea4fca6bd
child 47210 b1dd32b2a505
permissions -rw-r--r--
tuned proofs -- eliminated pointless chaining of facts after 'interpret';
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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`
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proof induct
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  show "P {}" by fact
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  fix x F assume F: "finite F" and P: "P F"
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  show "P (insert x F)"
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  proof cases
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    assume "x \<in> F"
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    hence "insert x F = F" by (rule insert_absorb)
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    with P show ?thesis by (simp only:)
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  next
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    assume "x \<notin> F"
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    from F this P show ?thesis by (rule insert)
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  qed
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qed
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subsubsection {* Choice principles *}
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lemma ex_new_if_finite: -- "does not depend on def of finite at all"
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  assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
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  shows "\<exists>a::'a. a \<notin> A"
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proof -
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  from assms have "A \<noteq> UNIV" by blast
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  then show ?thesis by blast
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qed
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text {* A finite choice principle. Does not need the SOME choice operator. *}
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lemma finite_set_choice:
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  "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
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proof (induct rule: finite_induct)
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  case empty then show ?case by simp
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next
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  case (insert a A)
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  then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
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  show ?case (is "EX f. ?P f")
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  proof
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    show "?P(%x. if x = a then b else f x)" using f ab by auto
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  qed
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qed
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subsubsection {* Finite sets are the images of initial segments of natural numbers *}
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lemma finite_imp_nat_seg_image_inj_on:
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  assumes "finite A" 
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  shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
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using assms
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proof induct
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  case empty
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  show ?case
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  proof
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    show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp 
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  qed
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next
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  case (insert a A)
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  have notinA: "a \<notin> A" by fact
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  from insert.hyps obtain n f
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    where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
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  hence "insert a A = f(n:=a) ` {i. i < Suc n}"
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        "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
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    by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
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  thus ?case by blast
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qed
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lemma nat_seg_image_imp_finite:
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  "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
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proof (induct n arbitrary: A)
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  case 0 thus ?case by simp
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next
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  case (Suc n)
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  let ?B = "f ` {i. i < n}"
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  have finB: "finite ?B" by(rule Suc.hyps[OF refl])
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  show ?case
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  proof cases
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    assume "\<exists>k<n. f n = f k"
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    hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  next
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    assume "\<not>(\<exists> k<n. f n = f k)"
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    hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  qed
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qed
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lemma finite_conv_nat_seg_image:
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  "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
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  by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
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lemma finite_imp_inj_to_nat_seg:
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  assumes "finite A"
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  shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
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proof -
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  from finite_imp_nat_seg_image_inj_on[OF `finite A`]
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  obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
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    by (auto simp:bij_betw_def)
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  let ?f = "the_inv_into {i. i<n} f"
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  have "inj_on ?f A & ?f ` A = {i. i<n}"
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    by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
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  thus ?thesis by blast
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qed
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lemma finite_Collect_less_nat [iff]:
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  "finite {n::nat. n < k}"
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  by (fastforce simp: finite_conv_nat_seg_image)
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lemma finite_Collect_le_nat [iff]:
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  "finite {n::nat. n \<le> k}"
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  by (simp add: le_eq_less_or_eq Collect_disj_eq)
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subsubsection {* Finiteness and common set operations *}
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lemma rev_finite_subset:
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  "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
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proof (induct arbitrary: A rule: finite_induct)
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  case empty
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  then show ?case by simp
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next
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  case (insert x F A)
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  have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
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  show "finite A"
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  proof cases
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    assume x: "x \<in> A"
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    with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
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    with r have "finite (A - {x})" .
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    hence "finite (insert x (A - {x}))" ..
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    also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
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    finally show ?thesis .
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  next
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    show "A \<subseteq> F ==> ?thesis" by fact
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    assume "x \<notin> A"
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    with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
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  qed
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qed
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lemma finite_subset:
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  "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
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  by (rule rev_finite_subset)
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lemma finite_UnI:
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  assumes "finite F" and "finite G"
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  shows "finite (F \<union> G)"
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  using assms by induct simp_all
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lemma finite_Un [iff]:
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  "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
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  by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
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lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
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proof -
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  have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
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  then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
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  then show ?thesis by simp
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qed
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lemma finite_Int [simp, intro]:
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  "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
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  by (blast intro: finite_subset)
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lemma finite_Collect_conjI [simp, intro]:
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  "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
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  by (simp add: Collect_conj_eq)
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lemma finite_Collect_disjI [simp]:
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  "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
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  by (simp add: Collect_disj_eq)
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lemma finite_Diff [simp, intro]:
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  "finite A \<Longrightarrow> finite (A - B)"
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  by (rule finite_subset, rule Diff_subset)
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lemma finite_Diff2 [simp]:
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  assumes "finite B"
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  shows "finite (A - B) \<longleftrightarrow> finite A"
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proof -
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  have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
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  also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
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  finally show ?thesis ..
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qed
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lemma finite_Diff_insert [iff]:
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  "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
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proof -
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  have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
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  moreover have "A - insert a B = A - B - {a}" by auto
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  ultimately show ?thesis by simp
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qed
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lemma finite_compl[simp]:
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  "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
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  by (simp add: Compl_eq_Diff_UNIV)
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lemma finite_Collect_not[simp]:
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  "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
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  by (simp add: Collect_neg_eq)
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lemma finite_Union [simp, intro]:
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  "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
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  by (induct rule: finite_induct) simp_all
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lemma finite_UN_I [intro]:
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  "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
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  by (induct rule: finite_induct) simp_all
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lemma finite_UN [simp]:
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  "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
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  by (blast intro: finite_subset)
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lemma finite_Inter [intro]:
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  "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
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  by (blast intro: Inter_lower finite_subset)
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lemma finite_INT [intro]:
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  "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
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  by (blast intro: INT_lower finite_subset)
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lemma finite_imageI [simp, intro]:
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  "finite F \<Longrightarrow> finite (h ` F)"
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  by (induct rule: finite_induct) simp_all
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lemma finite_image_set [simp]:
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  "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
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  by (simp add: image_Collect [symmetric])
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lemma finite_imageD:
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  assumes "finite (f ` A)" and "inj_on f A"
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  shows "finite A"
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using assms
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proof (induct "f ` A" arbitrary: A)
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  case empty then show ?case by simp
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next
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  case (insert x B)
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  then have B_A: "insert x B = f ` A" by simp
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  then obtain y where "x = f y" and "y \<in> A" by blast
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  from B_A `x \<notin> B` have "B = f ` A - {x}" by blast
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  with B_A `x \<notin> B` `x = f y` `inj_on f A` `y \<in> A` have "B = f ` (A - {y})" by (simp add: inj_on_image_set_diff)
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  moreover from `inj_on f A` have "inj_on f (A - {y})" by (rule inj_on_diff)
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  ultimately have "finite (A - {y})" by (rule insert.hyps)
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  then show "finite A" by simp
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qed
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lemma finite_surj:
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  "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
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  by (erule finite_subset) (rule finite_imageI)
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lemma finite_range_imageI:
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  "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
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  by (drule finite_imageI) (simp add: range_composition)
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lemma finite_subset_image:
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  assumes "finite B"
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  shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
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using assms
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proof induct
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  case empty then show ?case by simp
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next
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  case insert then show ?case
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    by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
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       blast
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qed
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lemma finite_vimage_IntI:
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  "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
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  apply (induct rule: finite_induct)
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   apply simp_all
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  apply (subst vimage_insert)
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  apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
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  done
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lemma finite_vimageI:
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  "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
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  using finite_vimage_IntI[of F h UNIV] by auto
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lemma finite_vimageD:
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  assumes fin: "finite (h -` F)" and surj: "surj h"
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  shows "finite F"
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proof -
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  have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
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  also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
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  finally show "finite F" .
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qed
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lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
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  unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
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lemma finite_Collect_bex [simp]:
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  assumes "finite A"
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  shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
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proof -
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  have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
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  with assms show ?thesis by simp
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qed
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lemma finite_Collect_bounded_ex [simp]:
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  assumes "finite {y. P y}"
haftmann@41656
   321
  shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
haftmann@41656
   322
proof -
haftmann@41656
   323
  have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
haftmann@41656
   324
  with assms show ?thesis by simp
haftmann@41656
   325
qed
nipkow@29920
   326
haftmann@41656
   327
lemma finite_Plus:
haftmann@41656
   328
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
haftmann@41656
   329
  by (simp add: Plus_def)
nipkow@17022
   330
nipkow@31080
   331
lemma finite_PlusD: 
nipkow@31080
   332
  fixes A :: "'a set" and B :: "'b set"
nipkow@31080
   333
  assumes fin: "finite (A <+> B)"
nipkow@31080
   334
  shows "finite A" "finite B"
nipkow@31080
   335
proof -
nipkow@31080
   336
  have "Inl ` A \<subseteq> A <+> B" by auto
haftmann@41656
   337
  then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
haftmann@41656
   338
  then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
nipkow@31080
   339
next
nipkow@31080
   340
  have "Inr ` B \<subseteq> A <+> B" by auto
haftmann@41656
   341
  then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
haftmann@41656
   342
  then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
nipkow@31080
   343
qed
nipkow@31080
   344
haftmann@41656
   345
lemma finite_Plus_iff [simp]:
haftmann@41656
   346
  "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
haftmann@41656
   347
  by (auto intro: finite_PlusD finite_Plus)
nipkow@31080
   348
haftmann@41656
   349
lemma finite_Plus_UNIV_iff [simp]:
haftmann@41656
   350
  "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
haftmann@41656
   351
  by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
wenzelm@12396
   352
nipkow@40786
   353
lemma finite_SigmaI [simp, intro]:
haftmann@41656
   354
  "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
nipkow@40786
   355
  by (unfold Sigma_def) blast
wenzelm@12396
   356
haftmann@41656
   357
lemma finite_cartesian_product:
haftmann@41656
   358
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
nipkow@15402
   359
  by (rule finite_SigmaI)
nipkow@15402
   360
wenzelm@12396
   361
lemma finite_Prod_UNIV:
haftmann@41656
   362
  "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
haftmann@41656
   363
  by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
wenzelm@12396
   364
paulson@15409
   365
lemma finite_cartesian_productD1:
haftmann@42207
   366
  assumes "finite (A \<times> B)" and "B \<noteq> {}"
haftmann@42207
   367
  shows "finite A"
haftmann@42207
   368
proof -
haftmann@42207
   369
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
haftmann@42207
   370
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   371
  then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
haftmann@42207
   372
  with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
haftmann@42207
   373
    by (simp add: image_compose)
haftmann@42207
   374
  then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
haftmann@42207
   375
  then show ?thesis
haftmann@42207
   376
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   377
qed
paulson@15409
   378
paulson@15409
   379
lemma finite_cartesian_productD2:
haftmann@42207
   380
  assumes "finite (A \<times> B)" and "A \<noteq> {}"
haftmann@42207
   381
  shows "finite B"
haftmann@42207
   382
proof -
haftmann@42207
   383
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
haftmann@42207
   384
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   385
  then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
haftmann@42207
   386
  with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
haftmann@42207
   387
    by (simp add: image_compose)
haftmann@42207
   388
  then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
haftmann@42207
   389
  then show ?thesis
haftmann@42207
   390
    by (auto simp add: finite_conv_nat_seg_image)
haftmann@42207
   391
qed
paulson@15409
   392
haftmann@41656
   393
lemma finite_Pow_iff [iff]:
haftmann@41656
   394
  "finite (Pow A) \<longleftrightarrow> finite A"
wenzelm@12396
   395
proof
wenzelm@12396
   396
  assume "finite (Pow A)"
haftmann@41656
   397
  then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
haftmann@41656
   398
  then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
wenzelm@12396
   399
next
wenzelm@12396
   400
  assume "finite A"
haftmann@41656
   401
  then show "finite (Pow A)"
huffman@35216
   402
    by induct (simp_all add: Pow_insert)
wenzelm@12396
   403
qed
wenzelm@12396
   404
haftmann@41656
   405
corollary finite_Collect_subsets [simp, intro]:
haftmann@41656
   406
  "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
haftmann@41656
   407
  by (simp add: Pow_def [symmetric])
nipkow@29918
   408
nipkow@15392
   409
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
haftmann@41656
   410
  by (blast intro: finite_subset [OF subset_Pow_Union])
nipkow@15392
   411
nipkow@15392
   412
haftmann@41656
   413
subsubsection {* Further induction rules on finite sets *}
haftmann@41656
   414
haftmann@41656
   415
lemma finite_ne_induct [case_names singleton insert, consumes 2]:
haftmann@41656
   416
  assumes "finite F" and "F \<noteq> {}"
haftmann@41656
   417
  assumes "\<And>x. P {x}"
haftmann@41656
   418
    and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
haftmann@41656
   419
  shows "P F"
wenzelm@46898
   420
using assms
wenzelm@46898
   421
proof induct
haftmann@41656
   422
  case empty then show ?case by simp
haftmann@41656
   423
next
haftmann@41656
   424
  case (insert x F) then show ?case by cases auto
haftmann@41656
   425
qed
haftmann@41656
   426
haftmann@41656
   427
lemma finite_subset_induct [consumes 2, case_names empty insert]:
haftmann@41656
   428
  assumes "finite F" and "F \<subseteq> A"
haftmann@41656
   429
  assumes empty: "P {}"
haftmann@41656
   430
    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
   431
  shows "P F"
wenzelm@46898
   432
using `finite F` `F \<subseteq> A`
wenzelm@46898
   433
proof induct
haftmann@41656
   434
  show "P {}" by fact
nipkow@31441
   435
next
haftmann@41656
   436
  fix x F
haftmann@41656
   437
  assume "finite F" and "x \<notin> F" and
haftmann@41656
   438
    P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
haftmann@41656
   439
  show "P (insert x F)"
haftmann@41656
   440
  proof (rule insert)
haftmann@41656
   441
    from i show "x \<in> A" by blast
haftmann@41656
   442
    from i have "F \<subseteq> A" by blast
haftmann@41656
   443
    with P show "P F" .
haftmann@41656
   444
    show "finite F" by fact
haftmann@41656
   445
    show "x \<notin> F" by fact
haftmann@41656
   446
  qed
haftmann@41656
   447
qed
haftmann@41656
   448
haftmann@41656
   449
lemma finite_empty_induct:
haftmann@41656
   450
  assumes "finite A"
haftmann@41656
   451
  assumes "P A"
haftmann@41656
   452
    and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
haftmann@41656
   453
  shows "P {}"
haftmann@41656
   454
proof -
haftmann@41656
   455
  have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
haftmann@41656
   456
  proof -
haftmann@41656
   457
    fix B :: "'a set"
haftmann@41656
   458
    assume "B \<subseteq> A"
haftmann@41656
   459
    with `finite A` have "finite B" by (rule rev_finite_subset)
haftmann@41656
   460
    from this `B \<subseteq> A` show "P (A - B)"
haftmann@41656
   461
    proof induct
haftmann@41656
   462
      case empty
haftmann@41656
   463
      from `P A` show ?case by simp
haftmann@41656
   464
    next
haftmann@41656
   465
      case (insert b B)
haftmann@41656
   466
      have "P (A - B - {b})"
haftmann@41656
   467
      proof (rule remove)
haftmann@41656
   468
        from `finite A` show "finite (A - B)" by induct auto
haftmann@41656
   469
        from insert show "b \<in> A - B" by simp
haftmann@41656
   470
        from insert show "P (A - B)" by simp
haftmann@41656
   471
      qed
haftmann@41656
   472
      also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
haftmann@41656
   473
      finally show ?case .
haftmann@41656
   474
    qed
haftmann@41656
   475
  qed
haftmann@41656
   476
  then have "P (A - A)" by blast
haftmann@41656
   477
  then show ?thesis by simp
nipkow@31441
   478
qed
nipkow@31441
   479
nipkow@31441
   480
haftmann@26441
   481
subsection {* Class @{text finite}  *}
haftmann@26041
   482
haftmann@29797
   483
class finite =
haftmann@26041
   484
  assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
huffman@27430
   485
begin
huffman@27430
   486
huffman@27430
   487
lemma finite [simp]: "finite (A \<Colon> 'a set)"
haftmann@26441
   488
  by (rule subset_UNIV finite_UNIV finite_subset)+
haftmann@26041
   489
haftmann@43866
   490
lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
bulwahn@40922
   491
  by simp
bulwahn@40922
   492
huffman@27430
   493
end
huffman@27430
   494
wenzelm@46898
   495
instance prod :: (finite, finite) finite
wenzelm@46898
   496
  by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
haftmann@26146
   497
haftmann@26041
   498
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
nipkow@39302
   499
  by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
haftmann@26041
   500
haftmann@26146
   501
instance "fun" :: (finite, finite) finite
haftmann@26146
   502
proof
haftmann@26041
   503
  show "finite (UNIV :: ('a => 'b) set)"
haftmann@26041
   504
  proof (rule finite_imageD)
haftmann@26041
   505
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
berghofe@26792
   506
    have "range ?graph \<subseteq> Pow UNIV" by simp
berghofe@26792
   507
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
berghofe@26792
   508
      by (simp only: finite_Pow_iff finite)
berghofe@26792
   509
    ultimately show "finite (range ?graph)"
berghofe@26792
   510
      by (rule finite_subset)
haftmann@26041
   511
    show "inj ?graph" by (rule inj_graph)
haftmann@26041
   512
  qed
haftmann@26041
   513
qed
haftmann@26041
   514
wenzelm@46898
   515
instance bool :: finite
wenzelm@46898
   516
  by default (simp add: UNIV_bool)
haftmann@44831
   517
haftmann@45962
   518
instance set :: (finite) finite
haftmann@45962
   519
  by default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
haftmann@45962
   520
wenzelm@46898
   521
instance unit :: finite
wenzelm@46898
   522
  by default (simp add: UNIV_unit)
haftmann@44831
   523
wenzelm@46898
   524
instance sum :: (finite, finite) finite
wenzelm@46898
   525
  by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
haftmann@27981
   526
haftmann@44831
   527
lemma finite_option_UNIV [simp]:
haftmann@44831
   528
  "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
haftmann@44831
   529
  by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
haftmann@44831
   530
wenzelm@46898
   531
instance option :: (finite) finite
wenzelm@46898
   532
  by default (simp add: UNIV_option_conv)
haftmann@44831
   533
haftmann@26041
   534
haftmann@35817
   535
subsection {* A basic fold functional for finite sets *}
nipkow@15392
   536
nipkow@15392
   537
text {* The intended behaviour is
wenzelm@31916
   538
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
nipkow@28853
   539
if @{text f} is ``left-commutative'':
nipkow@15392
   540
*}
nipkow@15392
   541
haftmann@42871
   542
locale comp_fun_commute =
nipkow@28853
   543
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@42871
   544
  assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
nipkow@28853
   545
begin
nipkow@28853
   546
haftmann@42809
   547
lemma fun_left_comm: "f x (f y z) = f y (f x z)"
haftmann@42871
   548
  using comp_fun_commute by (simp add: fun_eq_iff)
nipkow@28853
   549
nipkow@28853
   550
end
nipkow@28853
   551
nipkow@28853
   552
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
nipkow@28853
   553
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
nipkow@28853
   554
  emptyI [intro]: "fold_graph f z {} z" |
nipkow@28853
   555
  insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
nipkow@28853
   556
      \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
nipkow@28853
   557
nipkow@28853
   558
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
nipkow@28853
   559
nipkow@28853
   560
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
haftmann@37767
   561
  "fold f z A = (THE y. fold_graph f z A y)"
nipkow@15392
   562
paulson@15498
   563
text{*A tempting alternative for the definiens is
nipkow@28853
   564
@{term "if finite A then THE y. fold_graph f z A y else e"}.
paulson@15498
   565
It allows the removal of finiteness assumptions from the theorems
nipkow@28853
   566
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
nipkow@28853
   567
The proofs become ugly. It is not worth the effort. (???) *}
nipkow@28853
   568
nipkow@28853
   569
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
haftmann@41656
   570
by (induct rule: finite_induct) auto
nipkow@28853
   571
nipkow@28853
   572
nipkow@28853
   573
subsubsection{*From @{const fold_graph} to @{term fold}*}
nipkow@15392
   574
haftmann@42871
   575
context comp_fun_commute
haftmann@26041
   576
begin
haftmann@26041
   577
huffman@36045
   578
lemma fold_graph_insertE_aux:
huffman@36045
   579
  "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
   580
proof (induct set: fold_graph)
huffman@36045
   581
  case (insertI x A y) show ?case
huffman@36045
   582
  proof (cases "x = a")
huffman@36045
   583
    assume "x = a" with insertI show ?case by auto
nipkow@28853
   584
  next
huffman@36045
   585
    assume "x \<noteq> a"
huffman@36045
   586
    then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
huffman@36045
   587
      using insertI by auto
haftmann@42875
   588
    have "f x y = f a (f x y')"
huffman@36045
   589
      unfolding y by (rule fun_left_comm)
haftmann@42875
   590
    moreover have "fold_graph f z (insert x A - {a}) (f x y')"
huffman@36045
   591
      using y' and `x \<noteq> a` and `x \<notin> A`
huffman@36045
   592
      by (simp add: insert_Diff_if fold_graph.insertI)
haftmann@42875
   593
    ultimately show ?case by fast
nipkow@15392
   594
  qed
huffman@36045
   595
qed simp
huffman@36045
   596
huffman@36045
   597
lemma fold_graph_insertE:
huffman@36045
   598
  assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
huffman@36045
   599
  obtains y where "v = f x y" and "fold_graph f z A y"
huffman@36045
   600
using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
nipkow@28853
   601
nipkow@28853
   602
lemma fold_graph_determ:
nipkow@28853
   603
  "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
huffman@36045
   604
proof (induct arbitrary: y set: fold_graph)
huffman@36045
   605
  case (insertI x A y v)
huffman@36045
   606
  from `fold_graph f z (insert x A) v` and `x \<notin> A`
huffman@36045
   607
  obtain y' where "v = f x y'" and "fold_graph f z A y'"
huffman@36045
   608
    by (rule fold_graph_insertE)
huffman@36045
   609
  from `fold_graph f z A y'` have "y' = y" by (rule insertI)
huffman@36045
   610
  with `v = f x y'` show "v = f x y" by simp
huffman@36045
   611
qed fast
nipkow@15392
   612
nipkow@28853
   613
lemma fold_equality:
nipkow@28853
   614
  "fold_graph f z A y \<Longrightarrow> fold f z A = y"
nipkow@28853
   615
by (unfold fold_def) (blast intro: fold_graph_determ)
nipkow@15392
   616
haftmann@42272
   617
lemma fold_graph_fold:
haftmann@42272
   618
  assumes "finite A"
haftmann@42272
   619
  shows "fold_graph f z A (fold f z A)"
haftmann@42272
   620
proof -
haftmann@42272
   621
  from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
haftmann@42272
   622
  moreover note fold_graph_determ
haftmann@42272
   623
  ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
haftmann@42272
   624
  then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
haftmann@42272
   625
  then show ?thesis by (unfold fold_def)
haftmann@42272
   626
qed
huffman@36045
   627
nipkow@15392
   628
text{* The base case for @{text fold}: *}
nipkow@15392
   629
nipkow@28853
   630
lemma (in -) fold_empty [simp]: "fold f z {} = z"
nipkow@28853
   631
by (unfold fold_def) blast
nipkow@28853
   632
nipkow@28853
   633
text{* The various recursion equations for @{const fold}: *}
nipkow@28853
   634
haftmann@26041
   635
lemma fold_insert [simp]:
haftmann@42875
   636
  assumes "finite A" and "x \<notin> A"
haftmann@42875
   637
  shows "fold f z (insert x A) = f x (fold f z A)"
haftmann@42875
   638
proof (rule fold_equality)
haftmann@42875
   639
  from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
haftmann@42875
   640
  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
   641
qed
nipkow@28853
   642
nipkow@28853
   643
lemma fold_fun_comm:
nipkow@28853
   644
  "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
nipkow@28853
   645
proof (induct rule: finite_induct)
nipkow@28853
   646
  case empty then show ?case by simp
nipkow@28853
   647
next
nipkow@28853
   648
  case (insert y A) then show ?case
nipkow@28853
   649
    by (simp add: fun_left_comm[of x])
nipkow@28853
   650
qed
nipkow@28853
   651
nipkow@28853
   652
lemma fold_insert2:
nipkow@28853
   653
  "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
huffman@35216
   654
by (simp add: fold_fun_comm)
nipkow@15392
   655
haftmann@26041
   656
lemma fold_rec:
haftmann@42875
   657
  assumes "finite A" and "x \<in> A"
haftmann@42875
   658
  shows "fold f z A = f x (fold f z (A - {x}))"
nipkow@28853
   659
proof -
nipkow@28853
   660
  have A: "A = insert x (A - {x})" using `x \<in> A` by blast
nipkow@28853
   661
  then have "fold f z A = fold f z (insert x (A - {x}))" by simp
nipkow@28853
   662
  also have "\<dots> = f x (fold f z (A - {x}))"
nipkow@28853
   663
    by (rule fold_insert) (simp add: `finite A`)+
nipkow@15535
   664
  finally show ?thesis .
nipkow@15535
   665
qed
nipkow@15535
   666
nipkow@28853
   667
lemma fold_insert_remove:
nipkow@28853
   668
  assumes "finite A"
nipkow@28853
   669
  shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
nipkow@28853
   670
proof -
nipkow@28853
   671
  from `finite A` have "finite (insert x A)" by auto
nipkow@28853
   672
  moreover have "x \<in> insert x A" by auto
nipkow@28853
   673
  ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
nipkow@28853
   674
    by (rule fold_rec)
nipkow@28853
   675
  then show ?thesis by simp
nipkow@28853
   676
qed
nipkow@28853
   677
haftmann@26041
   678
end
nipkow@15392
   679
nipkow@15480
   680
text{* A simplified version for idempotent functions: *}
nipkow@15480
   681
haftmann@42871
   682
locale comp_fun_idem = comp_fun_commute +
haftmann@42871
   683
  assumes comp_fun_idem: "f x o f x = f x"
haftmann@26041
   684
begin
haftmann@26041
   685
haftmann@42869
   686
lemma fun_left_idem: "f x (f x z) = f x z"
haftmann@42871
   687
  using comp_fun_idem by (simp add: fun_eq_iff)
nipkow@28853
   688
haftmann@26041
   689
lemma fold_insert_idem:
nipkow@28853
   690
  assumes fin: "finite A"
nipkow@28853
   691
  shows "fold f z (insert x A) = f x (fold f z A)"
nipkow@15480
   692
proof cases
nipkow@28853
   693
  assume "x \<in> A"
nipkow@28853
   694
  then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
nipkow@28853
   695
  then show ?thesis using assms by (simp add:fun_left_idem)
nipkow@15480
   696
next
nipkow@28853
   697
  assume "x \<notin> A" then show ?thesis using assms by simp
nipkow@15480
   698
qed
nipkow@15480
   699
nipkow@28853
   700
declare fold_insert[simp del] fold_insert_idem[simp]
nipkow@28853
   701
nipkow@28853
   702
lemma fold_insert_idem2:
nipkow@28853
   703
  "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
nipkow@28853
   704
by(simp add:fold_fun_comm)
nipkow@15484
   705
haftmann@26041
   706
end
haftmann@26041
   707
haftmann@35817
   708
haftmann@35817
   709
subsubsection {* Expressing set operations via @{const fold} *}
haftmann@35817
   710
haftmann@42871
   711
lemma (in comp_fun_commute) comp_comp_fun_commute:
haftmann@42871
   712
  "comp_fun_commute (f \<circ> g)"
haftmann@35817
   713
proof
haftmann@42871
   714
qed (simp_all add: comp_fun_commute)
haftmann@35817
   715
haftmann@42871
   716
lemma (in comp_fun_idem) comp_comp_fun_idem:
haftmann@42871
   717
  "comp_fun_idem (f \<circ> g)"
haftmann@42871
   718
  by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
haftmann@42871
   719
    (simp_all add: comp_fun_idem)
haftmann@35817
   720
haftmann@42871
   721
lemma comp_fun_idem_insert:
haftmann@42871
   722
  "comp_fun_idem insert"
haftmann@35817
   723
proof
haftmann@35817
   724
qed auto
haftmann@35817
   725
haftmann@42871
   726
lemma comp_fun_idem_remove:
haftmann@46146
   727
  "comp_fun_idem Set.remove"
haftmann@35817
   728
proof
haftmann@35817
   729
qed auto
nipkow@31992
   730
haftmann@42871
   731
lemma (in semilattice_inf) comp_fun_idem_inf:
haftmann@42871
   732
  "comp_fun_idem inf"
haftmann@35817
   733
proof
haftmann@35817
   734
qed (auto simp add: inf_left_commute)
haftmann@35817
   735
haftmann@42871
   736
lemma (in semilattice_sup) comp_fun_idem_sup:
haftmann@42871
   737
  "comp_fun_idem sup"
haftmann@35817
   738
proof
haftmann@35817
   739
qed (auto simp add: sup_left_commute)
nipkow@31992
   740
haftmann@35817
   741
lemma union_fold_insert:
haftmann@35817
   742
  assumes "finite A"
haftmann@35817
   743
  shows "A \<union> B = fold insert B A"
haftmann@35817
   744
proof -
haftmann@42871
   745
  interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
haftmann@35817
   746
  from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
haftmann@35817
   747
qed
nipkow@31992
   748
haftmann@35817
   749
lemma minus_fold_remove:
haftmann@35817
   750
  assumes "finite A"
haftmann@46146
   751
  shows "B - A = fold Set.remove B A"
haftmann@35817
   752
proof -
haftmann@46146
   753
  interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
haftmann@46146
   754
  from `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
haftmann@46146
   755
  then show ?thesis ..
haftmann@35817
   756
qed
haftmann@35817
   757
haftmann@35817
   758
context complete_lattice
nipkow@31992
   759
begin
nipkow@31992
   760
haftmann@35817
   761
lemma inf_Inf_fold_inf:
haftmann@35817
   762
  assumes "finite A"
haftmann@35817
   763
  shows "inf B (Inf A) = fold inf B A"
haftmann@35817
   764
proof -
haftmann@42871
   765
  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
haftmann@35817
   766
  from `finite A` show ?thesis by (induct A arbitrary: B)
noschinl@44919
   767
    (simp_all add: inf_commute fold_fun_comm)
haftmann@35817
   768
qed
nipkow@31992
   769
haftmann@35817
   770
lemma sup_Sup_fold_sup:
haftmann@35817
   771
  assumes "finite A"
haftmann@35817
   772
  shows "sup B (Sup A) = fold sup B A"
haftmann@35817
   773
proof -
haftmann@42871
   774
  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
haftmann@35817
   775
  from `finite A` show ?thesis by (induct A arbitrary: B)
noschinl@44919
   776
    (simp_all add: sup_commute fold_fun_comm)
nipkow@31992
   777
qed
nipkow@31992
   778
haftmann@35817
   779
lemma Inf_fold_inf:
haftmann@35817
   780
  assumes "finite A"
haftmann@35817
   781
  shows "Inf A = fold inf top A"
haftmann@35817
   782
  using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
haftmann@35817
   783
haftmann@35817
   784
lemma Sup_fold_sup:
haftmann@35817
   785
  assumes "finite A"
haftmann@35817
   786
  shows "Sup A = fold sup bot A"
haftmann@35817
   787
  using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
nipkow@31992
   788
haftmann@46146
   789
lemma inf_INF_fold_inf:
haftmann@35817
   790
  assumes "finite A"
haftmann@42873
   791
  shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold") 
haftmann@35817
   792
proof (rule sym)
haftmann@42871
   793
  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
haftmann@42871
   794
  interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
haftmann@42873
   795
  from `finite A` show "?fold = ?inf"
haftmann@42869
   796
    by (induct A arbitrary: B)
hoelzl@44928
   797
      (simp_all add: INF_def inf_left_commute)
haftmann@35817
   798
qed
nipkow@31992
   799
haftmann@46146
   800
lemma sup_SUP_fold_sup:
haftmann@35817
   801
  assumes "finite A"
haftmann@42873
   802
  shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold") 
haftmann@35817
   803
proof (rule sym)
haftmann@42871
   804
  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
haftmann@42871
   805
  interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
haftmann@42873
   806
  from `finite A` show "?fold = ?sup"
haftmann@42869
   807
    by (induct A arbitrary: B)
hoelzl@44928
   808
      (simp_all add: SUP_def sup_left_commute)
haftmann@35817
   809
qed
nipkow@31992
   810
haftmann@46146
   811
lemma INF_fold_inf:
haftmann@35817
   812
  assumes "finite A"
haftmann@42873
   813
  shows "INFI A f = fold (inf \<circ> f) top A"
haftmann@46146
   814
  using assms inf_INF_fold_inf [of A top] by simp
nipkow@31992
   815
haftmann@46146
   816
lemma SUP_fold_sup:
haftmann@35817
   817
  assumes "finite A"
haftmann@42873
   818
  shows "SUPR A f = fold (sup \<circ> f) bot A"
haftmann@46146
   819
  using assms sup_SUP_fold_sup [of A bot] by simp
nipkow@31992
   820
nipkow@31992
   821
end
nipkow@31992
   822
nipkow@31992
   823
haftmann@35817
   824
subsection {* The derived combinator @{text fold_image} *}
nipkow@28853
   825
nipkow@28853
   826
definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
haftmann@42875
   827
  where "fold_image f g = fold (\<lambda>x y. f (g x) y)"
nipkow@28853
   828
nipkow@28853
   829
lemma fold_image_empty[simp]: "fold_image f g z {} = z"
haftmann@42875
   830
  by (simp add:fold_image_def)
nipkow@15392
   831
haftmann@26041
   832
context ab_semigroup_mult
haftmann@26041
   833
begin
haftmann@26041
   834
nipkow@28853
   835
lemma fold_image_insert[simp]:
haftmann@42875
   836
  assumes "finite A" and "a \<notin> A"
haftmann@42875
   837
  shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
nipkow@28853
   838
proof -
wenzelm@46898
   839
  interpret comp_fun_commute "%x y. (g x) * y"
wenzelm@46898
   840
    by default (simp add: fun_eq_iff mult_ac)
wenzelm@46898
   841
  from assms show ?thesis by (simp add: fold_image_def)
nipkow@28853
   842
qed
nipkow@28853
   843
nipkow@28853
   844
lemma fold_image_reindex:
haftmann@42875
   845
  assumes "finite A"
haftmann@42875
   846
  shows "inj_on h A \<Longrightarrow> fold_image times g z (h ` A) = fold_image times (g \<circ> h) z A"
haftmann@42875
   847
  using assms by induct auto
nipkow@28853
   848
nipkow@28853
   849
lemma fold_image_cong:
haftmann@42875
   850
  assumes "finite A" and g_h: "\<And>x. x\<in>A \<Longrightarrow> g x = h x"
haftmann@42875
   851
  shows "fold_image times g z A = fold_image times h z A"
haftmann@42875
   852
proof -
haftmann@42875
   853
  from `finite A`
haftmann@42875
   854
  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
   855
  proof (induct arbitrary: C)
haftmann@42875
   856
    case empty then show ?case by simp
haftmann@42875
   857
  next
haftmann@42875
   858
    case (insert x F) then show ?case apply -
haftmann@42875
   859
    apply (simp add: subset_insert_iff, clarify)
haftmann@42875
   860
    apply (subgoal_tac "finite C")
haftmann@42875
   861
      prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
haftmann@42875
   862
    apply (subgoal_tac "C = insert x (C - {x})")
haftmann@42875
   863
      prefer 2 apply blast
haftmann@42875
   864
    apply (erule ssubst)
haftmann@42875
   865
    apply (simp add: Ball_def del: insert_Diff_single)
haftmann@42875
   866
    done
haftmann@42875
   867
  qed
haftmann@42875
   868
  with g_h show ?thesis by simp
haftmann@42875
   869
qed
nipkow@15392
   870
haftmann@26041
   871
end
haftmann@26041
   872
haftmann@26041
   873
context comm_monoid_mult
haftmann@26041
   874
begin
haftmann@26041
   875
haftmann@35817
   876
lemma fold_image_1:
haftmann@35817
   877
  "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
haftmann@41656
   878
  apply (induct rule: finite_induct)
haftmann@35817
   879
  apply simp by auto
haftmann@35817
   880
nipkow@28853
   881
lemma fold_image_Un_Int:
haftmann@26041
   882
  "finite A ==> finite B ==>
nipkow@28853
   883
    fold_image times g 1 A * fold_image times g 1 B =
nipkow@28853
   884
    fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
haftmann@41656
   885
  apply (induct rule: finite_induct)
nipkow@28853
   886
by (induct set: finite) 
nipkow@28853
   887
   (auto simp add: mult_ac insert_absorb Int_insert_left)
haftmann@26041
   888
haftmann@35817
   889
lemma fold_image_Un_one:
haftmann@35817
   890
  assumes fS: "finite S" and fT: "finite T"
haftmann@35817
   891
  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
haftmann@35817
   892
  shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
haftmann@35817
   893
proof-
haftmann@35817
   894
  have "fold_image op * f 1 (S \<inter> T) = 1" 
haftmann@35817
   895
    apply (rule fold_image_1)
haftmann@35817
   896
    using fS fT I0 by auto 
haftmann@35817
   897
  with fold_image_Un_Int[OF fS fT] show ?thesis by simp
haftmann@35817
   898
qed
haftmann@35817
   899
haftmann@26041
   900
corollary fold_Un_disjoint:
haftmann@26041
   901
  "finite A ==> finite B ==> A Int B = {} ==>
nipkow@28853
   902
   fold_image times g 1 (A Un B) =
nipkow@28853
   903
   fold_image times g 1 A * fold_image times g 1 B"
nipkow@28853
   904
by (simp add: fold_image_Un_Int)
nipkow@28853
   905
nipkow@28853
   906
lemma fold_image_UN_disjoint:
haftmann@26041
   907
  "\<lbrakk> finite I; ALL i:I. finite (A i);
haftmann@26041
   908
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
nipkow@28853
   909
   \<Longrightarrow> fold_image times g 1 (UNION I A) =
nipkow@28853
   910
       fold_image times (%i. fold_image times g 1 (A i)) 1 I"
haftmann@41656
   911
apply (induct rule: finite_induct)
haftmann@41656
   912
apply simp
haftmann@41656
   913
apply atomize
nipkow@28853
   914
apply (subgoal_tac "ALL i:F. x \<noteq> i")
nipkow@28853
   915
 prefer 2 apply blast
nipkow@28853
   916
apply (subgoal_tac "A x Int UNION F A = {}")
nipkow@28853
   917
 prefer 2 apply blast
nipkow@28853
   918
apply (simp add: fold_Un_disjoint)
nipkow@28853
   919
done
nipkow@28853
   920
nipkow@28853
   921
lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@28853
   922
  fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
nipkow@28853
   923
  fold_image times (split g) 1 (SIGMA x:A. B x)"
nipkow@15392
   924
apply (subst Sigma_def)
nipkow@28853
   925
apply (subst fold_image_UN_disjoint, assumption, simp)
nipkow@15392
   926
 apply blast
nipkow@28853
   927
apply (erule fold_image_cong)
nipkow@28853
   928
apply (subst fold_image_UN_disjoint, simp, simp)
nipkow@15392
   929
 apply blast
paulson@15506
   930
apply simp
nipkow@15392
   931
done
nipkow@15392
   932
nipkow@28853
   933
lemma fold_image_distrib: "finite A \<Longrightarrow>
nipkow@28853
   934
   fold_image times (%x. g x * h x) 1 A =
nipkow@28853
   935
   fold_image times g 1 A *  fold_image times h 1 A"
nipkow@28853
   936
by (erule finite_induct) (simp_all add: mult_ac)
haftmann@26041
   937
chaieb@30260
   938
lemma fold_image_related: 
chaieb@30260
   939
  assumes Re: "R e e" 
chaieb@30260
   940
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
chaieb@30260
   941
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
chaieb@30260
   942
  shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
chaieb@30260
   943
  using fS by (rule finite_subset_induct) (insert assms, auto)
chaieb@30260
   944
chaieb@30260
   945
lemma  fold_image_eq_general:
chaieb@30260
   946
  assumes fS: "finite S"
chaieb@30260
   947
  and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 
chaieb@30260
   948
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
chaieb@30260
   949
  shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
chaieb@30260
   950
proof-
chaieb@30260
   951
  from h f12 have hS: "h ` S = S'" by auto
chaieb@30260
   952
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
chaieb@30260
   953
    from f12 h H  have "x = y" by auto }
chaieb@30260
   954
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
chaieb@30260
   955
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
chaieb@30260
   956
  from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
chaieb@30260
   957
  also have "\<dots> = fold_image (op *) (f2 o h) e S" 
chaieb@30260
   958
    using fold_image_reindex[OF fS hinj, of f2 e] .
chaieb@30260
   959
  also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
chaieb@30260
   960
    by blast
chaieb@30260
   961
  finally show ?thesis ..
chaieb@30260
   962
qed
chaieb@30260
   963
chaieb@30260
   964
lemma fold_image_eq_general_inverses:
chaieb@30260
   965
  assumes fS: "finite S" 
chaieb@30260
   966
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
chaieb@30260
   967
  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
   968
  shows "fold_image (op *) f e S = fold_image (op *) g e T"
chaieb@30260
   969
  (* metis solves it, but not yet available here *)
chaieb@30260
   970
  apply (rule fold_image_eq_general[OF fS, of T h g f e])
chaieb@30260
   971
  apply (rule ballI)
chaieb@30260
   972
  apply (frule kh)
chaieb@30260
   973
  apply (rule ex1I[])
chaieb@30260
   974
  apply blast
chaieb@30260
   975
  apply clarsimp
chaieb@30260
   976
  apply (drule hk) apply simp
chaieb@30260
   977
  apply (rule sym)
chaieb@30260
   978
  apply (erule conjunct1[OF conjunct2[OF hk]])
chaieb@30260
   979
  apply (rule ballI)
chaieb@30260
   980
  apply (drule  hk)
chaieb@30260
   981
  apply blast
chaieb@30260
   982
  done
chaieb@30260
   983
haftmann@26041
   984
end
haftmann@22917
   985
nipkow@25162
   986
haftmann@35817
   987
subsection {* A fold functional for non-empty sets *}
nipkow@15392
   988
nipkow@15392
   989
text{* Does not require start value. *}
wenzelm@12396
   990
berghofe@23736
   991
inductive
berghofe@22262
   992
  fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
berghofe@22262
   993
  for f :: "'a => 'a => 'a"
berghofe@22262
   994
where
paulson@15506
   995
  fold1Set_insertI [intro]:
nipkow@28853
   996
   "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
wenzelm@12396
   997
haftmann@35416
   998
definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
berghofe@22262
   999
  "fold1 f A == THE x. fold1Set f A x"
paulson@15506
  1000
paulson@15506
  1001
lemma fold1Set_nonempty:
haftmann@22917
  1002
  "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
nipkow@28853
  1003
by(erule fold1Set.cases, simp_all)
nipkow@15392
  1004
berghofe@23736
  1005
inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
berghofe@23736
  1006
berghofe@23736
  1007
inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
berghofe@22262
  1008
berghofe@22262
  1009
berghofe@22262
  1010
lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
huffman@35216
  1011
by (blast elim: fold_graph.cases)
nipkow@15392
  1012
haftmann@22917
  1013
lemma fold1_singleton [simp]: "fold1 f {a} = a"
nipkow@28853
  1014
by (unfold fold1_def) blast
wenzelm@12396
  1015
paulson@15508
  1016
lemma finite_nonempty_imp_fold1Set:
berghofe@22262
  1017
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
paulson@15508
  1018
apply (induct A rule: finite_induct)
nipkow@28853
  1019
apply (auto dest: finite_imp_fold_graph [of _ f])
paulson@15508
  1020
done
paulson@15506
  1021
nipkow@28853
  1022
text{*First, some lemmas about @{const fold_graph}.*}
nipkow@15392
  1023
haftmann@26041
  1024
context ab_semigroup_mult
haftmann@26041
  1025
begin
haftmann@26041
  1026
wenzelm@46898
  1027
lemma comp_fun_commute: "comp_fun_commute (op *)"
wenzelm@46898
  1028
  by default (simp add: fun_eq_iff mult_ac)
nipkow@28853
  1029
nipkow@28853
  1030
lemma fold_graph_insert_swap:
nipkow@28853
  1031
assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
nipkow@28853
  1032
shows "fold_graph times z (insert b A) (z * y)"
nipkow@28853
  1033
proof -
haftmann@42871
  1034
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
nipkow@28853
  1035
from assms show ?thesis
nipkow@28853
  1036
proof (induct rule: fold_graph.induct)
huffman@36045
  1037
  case emptyI show ?case by (subst mult_commute [of z b], fast)
paulson@15508
  1038
next
berghofe@22262
  1039
  case (insertI x A y)
nipkow@28853
  1040
    have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
paulson@15521
  1041
      using insertI by force  --{*how does @{term id} get unfolded?*}
haftmann@26041
  1042
    thus ?case by (simp add: insert_commute mult_ac)
paulson@15508
  1043
qed
nipkow@28853
  1044
qed
nipkow@28853
  1045
nipkow@28853
  1046
lemma fold_graph_permute_diff:
nipkow@28853
  1047
assumes fold: "fold_graph times b A x"
nipkow@28853
  1048
shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
paulson@15508
  1049
using fold
nipkow@28853
  1050
proof (induct rule: fold_graph.induct)
paulson@15508
  1051
  case emptyI thus ?case by simp
paulson@15508
  1052
next
berghofe@22262
  1053
  case (insertI x A y)
paulson@15521
  1054
  have "a = x \<or> a \<in> A" using insertI by simp
paulson@15521
  1055
  thus ?case
paulson@15521
  1056
  proof
paulson@15521
  1057
    assume "a = x"
paulson@15521
  1058
    with insertI show ?thesis
nipkow@28853
  1059
      by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
paulson@15521
  1060
  next
paulson@15521
  1061
    assume ainA: "a \<in> A"
nipkow@28853
  1062
    hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
nipkow@28853
  1063
      using insertI by force
paulson@15521
  1064
    moreover
paulson@15521
  1065
    have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
paulson@15521
  1066
      using ainA insertI by blast
nipkow@28853
  1067
    ultimately show ?thesis by simp
paulson@15508
  1068
  qed
paulson@15508
  1069
qed
paulson@15508
  1070
haftmann@26041
  1071
lemma fold1_eq_fold:
nipkow@28853
  1072
assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
nipkow@28853
  1073
proof -
haftmann@42871
  1074
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
nipkow@28853
  1075
  from assms show ?thesis
nipkow@28853
  1076
apply (simp add: fold1_def fold_def)
paulson@15508
  1077
apply (rule the_equality)
nipkow@28853
  1078
apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
paulson@15508
  1079
apply (rule sym, clarify)
paulson@15508
  1080
apply (case_tac "Aa=A")
huffman@35216
  1081
 apply (best intro: fold_graph_determ)
nipkow@28853
  1082
apply (subgoal_tac "fold_graph times a A x")
huffman@35216
  1083
 apply (best intro: fold_graph_determ)
nipkow@28853
  1084
apply (subgoal_tac "insert aa (Aa - {a}) = A")
nipkow@28853
  1085
 prefer 2 apply (blast elim: equalityE)
nipkow@28853
  1086
apply (auto dest: fold_graph_permute_diff [where a=a])
paulson@15508
  1087
done
nipkow@28853
  1088
qed
paulson@15508
  1089
paulson@15521
  1090
lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
paulson@15521
  1091
apply safe
nipkow@28853
  1092
 apply simp
nipkow@28853
  1093
 apply (drule_tac x=x in spec)
nipkow@28853
  1094
 apply (drule_tac x="A-{x}" in spec, auto)
paulson@15508
  1095
done
paulson@15508
  1096
haftmann@26041
  1097
lemma fold1_insert:
paulson@15521
  1098
  assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
haftmann@26041
  1099
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1100
proof -
haftmann@42871
  1101
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
nipkow@28853
  1102
  from nonempty obtain a A' where "A = insert a A' & a ~: A'"
paulson@15521
  1103
    by (auto simp add: nonempty_iff)
paulson@15521
  1104
  with A show ?thesis
nipkow@28853
  1105
    by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
paulson@15521
  1106
qed
paulson@15521
  1107
haftmann@26041
  1108
end
haftmann@26041
  1109
haftmann@26041
  1110
context ab_semigroup_idem_mult
haftmann@26041
  1111
begin
haftmann@26041
  1112
wenzelm@46898
  1113
lemma comp_fun_idem: "comp_fun_idem (op *)"
wenzelm@46898
  1114
  by default (simp_all add: fun_eq_iff mult_left_commute)
haftmann@35817
  1115
haftmann@26041
  1116
lemma fold1_insert_idem [simp]:
paulson@15521
  1117
  assumes nonempty: "A \<noteq> {}" and A: "finite A" 
haftmann@26041
  1118
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1119
proof -
haftmann@42871
  1120
  interpret comp_fun_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@42871
  1121
    by (rule comp_fun_idem)
nipkow@28853
  1122
  from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
paulson@15521
  1123
    by (auto simp add: nonempty_iff)
paulson@15521
  1124
  show ?thesis
paulson@15521
  1125
  proof cases
wenzelm@41550
  1126
    assume a: "a = x"
wenzelm@41550
  1127
    show ?thesis
paulson@15521
  1128
    proof cases
paulson@15521
  1129
      assume "A' = {}"
wenzelm@41550
  1130
      with A' a show ?thesis by simp
paulson@15521
  1131
    next
paulson@15521
  1132
      assume "A' \<noteq> {}"
wenzelm@41550
  1133
      with A A' a show ?thesis
huffman@35216
  1134
        by (simp add: fold1_insert mult_assoc [symmetric])
paulson@15521
  1135
    qed
paulson@15521
  1136
  next
paulson@15521
  1137
    assume "a \<noteq> x"
wenzelm@41550
  1138
    with A A' show ?thesis
huffman@35216
  1139
      by (simp add: insert_commute fold1_eq_fold)
paulson@15521
  1140
  qed
paulson@15521
  1141
qed
paulson@15506
  1142
haftmann@26041
  1143
lemma hom_fold1_commute:
haftmann@26041
  1144
assumes hom: "!!x y. h (x * y) = h x * h y"
haftmann@26041
  1145
and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
wenzelm@46898
  1146
using N
wenzelm@46898
  1147
proof (induct rule: finite_ne_induct)
haftmann@22917
  1148
  case singleton thus ?case by simp
haftmann@22917
  1149
next
haftmann@22917
  1150
  case (insert n N)
haftmann@26041
  1151
  then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
haftmann@26041
  1152
  also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
haftmann@26041
  1153
  also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
haftmann@26041
  1154
  also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
haftmann@22917
  1155
    using insert by(simp)
haftmann@22917
  1156
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@22917
  1157
  finally show ?case .
haftmann@22917
  1158
qed
haftmann@22917
  1159
haftmann@32679
  1160
lemma fold1_eq_fold_idem:
haftmann@32679
  1161
  assumes "finite A"
haftmann@32679
  1162
  shows "fold1 times (insert a A) = fold times a A"
haftmann@32679
  1163
proof (cases "a \<in> A")
haftmann@32679
  1164
  case False
haftmann@32679
  1165
  with assms show ?thesis by (simp add: fold1_eq_fold)
haftmann@32679
  1166
next
haftmann@42871
  1167
  interpret comp_fun_idem times by (fact comp_fun_idem)
haftmann@32679
  1168
  case True then obtain b B
haftmann@32679
  1169
    where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
haftmann@32679
  1170
  with assms have "finite B" by auto
haftmann@32679
  1171
  then have "fold times a (insert a B) = fold times (a * a) B"
haftmann@32679
  1172
    using `a \<notin> B` by (rule fold_insert2)
haftmann@32679
  1173
  then show ?thesis
haftmann@32679
  1174
    using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
haftmann@32679
  1175
qed
haftmann@32679
  1176
haftmann@26041
  1177
end
haftmann@26041
  1178
paulson@15506
  1179
paulson@15508
  1180
text{* Now the recursion rules for definitions: *}
paulson@15508
  1181
haftmann@22917
  1182
lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
huffman@35216
  1183
by simp
paulson@15508
  1184
haftmann@26041
  1185
lemma (in ab_semigroup_mult) fold1_insert_def:
haftmann@26041
  1186
  "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1187
by (simp add:fold1_insert)
haftmann@26041
  1188
haftmann@26041
  1189
lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
haftmann@26041
  1190
  "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1191
by simp
paulson@15508
  1192
paulson@15508
  1193
subsubsection{* Determinacy for @{term fold1Set} *}
paulson@15508
  1194
nipkow@28853
  1195
(*Not actually used!!*)
nipkow@28853
  1196
(*
haftmann@26041
  1197
context ab_semigroup_mult
haftmann@26041
  1198
begin
haftmann@26041
  1199
nipkow@28853
  1200
lemma fold_graph_permute:
nipkow@28853
  1201
  "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
nipkow@28853
  1202
   ==> fold_graph times id a (insert b A) x"
haftmann@26041
  1203
apply (cases "a=b") 
nipkow@28853
  1204
apply (auto dest: fold_graph_permute_diff) 
paulson@15506
  1205
done
nipkow@15376
  1206
haftmann@26041
  1207
lemma fold1Set_determ:
haftmann@26041
  1208
  "fold1Set times A x ==> fold1Set times A y ==> y = x"
paulson@15506
  1209
proof (clarify elim!: fold1Set.cases)
paulson@15506
  1210
  fix A x B y a b
nipkow@28853
  1211
  assume Ax: "fold_graph times id a A x"
nipkow@28853
  1212
  assume By: "fold_graph times id b B y"
paulson@15506
  1213
  assume anotA:  "a \<notin> A"
paulson@15506
  1214
  assume bnotB:  "b \<notin> B"
paulson@15506
  1215
  assume eq: "insert a A = insert b B"
paulson@15506
  1216
  show "y=x"
paulson@15506
  1217
  proof cases
paulson@15506
  1218
    assume same: "a=b"
paulson@15506
  1219
    hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
nipkow@28853
  1220
    thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
nipkow@15392
  1221
  next
paulson@15506
  1222
    assume diff: "a\<noteq>b"
paulson@15506
  1223
    let ?D = "B - {a}"
paulson@15506
  1224
    have B: "B = insert a ?D" and A: "A = insert b ?D"
paulson@15506
  1225
     and aB: "a \<in> B" and bA: "b \<in> A"
paulson@15506
  1226
      using eq anotA bnotB diff by (blast elim!:equalityE)+
paulson@15506
  1227
    with aB bnotB By
nipkow@28853
  1228
    have "fold_graph times id a (insert b ?D) y" 
nipkow@28853
  1229
      by (auto intro: fold_graph_permute simp add: insert_absorb)
paulson@15506
  1230
    moreover
nipkow@28853
  1231
    have "fold_graph times id a (insert b ?D) x"
paulson@15506
  1232
      by (simp add: A [symmetric] Ax) 
nipkow@28853
  1233
    ultimately show ?thesis by (blast intro: fold_graph_determ) 
nipkow@15392
  1234
  qed
wenzelm@12396
  1235
qed
wenzelm@12396
  1236
haftmann@26041
  1237
lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
paulson@15506
  1238
  by (unfold fold1_def) (blast intro: fold1Set_determ)
paulson@15506
  1239
haftmann@26041
  1240
end
nipkow@28853
  1241
*)
haftmann@26041
  1242
paulson@15506
  1243
declare
nipkow@28853
  1244
  empty_fold_graphE [rule del]  fold_graph.intros [rule del]
paulson@15506
  1245
  empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
ballarin@19931
  1246
  -- {* No more proofs involve these relations. *}
nipkow@15376
  1247
haftmann@26041
  1248
subsubsection {* Lemmas about @{text fold1} *}
haftmann@26041
  1249
haftmann@26041
  1250
context ab_semigroup_mult
haftmann@22917
  1251
begin
haftmann@22917
  1252
haftmann@26041
  1253
lemma fold1_Un:
nipkow@15484
  1254
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  1255
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
haftmann@26041
  1256
       fold1 times (A Un B) = fold1 times A * fold1 times B"
haftmann@26041
  1257
using A by (induct rule: finite_ne_induct)
haftmann@26041
  1258
  (simp_all add: fold1_insert mult_assoc)
haftmann@26041
  1259
haftmann@26041
  1260
lemma fold1_in:
haftmann@26041
  1261
  assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
haftmann@26041
  1262
  shows "fold1 times A \<in> A"
nipkow@15484
  1263
using A
nipkow@15484
  1264
proof (induct rule:finite_ne_induct)
paulson@15506
  1265
  case singleton thus ?case by simp
nipkow@15484
  1266
next
nipkow@15484
  1267
  case insert thus ?case using elem by (force simp add:fold1_insert)
nipkow@15484
  1268
qed
nipkow@15484
  1269
haftmann@26041
  1270
end
haftmann@26041
  1271
haftmann@26041
  1272
lemma (in ab_semigroup_idem_mult) fold1_Un2:
nipkow@15497
  1273
assumes A: "finite A" "A \<noteq> {}"
haftmann@26041
  1274
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
haftmann@26041
  1275
       fold1 times (A Un B) = fold1 times A * fold1 times B"
nipkow@15497
  1276
using A
haftmann@26041
  1277
proof(induct rule:finite_ne_induct)
nipkow@15497
  1278
  case singleton thus ?case by simp
nipkow@15484
  1279
next
haftmann@26041
  1280
  case insert thus ?case by (simp add: mult_assoc)
nipkow@18423
  1281
qed
nipkow@18423
  1282
nipkow@18423
  1283
haftmann@35817
  1284
subsection {* Locales as mini-packages for fold operations *}
haftmann@34007
  1285
haftmann@35817
  1286
subsubsection {* The natural case *}
haftmann@35719
  1287
haftmann@35719
  1288
locale folding =
haftmann@35719
  1289
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@35719
  1290
  fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@42871
  1291
  assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
haftmann@35722
  1292
  assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
haftmann@35719
  1293
begin
haftmann@35719
  1294
haftmann@35719
  1295
lemma empty [simp]:
haftmann@35719
  1296
  "F {} = id"
nipkow@39302
  1297
  by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1298
haftmann@35719
  1299
lemma insert [simp]:
haftmann@35719
  1300
  assumes "finite A" and "x \<notin> A"
haftmann@35719
  1301
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35719
  1302
proof -
wenzelm@46898
  1303
  interpret comp_fun_commute f
wenzelm@46898
  1304
    by default (insert comp_fun_commute, simp add: fun_eq_iff)
haftmann@35719
  1305
  from fold_insert2 assms
haftmann@35722
  1306
  have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
nipkow@39302
  1307
  with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1308
qed
haftmann@35719
  1309
haftmann@35719
  1310
lemma remove:
haftmann@35719
  1311
  assumes "finite A" and "x \<in> A"
haftmann@35719
  1312
  shows "F A = F (A - {x}) \<circ> f x"
haftmann@35719
  1313
proof -
haftmann@35719
  1314
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35719
  1315
    by (auto dest: mk_disjoint_insert)
haftmann@35719
  1316
  moreover from `finite A` this have "finite B" by simp
haftmann@35719
  1317
  ultimately show ?thesis by simp
haftmann@35719
  1318
qed
haftmann@35719
  1319
haftmann@35719
  1320
lemma insert_remove:
haftmann@35719
  1321
  assumes "finite A"
haftmann@35719
  1322
  shows "F (insert x A) = F (A - {x}) \<circ> f x"
haftmann@35722
  1323
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35719
  1324
haftmann@35817
  1325
lemma commute_left_comp:
haftmann@35817
  1326
  "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
haftmann@42871
  1327
  by (simp add: o_assoc comp_fun_commute)
haftmann@35817
  1328
haftmann@42871
  1329
lemma comp_fun_commute':
haftmann@35719
  1330
  assumes "finite A"
haftmann@35719
  1331
  shows "f x \<circ> F A = F A \<circ> f x"
haftmann@35817
  1332
  using assms by (induct A)
haftmann@42871
  1333
    (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: o_assoc [symmetric] comp_fun_commute)
haftmann@35817
  1334
haftmann@35817
  1335
lemma commute_left_comp':
haftmann@35817
  1336
  assumes "finite A"
haftmann@35817
  1337
  shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
haftmann@42871
  1338
  using assms by (simp add: o_assoc comp_fun_commute')
haftmann@35817
  1339
haftmann@42871
  1340
lemma comp_fun_commute'':
haftmann@35817
  1341
  assumes "finite A" and "finite B"
haftmann@35817
  1342
  shows "F B \<circ> F A = F A \<circ> F B"
haftmann@35817
  1343
  using assms by (induct A)
haftmann@42871
  1344
    (simp_all add: o_assoc, simp add: o_assoc [symmetric] comp_fun_commute')
haftmann@35719
  1345
haftmann@35817
  1346
lemma commute_left_comp'':
haftmann@35817
  1347
  assumes "finite A" and "finite B"
haftmann@35817
  1348
  shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
haftmann@42871
  1349
  using assms by (simp add: o_assoc comp_fun_commute'')
haftmann@35817
  1350
haftmann@42871
  1351
lemmas comp_fun_commutes = o_assoc [symmetric] comp_fun_commute commute_left_comp
haftmann@42871
  1352
  comp_fun_commute' commute_left_comp' comp_fun_commute'' commute_left_comp''
haftmann@35817
  1353
haftmann@35817
  1354
lemma union_inter:
haftmann@35817
  1355
  assumes "finite A" and "finite B"
haftmann@35817
  1356
  shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
haftmann@35817
  1357
  using assms by (induct A)
haftmann@42871
  1358
    (simp_all del: o_apply add: insert_absorb Int_insert_left comp_fun_commutes,
haftmann@35817
  1359
      simp add: o_assoc)
haftmann@35719
  1360
haftmann@35719
  1361
lemma union:
haftmann@35719
  1362
  assumes "finite A" and "finite B"
haftmann@35719
  1363
  and "A \<inter> B = {}"
haftmann@35719
  1364
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1365
proof -
haftmann@35817
  1366
  from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
haftmann@35817
  1367
  with `A \<inter> B = {}` show ?thesis by simp
haftmann@35719
  1368
qed
haftmann@35719
  1369
haftmann@34007
  1370
end
haftmann@35719
  1371
haftmann@35817
  1372
haftmann@35817
  1373
subsubsection {* The natural case with idempotency *}
haftmann@35817
  1374
haftmann@35719
  1375
locale folding_idem = folding +
haftmann@35719
  1376
  assumes idem_comp: "f x \<circ> f x = f x"
haftmann@35719
  1377
begin
haftmann@35719
  1378
haftmann@35817
  1379
lemma idem_left_comp:
haftmann@35817
  1380
  "f x \<circ> (f x \<circ> g) = f x \<circ> g"
haftmann@35817
  1381
  by (simp add: o_assoc idem_comp)
haftmann@35817
  1382
haftmann@35817
  1383
lemma in_comp_idem:
haftmann@35817
  1384
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1385
  shows "F A \<circ> f x = F A"
haftmann@35817
  1386
using assms by (induct A)
haftmann@42871
  1387
  (auto simp add: comp_fun_commutes idem_comp, simp add: commute_left_comp' [symmetric] comp_fun_commute')
haftmann@35719
  1388
haftmann@35817
  1389
lemma subset_comp_idem:
haftmann@35817
  1390
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1391
  shows "F A \<circ> F B = F A"
haftmann@35817
  1392
proof -
haftmann@35817
  1393
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1394
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1395
    (simp_all add: o_assoc in_comp_idem `finite A`)
haftmann@35817
  1396
qed
haftmann@35719
  1397
haftmann@35817
  1398
declare insert [simp del]
haftmann@35719
  1399
haftmann@35719
  1400
lemma insert_idem [simp]:
haftmann@35719
  1401
  assumes "finite A"
haftmann@35719
  1402
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35817
  1403
  using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
haftmann@35719
  1404
haftmann@35719
  1405
lemma union_idem:
haftmann@35719
  1406
  assumes "finite A" and "finite B"
haftmann@35719
  1407
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1408
proof -
haftmann@35817
  1409
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1410
  then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
haftmann@35817
  1411
  with assms show ?thesis by (simp add: union_inter)
haftmann@35719
  1412
qed
haftmann@35719
  1413
haftmann@35719
  1414
end
haftmann@35719
  1415
haftmann@35817
  1416
haftmann@35817
  1417
subsubsection {* The image case with fixed function *}
haftmann@35817
  1418
haftmann@35796
  1419
no_notation times (infixl "*" 70)
haftmann@35796
  1420
no_notation Groups.one ("1")
haftmann@35722
  1421
haftmann@35722
  1422
locale folding_image_simple = comm_monoid +
haftmann@35722
  1423
  fixes g :: "('b \<Rightarrow> 'a)"
haftmann@35722
  1424
  fixes F :: "'b set \<Rightarrow> 'a"
haftmann@35817
  1425
  assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
haftmann@35722
  1426
begin
haftmann@35722
  1427
haftmann@35722
  1428
lemma empty [simp]:
haftmann@35722
  1429
  "F {} = 1"
haftmann@35817
  1430
  by (simp add: eq_fold_g)
haftmann@35722
  1431
haftmann@35722
  1432
lemma insert [simp]:
haftmann@35722
  1433
  assumes "finite A" and "x \<notin> A"
haftmann@35722
  1434
  shows "F (insert x A) = g x * F A"
haftmann@35722
  1435
proof -
wenzelm@46898
  1436
  interpret comp_fun_commute "%x y. (g x) * y"
wenzelm@46898
  1437
    by default (simp add: ac_simps fun_eq_iff)
wenzelm@46898
  1438
  from assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
haftmann@35722
  1439
    by (simp add: fold_image_def)
haftmann@35817
  1440
  with `finite A` show ?thesis by (simp add: eq_fold_g)
haftmann@35722
  1441
qed
haftmann@35722
  1442
haftmann@35722
  1443
lemma remove:
haftmann@35722
  1444
  assumes "finite A" and "x \<in> A"
haftmann@35722
  1445
  shows "F A = g x * F (A - {x})"
haftmann@35722
  1446
proof -
haftmann@35722
  1447
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35722
  1448
    by (auto dest: mk_disjoint_insert)
haftmann@35722
  1449
  moreover from `finite A` this have "finite B" by simp
haftmann@35722
  1450
  ultimately show ?thesis by simp
haftmann@35722
  1451
qed
haftmann@35722
  1452
haftmann@35722
  1453
lemma insert_remove:
haftmann@35722
  1454
  assumes "finite A"
haftmann@35722
  1455
  shows "F (insert x A) = g x * F (A - {x})"
haftmann@35722
  1456
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35722
  1457
haftmann@35817
  1458
lemma neutral:
haftmann@35817
  1459
  assumes "finite A" and "\<forall>x\<in>A. g x = 1"
haftmann@35817
  1460
  shows "F A = 1"
haftmann@35817
  1461
  using assms by (induct A) simp_all
haftmann@35817
  1462
haftmann@35722
  1463
lemma union_inter:
haftmann@35722
  1464
  assumes "finite A" and "finite B"
haftmann@35817
  1465
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35722
  1466
using assms proof (induct A)
haftmann@35722
  1467
  case empty then show ?case by simp
haftmann@35722
  1468
next
haftmann@35722
  1469
  case (insert x A) then show ?case
haftmann@35722
  1470
    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
haftmann@35722
  1471
qed
haftmann@35722
  1472
haftmann@35817
  1473
corollary union_inter_neutral:
haftmann@35817
  1474
  assumes "finite A" and "finite B"
haftmann@35817
  1475
  and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
haftmann@35817
  1476
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1477
  using assms by (simp add: union_inter [symmetric] neutral)
haftmann@35817
  1478
haftmann@35722
  1479
corollary union_disjoint:
haftmann@35722
  1480
  assumes "finite A" and "finite B"
haftmann@35722
  1481
  assumes "A \<inter> B = {}"
haftmann@35722
  1482
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1483
  using assms by (simp add: union_inter_neutral)
haftmann@35722
  1484
haftmann@35719
  1485
end
haftmann@35722
  1486
haftmann@35817
  1487
haftmann@35817
  1488
subsubsection {* The image case with flexible function *}
haftmann@35817
  1489
haftmann@35722
  1490
locale folding_image = comm_monoid +
haftmann@35722
  1491
  fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@35722
  1492
  assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
haftmann@35722
  1493
haftmann@35722
  1494
sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
haftmann@35722
  1495
qed (fact eq_fold)
haftmann@35722
  1496
haftmann@35722
  1497
context folding_image
haftmann@35722
  1498
begin
haftmann@35722
  1499
haftmann@35817
  1500
lemma reindex: (* FIXME polymorhism *)
haftmann@35722
  1501
  assumes "finite A" and "inj_on h A"
haftmann@35722
  1502
  shows "F g (h ` A) = F (g \<circ> h) A"
haftmann@35722
  1503
  using assms by (induct A) auto
haftmann@35722
  1504
haftmann@35722
  1505
lemma cong:
haftmann@35722
  1506
  assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
haftmann@35722
  1507
  shows "F g A = F h A"
haftmann@35722
  1508
proof -
haftmann@35722
  1509
  from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
haftmann@35722
  1510
  apply - apply (erule finite_induct) apply simp
haftmann@35722
  1511
  apply (simp add: subset_insert_iff, clarify)
haftmann@35722
  1512
  apply (subgoal_tac "finite C")
haftmann@35722
  1513
  prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
haftmann@35722
  1514
  apply (subgoal_tac "C = insert x (C - {x})")
haftmann@35722
  1515
  prefer 2 apply blast
haftmann@35722
  1516
  apply (erule ssubst)
haftmann@35722
  1517
  apply (drule spec)
haftmann@35722
  1518
  apply (erule (1) notE impE)
haftmann@35722
  1519
  apply (simp add: Ball_def del: insert_Diff_single)
haftmann@35722
  1520
  done
haftmann@35722
  1521
  with assms show ?thesis by simp
haftmann@35722
  1522
qed
haftmann@35722
  1523
haftmann@35722
  1524
lemma UNION_disjoint:
haftmann@35722
  1525
  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
haftmann@35722
  1526
  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
haftmann@35722
  1527
  shows "F g (UNION I A) = F (F g \<circ> A) I"
haftmann@35722
  1528
apply (insert assms)
haftmann@41656
  1529
apply (induct rule: finite_induct)
haftmann@41656
  1530
apply simp
haftmann@41656
  1531
apply atomize
haftmann@35722
  1532
apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
haftmann@35722
  1533
 prefer 2 apply blast
haftmann@35722
  1534
apply (subgoal_tac "A x Int UNION Fa A = {}")
haftmann@35722
  1535
 prefer 2 apply blast
haftmann@35722
  1536
apply (simp add: union_disjoint)
haftmann@35722
  1537
done
haftmann@35722
  1538
haftmann@35722
  1539
lemma distrib:
haftmann@35722
  1540
  assumes "finite A"
haftmann@35722
  1541
  shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
haftmann@35722
  1542
  using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
haftmann@35722
  1543
haftmann@35722
  1544
lemma related: 
haftmann@35722
  1545
  assumes Re: "R 1 1" 
haftmann@35722
  1546
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
haftmann@35722
  1547
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
haftmann@35722
  1548
  shows "R (F h S) (F g S)"
haftmann@35722
  1549
  using fS by (rule finite_subset_induct) (insert assms, auto)
haftmann@35722
  1550
haftmann@35722
  1551
lemma eq_general:
haftmann@35722
  1552
  assumes fS: "finite S"
haftmann@35722
  1553
  and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 
haftmann@35722
  1554
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
haftmann@35722
  1555
  shows "F f1 S = F f2 S'"
haftmann@35722
  1556
proof-
haftmann@35722
  1557
  from h f12 have hS: "h ` S = S'" by blast
haftmann@35722
  1558
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
haftmann@35722
  1559
    from f12 h H  have "x = y" by auto }
haftmann@35722
  1560
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
haftmann@35722
  1561
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
haftmann@35722
  1562
  from hS have "F f2 S' = F f2 (h ` S)" by simp
haftmann@35722
  1563
  also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
haftmann@35722
  1564
  also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
haftmann@35722
  1565
    by blast
haftmann@35722
  1566
  finally show ?thesis ..
haftmann@35722
  1567
qed
haftmann@35722
  1568
haftmann@35722
  1569
lemma eq_general_inverses:
haftmann@35722
  1570
  assumes fS: "finite S" 
haftmann@35722
  1571
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
haftmann@35722
  1572
  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
  1573
  shows "F j S = F g T"
haftmann@35722
  1574
  (* metis solves it, but not yet available here *)
haftmann@35722
  1575
  apply (rule eq_general [OF fS, of T h g j])
haftmann@35722
  1576
  apply (rule ballI)
haftmann@35722
  1577
  apply (frule kh)
haftmann@35722
  1578
  apply (rule ex1I[])
haftmann@35722
  1579
  apply blast
haftmann@35722
  1580
  apply clarsimp
haftmann@35722
  1581
  apply (drule hk) apply simp
haftmann@35722
  1582
  apply (rule sym)
haftmann@35722
  1583
  apply (erule conjunct1[OF conjunct2[OF hk]])
haftmann@35722
  1584
  apply (rule ballI)
haftmann@35722
  1585
  apply (drule hk)
haftmann@35722
  1586
  apply blast
haftmann@35722
  1587
  done
haftmann@35722
  1588
haftmann@35722
  1589
end
haftmann@35722
  1590
haftmann@35817
  1591
haftmann@35817
  1592
subsubsection {* The image case with fixed function and idempotency *}
haftmann@35817
  1593
haftmann@35817
  1594
locale folding_image_simple_idem = folding_image_simple +
haftmann@35817
  1595
  assumes idem: "x * x = x"
haftmann@35817
  1596
haftmann@35817
  1597
sublocale folding_image_simple_idem < semilattice proof
haftmann@35817
  1598
qed (fact idem)
haftmann@35817
  1599
haftmann@35817
  1600
context folding_image_simple_idem
haftmann@35817
  1601
begin
haftmann@35817
  1602
haftmann@35817
  1603
lemma in_idem:
haftmann@35817
  1604
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1605
  shows "g x * F A = F A"
haftmann@35817
  1606
  using assms by (induct A) (auto simp add: left_commute)
haftmann@35817
  1607
haftmann@35817
  1608
lemma subset_idem:
haftmann@35817
  1609
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1610
  shows "F B * F A = F A"
haftmann@35817
  1611
proof -
haftmann@35817
  1612
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1613
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1614
    (auto simp add: assoc in_idem `finite A`)
haftmann@35817
  1615
qed
haftmann@35817
  1616
haftmann@35817
  1617
declare insert [simp del]
haftmann@35817
  1618
haftmann@35817
  1619
lemma insert_idem [simp]:
haftmann@35817
  1620
  assumes "finite A"
haftmann@35817
  1621
  shows "F (insert x A) = g x * F A"
haftmann@35817
  1622
  using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
haftmann@35817
  1623
haftmann@35817
  1624
lemma union_idem:
haftmann@35817
  1625
  assumes "finite A" and "finite B"
haftmann@35817
  1626
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1627
proof -
haftmann@35817
  1628
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1629
  then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
haftmann@35817
  1630
  with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1631
qed
haftmann@35817
  1632
haftmann@35817
  1633
end
haftmann@35817
  1634
haftmann@35817
  1635
haftmann@35817
  1636
subsubsection {* The image case with flexible function and idempotency *}
haftmann@35817
  1637
haftmann@35817
  1638
locale folding_image_idem = folding_image +
haftmann@35817
  1639
  assumes idem: "x * x = x"
haftmann@35817
  1640
haftmann@35817
  1641
sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
haftmann@35817
  1642
qed (fact idem)
haftmann@35817
  1643
haftmann@35817
  1644
haftmann@35817
  1645
subsubsection {* The neutral-less case *}
haftmann@35817
  1646
haftmann@35817
  1647
locale folding_one = abel_semigroup +
haftmann@35817
  1648
  fixes F :: "'a set \<Rightarrow> 'a"
haftmann@35817
  1649
  assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
haftmann@35817
  1650
begin
haftmann@35817
  1651
haftmann@35817
  1652
lemma singleton [simp]:
haftmann@35817
  1653
  "F {x} = x"
haftmann@35817
  1654
  by (simp add: eq_fold)
haftmann@35817
  1655
haftmann@35817
  1656
lemma eq_fold':
haftmann@35817
  1657
  assumes "finite A" and "x \<notin> A"
haftmann@35817
  1658
  shows "F (insert x A) = fold (op *) x A"
haftmann@35817
  1659
proof -
wenzelm@46898
  1660
  interpret ab_semigroup_mult "op *" by default (simp_all add: ac_simps)
wenzelm@46898
  1661
  from assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
haftmann@35817
  1662
qed
haftmann@35817
  1663
haftmann@35817
  1664
lemma insert [simp]:
haftmann@36637
  1665
  assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
haftmann@36637
  1666
  shows "F (insert x A) = x * F A"
haftmann@36637
  1667
proof -
haftmann@36637
  1668
  from `A \<noteq> {}` obtain b where "b \<in> A" by blast
haftmann@35817
  1669
  then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1670
  with `finite A` have "finite B" by simp
haftmann@35817
  1671
  interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
nipkow@39302
  1672
  qed (simp_all add: fun_eq_iff ac_simps)
haftmann@42871
  1673
  thm fold.comp_fun_commute' [of B b, simplified fun_eq_iff, simplified]
haftmann@42871
  1674
  from `finite B` fold.comp_fun_commute' [of B x]
haftmann@35817
  1675
    have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
nipkow@39302
  1676
  then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)
haftmann@35817
  1677
  from `finite B` * fold.insert [of B b]
haftmann@35817
  1678
    have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
nipkow@39302
  1679
  then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)
haftmann@35817
  1680
  from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
haftmann@35817
  1681
qed
haftmann@35817
  1682
haftmann@35817
  1683
lemma remove:
haftmann@35817
  1684
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1685
  shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1686
proof -
haftmann@35817
  1687
  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1688
  with assms show ?thesis by simp
haftmann@35817
  1689
qed
haftmann@35817
  1690
haftmann@35817
  1691
lemma insert_remove:
haftmann@35817
  1692
  assumes "finite A"
haftmann@35817
  1693
  shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1694
  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
haftmann@35817
  1695
haftmann@35817
  1696
lemma union_disjoint:
haftmann@35817
  1697
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
haftmann@35817
  1698
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1699
  using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
haftmann@35817
  1700
haftmann@35817
  1701
lemma union_inter:
haftmann@35817
  1702
  assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
haftmann@35817
  1703
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35817
  1704
proof -
haftmann@35817
  1705
  from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
haftmann@35817
  1706
  from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
haftmann@35817
  1707
    case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
haftmann@35817
  1708
  next
haftmann@35817
  1709
    case (insert x A) show ?case proof (cases "x \<in> B")
haftmann@35817
  1710
      case True then have "B \<noteq> {}" by auto
haftmann@35817
  1711
      with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
haftmann@35817
  1712
        (simp_all add: insert_absorb ac_simps union_disjoint)
haftmann@35817
  1713
    next
haftmann@35817
  1714
      case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
haftmann@35817
  1715
      moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
haftmann@35817
  1716
        by auto
haftmann@35817
  1717
      ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
haftmann@35817
  1718
    qed
haftmann@35817
  1719
  qed
haftmann@35817
  1720
qed
haftmann@35817
  1721
haftmann@35817
  1722
lemma closed:
haftmann@35817
  1723
  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
haftmann@35817
  1724
  shows "F A \<in> A"
haftmann@35817
  1725
using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
haftmann@35817
  1726
  case singleton then show ?case by simp
haftmann@35817
  1727
next
haftmann@35817
  1728
  case insert with elem show ?case by force
haftmann@35817
  1729
qed
haftmann@35817
  1730
haftmann@35817
  1731
end
haftmann@35817
  1732
haftmann@35817
  1733
haftmann@35817
  1734
subsubsection {* The neutral-less case with idempotency *}
haftmann@35817
  1735
haftmann@35817
  1736
locale folding_one_idem = folding_one +
haftmann@35817
  1737
  assumes idem: "x * x = x"
haftmann@35817
  1738
haftmann@35817
  1739
sublocale folding_one_idem < semilattice proof
haftmann@35817
  1740
qed (fact idem)
haftmann@35817
  1741
haftmann@35817
  1742
context folding_one_idem
haftmann@35817
  1743
begin
haftmann@35817
  1744
haftmann@35817
  1745
lemma in_idem:
haftmann@35817
  1746
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1747
  shows "x * F A = F A"
haftmann@35817
  1748
proof -
haftmann@35817
  1749
  from assms have "A \<noteq> {}" by auto
haftmann@35817
  1750
  with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
haftmann@35817
  1751
qed
haftmann@35817
  1752
haftmann@35817
  1753
lemma subset_idem:
haftmann@35817
  1754
  assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
haftmann@35817
  1755
  shows "F B * F A = F A"
haftmann@35817
  1756
proof -
haftmann@35817
  1757
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1758
  then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
haftmann@35817
  1759
    (simp_all add: assoc in_idem `finite A`)
haftmann@35817
  1760
qed
haftmann@35817
  1761
haftmann@35817
  1762
lemma eq_fold_idem':
haftmann@35817
  1763
  assumes "finite A"
haftmann@35817
  1764
  shows "F (insert a A) = fold (op *) a A"
haftmann@35817
  1765
proof -
wenzelm@46898
  1766
  interpret ab_semigroup_idem_mult "op *" by default (simp_all add: ac_simps)
wenzelm@46898
  1767
  from assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
haftmann@35817
  1768
qed
haftmann@35817
  1769
haftmann@35817
  1770
lemma insert_idem [simp]:
haftmann@36637
  1771
  assumes "finite A" and "A \<noteq> {}"
haftmann@36637
  1772
  shows "F (insert x A) = x * F A"
haftmann@35817
  1773
proof (cases "x \<in> A")
haftmann@36637
  1774
  case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
haftmann@35817
  1775
next
haftmann@36637
  1776
  case True
haftmann@36637
  1777
  from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
haftmann@35817
  1778
qed
haftmann@35817
  1779
  
haftmann@35817
  1780
lemma union_idem:
haftmann@35817
  1781
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
haftmann@35817
  1782
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1783
proof (cases "A \<inter> B = {}")
haftmann@35817
  1784
  case True with assms show ?thesis by (simp add: union_disjoint)
haftmann@35817
  1785
next
haftmann@35817
  1786
  case False
haftmann@35817
  1787
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1788
  with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
haftmann@35817
  1789
  with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1790
qed
haftmann@35817
  1791
haftmann@35817
  1792
lemma hom_commute:
haftmann@35817
  1793
  assumes hom: "\<And>x y. h (x * y) = h x * h y"
haftmann@35817
  1794
  and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
haftmann@35817
  1795
using N proof (induct rule: finite_ne_induct)
haftmann@35817
  1796
  case singleton thus ?case by simp
haftmann@35817
  1797
next
haftmann@35817
  1798
  case (insert n N)
haftmann@35817
  1799
  then have "h (F (insert n N)) = h (n * F N)" by simp
haftmann@35817
  1800
  also have "\<dots> = h n * h (F N)" by (rule hom)
haftmann@35817
  1801
  also have "h (F N) = F (h ` N)" by(rule insert)
haftmann@35817
  1802
  also have "h n * \<dots> = F (insert (h n) (h ` N))"
haftmann@35817
  1803
    using insert by(simp)
haftmann@35817
  1804
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@35817
  1805
  finally show ?case .
haftmann@35817
  1806
qed
haftmann@35817
  1807
haftmann@35817
  1808
end
haftmann@35817
  1809
haftmann@35796
  1810
notation times (infixl "*" 70)
haftmann@35796
  1811
notation Groups.one ("1")
haftmann@35722
  1812
haftmann@35722
  1813
haftmann@35722
  1814
subsection {* Finite cardinality *}
haftmann@35722
  1815
haftmann@35722
  1816
text {* This definition, although traditional, is ugly to work with:
haftmann@35722
  1817
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
haftmann@35722
  1818
But now that we have @{text fold_image} things are easy:
haftmann@35722
  1819
*}
haftmann@35722
  1820
haftmann@35722
  1821
definition card :: "'a set \<Rightarrow> nat" where
haftmann@35722
  1822
  "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
haftmann@35722
  1823
haftmann@37770
  1824
interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
haftmann@35722
  1825
qed (simp add: card_def)
haftmann@35722
  1826
haftmann@35722
  1827
lemma card_infinite [simp]:
haftmann@35722
  1828
  "\<not> finite A \<Longrightarrow> card A = 0"
haftmann@35722
  1829
  by (simp add: card_def)
haftmann@35722
  1830
haftmann@35722
  1831
lemma card_empty:
haftmann@35722
  1832
  "card {} = 0"
haftmann@35722
  1833
  by (fact card.empty)
haftmann@35722
  1834
haftmann@35722
  1835
lemma card_insert_disjoint:
haftmann@35722
  1836
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
haftmann@35722
  1837
  by simp
haftmann@35722
  1838
haftmann@35722
  1839
lemma card_insert_if:
haftmann@35722
  1840
  "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
haftmann@35722
  1841
  by auto (simp add: card.insert_remove card.remove)
haftmann@35722
  1842
haftmann@35722
  1843
lemma card_ge_0_finite:
haftmann@35722
  1844
  "card A > 0 \<Longrightarrow> finite A"
haftmann@35722
  1845
  by (rule ccontr) simp
haftmann@35722
  1846
blanchet@35828
  1847
lemma card_0_eq [simp, no_atp]:
haftmann@35722
  1848
  "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
haftmann@35722
  1849
  by (auto dest: mk_disjoint_insert)
haftmann@35722
  1850
haftmann@35722
  1851
lemma finite_UNIV_card_ge_0:
haftmann@35722
  1852
  "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
haftmann@35722
  1853
  by (rule ccontr) simp
haftmann@35722
  1854
haftmann@35722
  1855
lemma card_eq_0_iff:
haftmann@35722
  1856
  "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
haftmann@35722
  1857
  by auto
haftmann@35722
  1858
haftmann@35722
  1859
lemma card_gt_0_iff:
haftmann@35722
  1860
  "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
haftmann@35722
  1861
  by (simp add: neq0_conv [symmetric] card_eq_0_iff) 
haftmann@35722
  1862
haftmann@35722
  1863
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
haftmann@35722
  1864
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
haftmann@35722
  1865
apply(simp del:insert_Diff_single)
haftmann@35722
  1866
done
haftmann@35722
  1867
haftmann@35722
  1868
lemma card_Diff_singleton:
haftmann@35722
  1869
  "finite A ==> x: A ==> card (A - {x}) = card A - 1"
haftmann@35722
  1870
by (simp add: card_Suc_Diff1 [symmetric])
haftmann@35722
  1871
haftmann@35722
  1872
lemma card_Diff_singleton_if:
bulwahn@45166
  1873
  "finite A ==> card (A - {x}) = (if x : A then card A - 1 else card A)"
haftmann@35722
  1874
by (simp add: card_Diff_singleton)
haftmann@35722
  1875
haftmann@35722
  1876
lemma card_Diff_insert[simp]:
haftmann@35722
  1877
assumes "finite A" and "a:A" and "a ~: B"
haftmann@35722
  1878
shows "card(A - insert a B) = card(A - B) - 1"
haftmann@35722
  1879
proof -
haftmann@35722
  1880
  have "A - insert a B = (A - B) - {a}" using assms by blast
haftmann@35722
  1881
  then show ?thesis using assms by(simp add:card_Diff_singleton)
haftmann@35722
  1882
qed
haftmann@35722
  1883
haftmann@35722
  1884
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
haftmann@35722
  1885
by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  1886
haftmann@35722
  1887
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
haftmann@35722
  1888
by (simp add: card_insert_if)
haftmann@35722
  1889
nipkow@41987
  1890
lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
nipkow@41987
  1891
by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
nipkow@41987
  1892
nipkow@41988
  1893
lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
nipkow@41987
  1894
using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
nipkow@41987
  1895
haftmann@35722
  1896
lemma card_mono:
haftmann@35722
  1897
  assumes "finite B" and "A \<subseteq> B"
haftmann@35722
  1898
  shows "card A \<le> card B"
haftmann@35722
  1899
proof -
haftmann@35722
  1900
  from assms have "finite A" by (auto intro: finite_subset)
haftmann@35722
  1901
  then show ?thesis using assms proof (induct A arbitrary: B)
haftmann@35722
  1902
    case empty then show ?case by simp
haftmann@35722
  1903
  next
haftmann@35722
  1904
    case (insert x A)
haftmann@35722
  1905
    then have "x \<in> B" by simp
haftmann@35722
  1906
    from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
haftmann@35722
  1907
    with insert.hyps have "card A \<le> card (B - {x})" by auto
haftmann@35722
  1908
    with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
haftmann@35722
  1909
  qed
haftmann@35722
  1910
qed
haftmann@35722
  1911
haftmann@35722
  1912
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
haftmann@41656
  1913
apply (induct rule: finite_induct)
haftmann@41656
  1914
apply simp
haftmann@41656
  1915
apply clarify
haftmann@35722
  1916
apply (subgoal_tac "finite A & A - {x} <= F")
haftmann@35722
  1917
 prefer 2 apply (blast intro: finite_subset, atomize)
haftmann@35722
  1918
apply (drule_tac x = "A - {x}" in spec)
haftmann@35722
  1919
apply (simp add: card_Diff_singleton_if split add: split_if_asm)
haftmann@35722
  1920
apply (case_tac "card A", auto)
haftmann@35722
  1921
done
haftmann@35722
  1922
haftmann@35722
  1923
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
haftmann@35722
  1924
apply (simp add: psubset_eq linorder_not_le [symmetric])
haftmann@35722
  1925
apply (blast dest: card_seteq)
haftmann@35722
  1926
done
haftmann@35722
  1927
haftmann@35722
  1928
lemma card_Un_Int: "finite A ==> finite B
haftmann@35722
  1929
    ==> card A + card B = card (A Un B) + card (A Int B)"
haftmann@35817
  1930
  by (fact card.union_inter [symmetric])
haftmann@35722
  1931
haftmann@35722
  1932
lemma card_Un_disjoint: "finite A ==> finite B
haftmann@35722
  1933
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
haftmann@35722
  1934
  by (fact card.union_disjoint)
haftmann@35722
  1935
haftmann@35722
  1936
lemma card_Diff_subset:
haftmann@35722
  1937
  assumes "finite B" and "B \<subseteq> A"
haftmann@35722
  1938
  shows "card (A - B) = card A - card B"
haftmann@35722
  1939
proof (cases "finite A")
haftmann@35722
  1940
  case False with assms show ?thesis by simp
haftmann@35722
  1941
next
haftmann@35722
  1942
  case True with assms show ?thesis by (induct B arbitrary: A) simp_all
haftmann@35722
  1943
qed
haftmann@35722
  1944
haftmann@35722
  1945
lemma card_Diff_subset_Int:
haftmann@35722
  1946
  assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
haftmann@35722
  1947
proof -
haftmann@35722
  1948
  have "A - B = A - A \<inter> B" by auto
haftmann@35722
  1949
  thus ?thesis
haftmann@35722
  1950
    by (simp add: card_Diff_subset AB) 
haftmann@35722
  1951
qed
haftmann@35722
  1952
nipkow@40716
  1953
lemma diff_card_le_card_Diff:
nipkow@40716
  1954
assumes "finite B" shows "card A - card B \<le> card(A - B)"
nipkow@40716
  1955
proof-
nipkow@40716
  1956
  have "card A - card B \<le> card A - card (A \<inter> B)"
nipkow@40716
  1957
    using card_mono[OF assms Int_lower2, of A] by arith
nipkow@40716
  1958
  also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
nipkow@40716
  1959
  finally show ?thesis .
nipkow@40716
  1960
qed
nipkow@40716
  1961
haftmann@35722
  1962
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
haftmann@35722
  1963
apply (rule Suc_less_SucD)
haftmann@35722
  1964
apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  1965
done
haftmann@35722
  1966
haftmann@35722
  1967
lemma card_Diff2_less:
haftmann@35722
  1968
  "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
haftmann@35722
  1969
apply (case_tac "x = y")
haftmann@35722
  1970
 apply (simp add: card_Diff1_less del:card_Diff_insert)
haftmann@35722
  1971
apply (rule less_trans)
haftmann@35722
  1972
 prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
haftmann@35722
  1973
done
haftmann@35722
  1974
haftmann@35722
  1975
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
haftmann@35722
  1976
apply (case_tac "x : A")
haftmann@35722
  1977
 apply (simp_all add: card_Diff1_less less_imp_le)
haftmann@35722
  1978
done
haftmann@35722
  1979
haftmann@35722
  1980
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
haftmann@35722
  1981
by (erule psubsetI, blast)
haftmann@35722
  1982
haftmann@35722
  1983
lemma insert_partition:
haftmann@35722
  1984
  "\<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
  1985
  \<Longrightarrow> x \<inter> \<Union> F = {}"
haftmann@35722
  1986
by auto
haftmann@35722
  1987
haftmann@35722
  1988
lemma finite_psubset_induct[consumes 1, case_names psubset]:
urbanc@36079
  1989
  assumes fin: "finite A" 
urbanc@36079
  1990
  and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A" 
urbanc@36079
  1991
  shows "P A"
urbanc@36079
  1992
using fin
urbanc@36079
  1993
proof (induct A taking: card rule: measure_induct_rule)
haftmann@35722
  1994
  case (less A)
urbanc@36079
  1995
  have fin: "finite A" by fact
urbanc@36079
  1996
  have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
urbanc@36079
  1997
  { fix B 
urbanc@36079
  1998
    assume asm: "B \<subset> A"
urbanc@36079
  1999
    from asm have "card B < card A" using psubset_card_mono fin by blast
urbanc@36079
  2000
    moreover
urbanc@36079
  2001
    from asm have "B \<subseteq> A" by auto
urbanc@36079
  2002
    then have "finite B" using fin finite_subset by blast
urbanc@36079
  2003
    ultimately 
urbanc@36079
  2004
    have "P B" using ih by simp
urbanc@36079
  2005
  }
urbanc@36079
  2006
  with fin show "P A" using major by blast
haftmann@35722
  2007
qed
haftmann@35722
  2008
haftmann@35722
  2009
text{* main cardinality theorem *}
haftmann@35722
  2010
lemma card_partition [rule_format]:
haftmann@35722
  2011
  "finite C ==>
haftmann@35722
  2012
     finite (\<Union> C) -->
haftmann@35722
  2013
     (\<forall>c\<in>C. card c = k) -->
haftmann@35722
  2014
     (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
haftmann@35722
  2015
     k * card(C) = card (\<Union> C)"
haftmann@35722
  2016
apply (erule finite_induct, simp)
haftmann@35722
  2017
apply (simp add: card_Un_disjoint insert_partition 
haftmann@35722
  2018
       finite_subset [of _ "\<Union> (insert x F)"])
haftmann@35722
  2019
done
haftmann@35722
  2020
haftmann@35722
  2021
lemma card_eq_UNIV_imp_eq_UNIV:
haftmann@35722
  2022
  assumes fin: "finite (UNIV :: 'a set)"
haftmann@35722
  2023
  and card: "card A = card (UNIV :: 'a set)"
haftmann@35722
  2024
  shows "A = (UNIV :: 'a set)"
haftmann@35722
  2025
proof
haftmann@35722
  2026
  show "A \<subseteq> UNIV" by simp
haftmann@35722
  2027
  show "UNIV \<subseteq> A"
haftmann@35722
  2028
  proof
haftmann@35722
  2029
    fix x
haftmann@35722
  2030
    show "x \<in> A"
haftmann@35722
  2031
    proof (rule ccontr)
haftmann@35722
  2032
      assume "x \<notin> A"
haftmann@35722
  2033
      then have "A \<subset> UNIV" by auto
haftmann@35722
  2034
      with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
haftmann@35722
  2035
      with card show False by simp
haftmann@35722
  2036
    qed
haftmann@35722
  2037
  qed
haftmann@35722
  2038
qed
haftmann@35722
  2039
haftmann@35722
  2040
text{*The form of a finite set of given cardinality*}
haftmann@35722
  2041
haftmann@35722
  2042
lemma card_eq_SucD:
haftmann@35722
  2043
assumes "card A = Suc k"
haftmann@35722
  2044
shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
haftmann@35722
  2045
proof -
haftmann@35722
  2046
  have fin: "finite A" using assms by (auto intro: ccontr)
haftmann@35722
  2047
  moreover have "card A \<noteq> 0" using assms by auto
haftmann@35722
  2048
  ultimately obtain b where b: "b \<in> A" by auto
haftmann@35722
  2049
  show ?thesis
haftmann@35722
  2050
  proof (intro exI conjI)
haftmann@35722
  2051
    show "A = insert b (A-{b})" using b by blast
haftmann@35722
  2052
    show "b \<notin> A - {b}" by blast
haftmann@35722
  2053
    show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
nipkow@44890
  2054
      using assms b fin by(fastforce dest:mk_disjoint_insert)+
haftmann@35722
  2055
  qed
haftmann@35722
  2056
qed
haftmann@35722
  2057
haftmann@35722
  2058
lemma card_Suc_eq:
haftmann@35722
  2059
  "(card A = Suc k) =
haftmann@35722
  2060
   (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
haftmann@35722
  2061
apply(rule iffI)
haftmann@35722
  2062
 apply(erule card_eq_SucD)
haftmann@35722
  2063
apply(auto)
haftmann@35722
  2064
apply(subst card_insert)
haftmann@35722
  2065
 apply(auto intro:ccontr)
haftmann@35722
  2066
done
haftmann@35722
  2067
nipkow@44744
  2068
lemma card_le_Suc_iff: "finite A \<Longrightarrow>
nipkow@44744
  2069
  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
  2070
by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
nipkow@44744
  2071
  dest: subset_singletonD split: nat.splits if_splits)
nipkow@44744
  2072
haftmann@35722
  2073
lemma finite_fun_UNIVD2:
haftmann@35722
  2074
  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
haftmann@35722
  2075
  shows "finite (UNIV :: 'b set)"
haftmann@35722
  2076
proof -
haftmann@46146
  2077
  from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
haftmann@46146
  2078
    by (rule finite_imageI)
haftmann@46146
  2079
  moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
haftmann@46146
  2080
    by (rule UNIV_eq_I) auto
haftmann@35722
  2081
  ultimately show "finite (UNIV :: 'b set)" by simp
haftmann@35722
  2082
qed
haftmann@35722
  2083
haftmann@35722
  2084
lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
haftmann@35722
  2085
  unfolding UNIV_unit by simp
haftmann@35722
  2086
haftmann@35722
  2087
haftmann@35722
  2088
subsubsection {* Cardinality of image *}
haftmann@35722
  2089
haftmann@35722
  2090
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
haftmann@41656
  2091
apply (induct rule: finite_induct)
haftmann@35722
  2092
 apply simp
haftmann@35722
  2093
apply (simp add: le_SucI card_insert_if)
haftmann@35722
  2094
done
haftmann@35722
  2095
haftmann@35722
  2096
lemma card_image:
haftmann@35722
  2097
  assumes "inj_on f A"
haftmann@35722
  2098
  shows "card (f ` A) = card A"
haftmann@35722
  2099
proof (cases "finite A")
haftmann@35722
  2100
  case True then show ?thesis using assms by (induct A) simp_all
haftmann@35722
  2101
next
haftmann@35722
  2102
  case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
haftmann@35722
  2103
  with False show ?thesis by simp
haftmann@35722
  2104
qed
haftmann@35722
  2105
haftmann@35722
  2106
lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
haftmann@35722
  2107
by(auto simp: card_image bij_betw_def)
haftmann@35722
  2108
haftmann@35722
  2109
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
haftmann@35722
  2110
by (simp add: card_seteq card_image)
haftmann@35722
  2111
haftmann@35722
  2112
lemma eq_card_imp_inj_on:
haftmann@35722
  2113
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
haftmann@35722
  2114
apply (induct rule:finite_induct)
haftmann@35722
  2115
apply simp
haftmann@35722
  2116
apply(frule card_image_le[where f = f])
haftmann@35722
  2117
apply(simp add:card_insert_if split:if_splits)
haftmann@35722
  2118
done
haftmann@35722
  2119
haftmann@35722
  2120
lemma inj_on_iff_eq_card:
haftmann@35722
  2121
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
haftmann@35722
  2122
by(blast intro: card_image eq_card_imp_inj_on)
haftmann@35722
  2123
haftmann@35722
  2124
haftmann@35722
  2125
lemma card_inj_on_le:
haftmann@35722
  2126
  "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
haftmann@35722
  2127
apply (subgoal_tac "finite A") 
haftmann@35722
  2128
 apply (force intro: card_mono simp add: card_image [symmetric])
haftmann@35722
  2129
apply (blast intro: finite_imageD dest: finite_subset) 
haftmann@35722
  2130
done
haftmann@35722
  2131
haftmann@35722
  2132
lemma card_bij_eq:
haftmann@35722
  2133
  "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
haftmann@35722
  2134
     finite A; finite B |] ==> card A = card B"
haftmann@35722
  2135
by (auto intro: le_antisym card_inj_on_le)
haftmann@35722
  2136
hoelzl@40703
  2137
lemma bij_betw_finite:
hoelzl@40703
  2138
  assumes "bij_betw f A B"
hoelzl@40703
  2139
  shows "finite A \<longleftrightarrow> finite B"
hoelzl@40703
  2140
using assms unfolding bij_betw_def
hoelzl@40703
  2141
using finite_imageD[of f A] by auto
haftmann@35722
  2142
haftmann@41656
  2143
nipkow@37466
  2144
subsubsection {* Pigeonhole Principles *}
nipkow@37466
  2145
nipkow@40311
  2146
lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
nipkow@37466
  2147
by (auto dest: card_image less_irrefl_nat)
nipkow@37466
  2148
nipkow@37466
  2149
lemma pigeonhole_infinite:
nipkow@37466
  2150
assumes  "~ finite A" and "finite(f`A)"
nipkow@37466
  2151
shows "EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  2152
proof -
nipkow@37466
  2153
  have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  2154
  proof(induct "f`A" arbitrary: A rule: finite_induct)
nipkow@37466
  2155
    case empty thus ?case by simp
nipkow@37466
  2156
  next
nipkow@37466
  2157
    case (insert b F)
nipkow@37466
  2158
    show ?case
nipkow@37466
  2159
    proof cases
nipkow@37466
  2160
      assume "finite{a:A. f a = b}"
nipkow@37466
  2161
      hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
nipkow@37466
  2162
      also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
nipkow@37466
  2163
      finally have "~ finite({a:A. f a \<noteq> b})" .
nipkow@37466
  2164
      from insert(3)[OF _ this]
nipkow@37466
  2165
      show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
nipkow@37466
  2166
    next
nipkow@37466
  2167
      assume 1: "~finite{a:A. f a = b}"
nipkow@37466
  2168
      hence "{a \<in> A. f a = b} \<noteq> {}" by force
nipkow@37466
  2169
      thus ?thesis using 1 by blast
nipkow@37466
  2170
    qed
nipkow@37466
  2171
  qed
nipkow@37466
  2172
  from this[OF assms(2,1)] show ?thesis .
nipkow@37466
  2173
qed
nipkow@37466
  2174
nipkow@37466
  2175
lemma pigeonhole_infinite_rel:
nipkow@37466
  2176
assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
nipkow@37466
  2177
shows "EX b:B. ~finite{a:A. R a b}"
nipkow@37466
  2178
proof -
nipkow@37466
  2179
   let ?F = "%a. {b:B. R a b}"
nipkow@37466
  2180
   from finite_Pow_iff[THEN iffD2, OF `finite B`]
nipkow@37466
  2181
   have "finite(?F ` A)" by(blast intro: rev_finite_subset)
nipkow@37466
  2182
   from pigeonhole_infinite[where f = ?F, OF assms(1) this]
nipkow@37466
  2183
   obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
nipkow@37466
  2184
   obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
nipkow@37466
  2185
   { assume "finite{a:A. R a b0}"
nipkow@37466
  2186
     then have "finite {a\<in>A. ?F a = ?F a0}"
nipkow@37466
  2187
       using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
nipkow@37466
  2188
   }
nipkow@37466
  2189
   with 1 `b0 : B` show ?thesis by blast
nipkow@37466
  2190
qed
nipkow@37466
  2191
nipkow@37466
  2192
haftmann@35722
  2193
subsubsection {* Cardinality of sums *}
haftmann@35722
  2194
haftmann@35722
  2195
lemma card_Plus:
haftmann@35722
  2196
  assumes "finite A" and "finite B"
haftmann@35722
  2197
  shows "card (A <+> B) = card A + card B"
haftmann@35722
  2198
proof -
haftmann@35722
  2199
  have "Inl`A \<inter> Inr`B = {}" by fast
haftmann@35722
  2200
  with assms show ?thesis
haftmann@35722
  2201
    unfolding Plus_def
haftmann@35722
  2202
    by (simp add: card_Un_disjoint card_image)
haftmann@35722
  2203
qed
haftmann@35722
  2204
haftmann@35722
  2205
lemma card_Plus_conv_if:
haftmann@35722
  2206
  "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
haftmann@35722
  2207
  by (auto simp add: card_Plus)
haftmann@35722
  2208
haftmann@35722
  2209
haftmann@35722
  2210
subsubsection {* Cardinality of the Powerset *}
haftmann@35722
  2211
haftmann@35722
  2212
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
haftmann@41656
  2213
apply (induct rule: finite_induct)
haftmann@35722
  2214
 apply (simp_all add: Pow_insert)
haftmann@35722
  2215
apply (subst card_Un_disjoint, blast)
nipkow@40786
  2216
  apply (blast, blast)
haftmann@35722
  2217
apply (subgoal_tac "inj_on (insert x) (Pow F)")
haftmann@35722
  2218
 apply (simp add: card_image Pow_insert)
haftmann@35722
  2219
apply (unfold inj_on_def)
haftmann@35722
  2220
apply (blast elim!: equalityE)
haftmann@35722
  2221
done
haftmann@35722
  2222
nipkow@41987
  2223
text {* Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.  *}
haftmann@35722
  2224
haftmann@35722
  2225
lemma dvd_partition:
haftmann@35722
  2226
  "finite (Union C) ==>
haftmann@35722
  2227
    ALL c : C. k dvd card c ==>
haftmann@35722
  2228
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
haftmann@35722
  2229
  k dvd card (Union C)"
haftmann@41656
  2230
apply (frule finite_UnionD)
haftmann@41656
  2231
apply (rotate_tac -1)
haftmann@41656
  2232
apply (induct rule: finite_induct)
haftmann@41656
  2233
apply simp_all
haftmann@41656
  2234
apply clarify
haftmann@35722
  2235
apply (subst card_Un_disjoint)
haftmann@35722
  2236
   apply (auto simp add: disjoint_eq_subset_Compl)
haftmann@35722
  2237
done
haftmann@35722
  2238
haftmann@35722
  2239
haftmann@35722
  2240
subsubsection {* Relating injectivity and surjectivity *}
haftmann@35722
  2241
haftmann@41656
  2242
lemma finite_surj_inj: "finite A \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> inj_on f A"
haftmann@35722
  2243
apply(rule eq_card_imp_inj_on, assumption)
haftmann@35722
  2244
apply(frule finite_imageI)
haftmann@35722
  2245
apply(drule (1) card_seteq)
haftmann@35722
  2246
 apply(erule card_image_le)
haftmann@35722
  2247
apply simp
haftmann@35722
  2248
done
haftmann@35722
  2249
haftmann@35722
  2250
lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2251
shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
hoelzl@40702
  2252
by (blast intro: finite_surj_inj subset_UNIV)
haftmann@35722
  2253
haftmann@35722
  2254
lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2255
shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
nipkow@44890
  2256
by(fastforce simp:surj_def dest!: endo_inj_surj)
haftmann@35722
  2257
haftmann@35722
  2258
corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
haftmann@35722
  2259
proof
haftmann@35722
  2260
  assume "finite(UNIV::nat set)"
haftmann@35722
  2261
  with finite_UNIV_inj_surj[of Suc]
haftmann@35722
  2262
  show False by simp (blast dest: Suc_neq_Zero surjD)
haftmann@35722
  2263
qed
haftmann@35722
  2264
blanchet@35828
  2265
(* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
blanchet@35828
  2266
lemma infinite_UNIV_char_0[no_atp]:
haftmann@35722
  2267
  "\<not> finite (UNIV::'a::semiring_char_0 set)"
haftmann@35722
  2268
proof
haftmann@35722
  2269
  assume "finite (UNIV::'a set)"
haftmann@35722
  2270
  with subset_UNIV have "finite (range of_nat::'a set)"
haftmann@35722
  2271
    by (rule finite_subset)
haftmann@35722
  2272
  moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
haftmann@35722
  2273
    by (simp add: inj_on_def)
haftmann@35722
  2274
  ultimately have "finite (UNIV::nat set)"
haftmann@35722
  2275
    by (rule finite_imageD)
haftmann@35722
  2276
  then show "False"
haftmann@35722
  2277
    by simp
haftmann@35722
  2278
qed
haftmann@35722
  2279
haftmann@46033
  2280
hide_const (open) Finite_Set.fold
haftmann@46033
  2281
haftmann@35722
  2282
end