src/HOL/Finite_Set.thy
author haftmann
Thu Mar 18 13:56:32 2010 +0100 (2010-03-18)
changeset 35817 d8b8527102f5
parent 35796 2d44d2a1f68e
child 35831 e31ec41a551b
permissions -rw-r--r--
added locales folding_one_(idem); various streamlining and tuning
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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 Power Option
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begin
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subsection {* Predicate for finite sets *}
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inductive finite :: "'a set => bool"
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  where
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    emptyI [simp, intro!]: "finite {}"
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  | insertI [simp, intro!]: "finite A ==> finite (insert a A)"
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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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  thus ?thesis by blast
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qed
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lemma finite_induct [case_names empty insert, induct set: finite]:
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  "finite F ==>
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    P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
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  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
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proof -
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  assume "P {}" and
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    insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
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  assume "finite F"
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  thus "P F"
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  proof induct
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    show "P {}" 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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qed
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lemma finite_ne_induct[case_names singleton insert, consumes 2]:
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assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>
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 \<lbrakk> \<And>x. P{x};
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   \<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk>
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 \<Longrightarrow> P F"
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using fin
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proof induct
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  case empty thus ?case by simp
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next
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  case (insert x F)
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  show ?case
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  proof cases
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    assume "F = {}"
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    thus ?thesis using `P {x}` by simp
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  next
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    assume "F \<noteq> {}"
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    thus ?thesis using insert by blast
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  qed
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qed
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lemma finite_subset_induct [consumes 2, case_names empty insert]:
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  assumes "finite F" and "F \<subseteq> A"
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    and empty: "P {}"
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    and insert: "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
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  shows "P F"
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proof -
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  from `finite F` and `F \<subseteq> A`
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  show ?thesis
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  proof induct
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    show "P {}" by fact
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  next
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    fix x F
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    assume "finite F" and "x \<notin> F" and
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      P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
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    show "P (insert x F)"
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    proof (rule insert)
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      from i show "x \<in> A" by blast
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      from i have "F \<subseteq> A" by blast
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      with P show "P F" .
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      show "finite F" by fact
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      show "x \<notin> F" by fact
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    qed
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  qed
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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> ALL x:A. (EX y. P x y) \<Longrightarrow> EX f. ALL x:A. P x (f x)"
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proof (induct set: finite)
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  case empty thus ?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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text{* 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 fin: "finite A" 
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  shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}"
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using fin
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proof induct
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  case empty
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  show ?case  
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  proof show "\<exists>f. {} = f ` {i::nat. i < 0} & 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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  "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
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proof (induct n)
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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 = (\<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 "EX f n::nat. f`A = {i. i<n} & 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]: "finite{n::nat. n<k}"
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by(fastsimp simp: finite_conv_nat_seg_image)
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text {* Finiteness and set theoretic constructions *}
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lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
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by (induct set: finite) simp_all
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lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
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  -- {* Every subset of a finite set is finite. *}
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proof -
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  assume "finite B"
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  thus "!!A. A \<subseteq> B ==> finite A"
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  proof induct
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    case empty
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    thus ?case by simp
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  next
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    case (insert x F A)
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    have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> 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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qed
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lemma rev_finite_subset: "finite B ==> A \<subseteq> B ==> finite A"
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by (rule finite_subset)
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lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
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by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
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lemma finite_Collect_disjI[simp]:
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  "finite{x. P x | Q x} = (finite{x. P x} & finite{x. Q x})"
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by(simp add:Collect_disj_eq)
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lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
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  -- {* The converse obviously fails. *}
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by (blast intro: finite_subset)
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lemma finite_Collect_conjI [simp, intro]:
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  "finite{x. P x} | finite{x. Q x} ==> finite{x. P x & Q x}"
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  -- {* The converse obviously fails. *}
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by(simp add:Collect_conj_eq)
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lemma finite_Collect_le_nat[iff]: "finite{n::nat. n<=k}"
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by(simp add: le_eq_less_or_eq)
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lemma finite_insert [simp]: "finite (insert a A) = finite A"
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  apply (subst insert_is_Un)
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  apply (simp only: finite_Un, blast)
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  done
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lemma finite_Union[simp, intro]:
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 "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
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by (induct rule:finite_induct) simp_all
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lemma finite_Inter[intro]: "EX A: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]: "EX x:I. finite(A x) \<Longrightarrow> finite(INT x:I. A x)"
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by (blast intro: INT_lower finite_subset)
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lemma finite_empty_induct:
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  assumes "finite A"
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    and "P A"
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    and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
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  shows "P {}"
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proof -
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  have "P (A - A)"
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  proof -
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    {
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      fix c b :: "'a set"
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      assume c: "finite c" and b: "finite b"
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        and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
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      have "c \<subseteq> b ==> P (b - c)"
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        using c
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      proof induct
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        case empty
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        from P1 show ?case by simp
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      next
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        case (insert x F)
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        have "P (b - F - {x})"
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        proof (rule P2)
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          from _ b show "finite (b - F)" by (rule finite_subset) blast
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          from insert show "x \<in> b - F" by simp
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          from insert show "P (b - F)" by simp
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        qed
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        also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
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        finally show ?case .
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      qed
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    }
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    then show ?thesis by this (simp_all add: assms)
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  qed
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  then show ?thesis by simp
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qed
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lemma finite_Diff [simp]: "finite A ==> finite (A - B)"
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by (rule Diff_subset [THEN finite_subset])
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lemma finite_Diff2 [simp]:
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  assumes "finite B" shows "finite (A - B) = finite A"
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proof -
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  have "finite A \<longleftrightarrow> finite((A-B) Un (A Int 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_compl[simp]:
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  "finite(A::'a set) \<Longrightarrow> finite(-A) = 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. ~P x} = finite(UNIV::'a set)"
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by(simp add:Collect_neg_eq)
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lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
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  apply (subst Diff_insert)
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  apply (case_tac "a : A - B")
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   apply (rule finite_insert [symmetric, THEN trans])
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   apply (subst insert_Diff, simp_all)
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  done
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text {* Image and Inverse Image over Finite Sets *}
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lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
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  -- {* The image of a finite set is finite. *}
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  by (induct set: finite) 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_surj: "finite A ==> B <= f ` A ==> finite B"
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  apply (frule finite_imageI)
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  apply (erule finite_subset, assumption)
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  done
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lemma finite_range_imageI:
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    "finite (range g) ==> finite (range (%x. f (g x)))"
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  apply (drule finite_imageI, simp add: range_composition)
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  done
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lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
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proof -
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  have aux: "!!A. finite (A - {}) = finite A" by simp
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  fix B :: "'a set"
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  assume "finite B"
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  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
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    apply induct
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     apply simp
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    apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
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     apply clarify
wenzelm@12396
   334
     apply (simp (no_asm_use) add: inj_on_def)
paulson@14208
   335
     apply (blast dest!: aux [THEN iffD1], atomize)
wenzelm@12396
   336
    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
paulson@14208
   337
    apply (frule subsetD [OF equalityD2 insertI1], clarify)
wenzelm@12396
   338
    apply (rule_tac x = xa in bexI)
wenzelm@12396
   339
     apply (simp_all add: inj_on_image_set_diff)
wenzelm@12396
   340
    done
wenzelm@12396
   341
qed (rule refl)
wenzelm@12396
   342
wenzelm@12396
   343
paulson@13825
   344
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
paulson@13825
   345
  -- {* The inverse image of a singleton under an injective function
paulson@13825
   346
         is included in a singleton. *}
paulson@14430
   347
  apply (auto simp add: inj_on_def)
paulson@14430
   348
  apply (blast intro: the_equality [symmetric])
paulson@13825
   349
  done
paulson@13825
   350
paulson@13825
   351
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
paulson@13825
   352
  -- {* The inverse image of a finite set under an injective function
paulson@13825
   353
         is finite. *}
berghofe@22262
   354
  apply (induct set: finite)
wenzelm@21575
   355
   apply simp_all
paulson@14430
   356
  apply (subst vimage_insert)
huffman@35216
   357
  apply (simp add: finite_subset [OF inj_vimage_singleton])
paulson@13825
   358
  done
paulson@13825
   359
huffman@34111
   360
lemma finite_vimageD:
huffman@34111
   361
  assumes fin: "finite (h -` F)" and surj: "surj h"
huffman@34111
   362
  shows "finite F"
huffman@34111
   363
proof -
huffman@34111
   364
  have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
huffman@34111
   365
  also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
huffman@34111
   366
  finally show "finite F" .
huffman@34111
   367
qed
huffman@34111
   368
huffman@34111
   369
lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
huffman@34111
   370
  unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
huffman@34111
   371
paulson@13825
   372
nipkow@15392
   373
text {* The finite UNION of finite sets *}
wenzelm@12396
   374
wenzelm@12396
   375
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
berghofe@22262
   376
  by (induct set: finite) simp_all
wenzelm@12396
   377
wenzelm@12396
   378
text {*
wenzelm@12396
   379
  Strengthen RHS to
paulson@14430
   380
  @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
wenzelm@12396
   381
wenzelm@12396
   382
  We'd need to prove
paulson@14430
   383
  @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
wenzelm@12396
   384
  by induction. *}
wenzelm@12396
   385
nipkow@29918
   386
lemma finite_UN [simp]:
nipkow@29918
   387
  "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
nipkow@29918
   388
by (blast intro: finite_UN_I finite_subset)
wenzelm@12396
   389
nipkow@29920
   390
lemma finite_Collect_bex[simp]: "finite A \<Longrightarrow>
nipkow@29920
   391
  finite{x. EX y:A. Q x y} = (ALL y:A. finite{x. Q x y})"
nipkow@29920
   392
apply(subgoal_tac "{x. EX y:A. Q x y} = UNION A (%y. {x. Q x y})")
nipkow@29920
   393
 apply auto
nipkow@29920
   394
done
nipkow@29920
   395
nipkow@29920
   396
lemma finite_Collect_bounded_ex[simp]: "finite{y. P y} \<Longrightarrow>
nipkow@29920
   397
  finite{x. EX y. P y & Q x y} = (ALL y. P y \<longrightarrow> finite{x. Q x y})"
nipkow@29920
   398
apply(subgoal_tac "{x. EX y. P y & Q x y} = UNION {y. P y} (%y. {x. Q x y})")
nipkow@29920
   399
 apply auto
nipkow@29920
   400
done
nipkow@29920
   401
nipkow@29920
   402
nipkow@17022
   403
lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"
nipkow@17022
   404
by (simp add: Plus_def)
nipkow@17022
   405
nipkow@31080
   406
lemma finite_PlusD: 
nipkow@31080
   407
  fixes A :: "'a set" and B :: "'b set"
nipkow@31080
   408
  assumes fin: "finite (A <+> B)"
nipkow@31080
   409
  shows "finite A" "finite B"
nipkow@31080
   410
proof -
nipkow@31080
   411
  have "Inl ` A \<subseteq> A <+> B" by auto
nipkow@31080
   412
  hence "finite (Inl ` A :: ('a + 'b) set)" using fin by(rule finite_subset)
nipkow@31080
   413
  thus "finite A" by(rule finite_imageD)(auto intro: inj_onI)
nipkow@31080
   414
next
nipkow@31080
   415
  have "Inr ` B \<subseteq> A <+> B" by auto
nipkow@31080
   416
  hence "finite (Inr ` B :: ('a + 'b) set)" using fin by(rule finite_subset)
nipkow@31080
   417
  thus "finite B" by(rule finite_imageD)(auto intro: inj_onI)
nipkow@31080
   418
qed
nipkow@31080
   419
nipkow@31080
   420
lemma finite_Plus_iff[simp]: "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
nipkow@31080
   421
by(auto intro: finite_PlusD finite_Plus)
nipkow@31080
   422
nipkow@31080
   423
lemma finite_Plus_UNIV_iff[simp]:
nipkow@31080
   424
  "finite (UNIV :: ('a + 'b) set) =
nipkow@31080
   425
  (finite (UNIV :: 'a set) & finite (UNIV :: 'b set))"
nipkow@31080
   426
by(subst UNIV_Plus_UNIV[symmetric])(rule finite_Plus_iff)
nipkow@31080
   427
nipkow@31080
   428
nipkow@15392
   429
text {* Sigma of finite sets *}
wenzelm@12396
   430
wenzelm@12396
   431
lemma finite_SigmaI [simp]:
wenzelm@12396
   432
    "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
wenzelm@12396
   433
  by (unfold Sigma_def) (blast intro!: finite_UN_I)
wenzelm@12396
   434
nipkow@15402
   435
lemma finite_cartesian_product: "[| finite A; finite B |] ==>
nipkow@15402
   436
    finite (A <*> B)"
nipkow@15402
   437
  by (rule finite_SigmaI)
nipkow@15402
   438
wenzelm@12396
   439
lemma finite_Prod_UNIV:
wenzelm@12396
   440
    "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
wenzelm@12396
   441
  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
wenzelm@12396
   442
   apply (erule ssubst)
paulson@14208
   443
   apply (erule finite_SigmaI, auto)
wenzelm@12396
   444
  done
wenzelm@12396
   445
paulson@15409
   446
lemma finite_cartesian_productD1:
paulson@15409
   447
     "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
paulson@15409
   448
apply (auto simp add: finite_conv_nat_seg_image) 
paulson@15409
   449
apply (drule_tac x=n in spec) 
paulson@15409
   450
apply (drule_tac x="fst o f" in spec) 
paulson@15409
   451
apply (auto simp add: o_def) 
paulson@15409
   452
 prefer 2 apply (force dest!: equalityD2) 
paulson@15409
   453
apply (drule equalityD1) 
paulson@15409
   454
apply (rename_tac y x)
paulson@15409
   455
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
paulson@15409
   456
 prefer 2 apply force
paulson@15409
   457
apply clarify
paulson@15409
   458
apply (rule_tac x=k in image_eqI, auto)
paulson@15409
   459
done
paulson@15409
   460
paulson@15409
   461
lemma finite_cartesian_productD2:
paulson@15409
   462
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
paulson@15409
   463
apply (auto simp add: finite_conv_nat_seg_image) 
paulson@15409
   464
apply (drule_tac x=n in spec) 
paulson@15409
   465
apply (drule_tac x="snd o f" in spec) 
paulson@15409
   466
apply (auto simp add: o_def) 
paulson@15409
   467
 prefer 2 apply (force dest!: equalityD2) 
paulson@15409
   468
apply (drule equalityD1)
paulson@15409
   469
apply (rename_tac x y)
paulson@15409
   470
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
paulson@15409
   471
 prefer 2 apply force
paulson@15409
   472
apply clarify
paulson@15409
   473
apply (rule_tac x=k in image_eqI, auto)
paulson@15409
   474
done
paulson@15409
   475
paulson@15409
   476
nipkow@15392
   477
text {* The powerset of a finite set *}
wenzelm@12396
   478
wenzelm@12396
   479
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
wenzelm@12396
   480
proof
wenzelm@12396
   481
  assume "finite (Pow A)"
wenzelm@12396
   482
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
wenzelm@12396
   483
  thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
wenzelm@12396
   484
next
wenzelm@12396
   485
  assume "finite A"
wenzelm@12396
   486
  thus "finite (Pow A)"
huffman@35216
   487
    by induct (simp_all add: Pow_insert)
wenzelm@12396
   488
qed
wenzelm@12396
   489
nipkow@29916
   490
lemma finite_Collect_subsets[simp,intro]: "finite A \<Longrightarrow> finite{B. B \<subseteq> A}"
nipkow@29916
   491
by(simp add: Pow_def[symmetric])
nipkow@15392
   492
nipkow@29918
   493
nipkow@15392
   494
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
nipkow@15392
   495
by(blast intro: finite_subset[OF subset_Pow_Union])
nipkow@15392
   496
nipkow@15392
   497
nipkow@31441
   498
lemma finite_subset_image:
nipkow@31441
   499
  assumes "finite B"
nipkow@31441
   500
  shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
nipkow@31441
   501
using assms proof(induct)
nipkow@31441
   502
  case empty thus ?case by simp
nipkow@31441
   503
next
nipkow@31441
   504
  case insert thus ?case
nipkow@31441
   505
    by (clarsimp simp del: image_insert simp add: image_insert[symmetric])
nipkow@31441
   506
       blast
nipkow@31441
   507
qed
nipkow@31441
   508
nipkow@31441
   509
haftmann@26441
   510
subsection {* Class @{text finite}  *}
haftmann@26041
   511
haftmann@26041
   512
setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*}
haftmann@29797
   513
class finite =
haftmann@26041
   514
  assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
haftmann@26041
   515
setup {* Sign.parent_path *}
haftmann@26041
   516
hide const finite
haftmann@26041
   517
huffman@27430
   518
context finite
huffman@27430
   519
begin
huffman@27430
   520
huffman@27430
   521
lemma finite [simp]: "finite (A \<Colon> 'a set)"
haftmann@26441
   522
  by (rule subset_UNIV finite_UNIV finite_subset)+
haftmann@26041
   523
huffman@27430
   524
end
huffman@27430
   525
haftmann@26146
   526
lemma UNIV_unit [noatp]:
haftmann@26041
   527
  "UNIV = {()}" by auto
haftmann@26041
   528
haftmann@35719
   529
instance unit :: finite proof
haftmann@35719
   530
qed (simp add: UNIV_unit)
haftmann@26146
   531
haftmann@26146
   532
lemma UNIV_bool [noatp]:
haftmann@26041
   533
  "UNIV = {False, True}" by auto
haftmann@26041
   534
haftmann@35719
   535
instance bool :: finite proof
haftmann@35719
   536
qed (simp add: UNIV_bool)
haftmann@35719
   537
haftmann@35719
   538
instance * :: (finite, finite) finite proof
haftmann@35719
   539
qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
haftmann@26146
   540
haftmann@35719
   541
lemma finite_option_UNIV [simp]:
haftmann@35719
   542
  "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
haftmann@35719
   543
  by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
haftmann@35719
   544
haftmann@35719
   545
instance option :: (finite) finite proof
haftmann@35719
   546
qed (simp add: UNIV_option_conv)
haftmann@26146
   547
haftmann@26041
   548
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
haftmann@26041
   549
  by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
haftmann@26041
   550
haftmann@26146
   551
instance "fun" :: (finite, finite) finite
haftmann@26146
   552
proof
haftmann@26041
   553
  show "finite (UNIV :: ('a => 'b) set)"
haftmann@26041
   554
  proof (rule finite_imageD)
haftmann@26041
   555
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
berghofe@26792
   556
    have "range ?graph \<subseteq> Pow UNIV" by simp
berghofe@26792
   557
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
berghofe@26792
   558
      by (simp only: finite_Pow_iff finite)
berghofe@26792
   559
    ultimately show "finite (range ?graph)"
berghofe@26792
   560
      by (rule finite_subset)
haftmann@26041
   561
    show "inj ?graph" by (rule inj_graph)
haftmann@26041
   562
  qed
haftmann@26041
   563
qed
haftmann@26041
   564
haftmann@35719
   565
instance "+" :: (finite, finite) finite proof
haftmann@35719
   566
qed (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
haftmann@27981
   567
haftmann@26041
   568
haftmann@35817
   569
subsection {* A basic fold functional for finite sets *}
nipkow@15392
   570
nipkow@15392
   571
text {* The intended behaviour is
wenzelm@31916
   572
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
nipkow@28853
   573
if @{text f} is ``left-commutative'':
nipkow@15392
   574
*}
nipkow@15392
   575
nipkow@28853
   576
locale fun_left_comm =
nipkow@28853
   577
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@28853
   578
  assumes fun_left_comm: "f x (f y z) = f y (f x z)"
nipkow@28853
   579
begin
nipkow@28853
   580
nipkow@28853
   581
text{* On a functional level it looks much nicer: *}
nipkow@28853
   582
nipkow@28853
   583
lemma fun_comp_comm:  "f x \<circ> f y = f y \<circ> f x"
nipkow@28853
   584
by (simp add: fun_left_comm expand_fun_eq)
nipkow@28853
   585
nipkow@28853
   586
end
nipkow@28853
   587
nipkow@28853
   588
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
nipkow@28853
   589
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
nipkow@28853
   590
  emptyI [intro]: "fold_graph f z {} z" |
nipkow@28853
   591
  insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
nipkow@28853
   592
      \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
nipkow@28853
   593
nipkow@28853
   594
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
nipkow@28853
   595
nipkow@28853
   596
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
nipkow@28853
   597
[code del]: "fold f z A = (THE y. fold_graph f z A y)"
nipkow@15392
   598
paulson@15498
   599
text{*A tempting alternative for the definiens is
nipkow@28853
   600
@{term "if finite A then THE y. fold_graph f z A y else e"}.
paulson@15498
   601
It allows the removal of finiteness assumptions from the theorems
nipkow@28853
   602
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
nipkow@28853
   603
The proofs become ugly. It is not worth the effort. (???) *}
nipkow@28853
   604
nipkow@28853
   605
nipkow@28853
   606
lemma Diff1_fold_graph:
nipkow@28853
   607
  "fold_graph f z (A - {x}) y \<Longrightarrow> x \<in> A \<Longrightarrow> fold_graph f z A (f x y)"
nipkow@28853
   608
by (erule insert_Diff [THEN subst], rule fold_graph.intros, auto)
nipkow@28853
   609
nipkow@28853
   610
lemma fold_graph_imp_finite: "fold_graph f z A x \<Longrightarrow> finite A"
nipkow@28853
   611
by (induct set: fold_graph) auto
nipkow@28853
   612
nipkow@28853
   613
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
nipkow@28853
   614
by (induct set: finite) auto
nipkow@28853
   615
nipkow@28853
   616
nipkow@28853
   617
subsubsection{*From @{const fold_graph} to @{term fold}*}
nipkow@15392
   618
paulson@15510
   619
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
wenzelm@19868
   620
  by (auto simp add: less_Suc_eq) 
paulson@15510
   621
paulson@15510
   622
lemma insert_image_inj_on_eq:
paulson@15510
   623
     "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; 
paulson@15510
   624
        inj_on h {i. i < Suc m}|] 
paulson@15510
   625
      ==> A = h ` {i. i < m}"
paulson@15510
   626
apply (auto simp add: image_less_Suc inj_on_def)
paulson@15510
   627
apply (blast intro: less_trans) 
paulson@15510
   628
done
paulson@15510
   629
paulson@15510
   630
lemma insert_inj_onE:
paulson@15510
   631
  assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" 
paulson@15510
   632
      and inj_on: "inj_on h {i::nat. i<n}"
paulson@15510
   633
  shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"
paulson@15510
   634
proof (cases n)
paulson@15510
   635
  case 0 thus ?thesis using aA by auto
paulson@15510
   636
next
paulson@15510
   637
  case (Suc m)
wenzelm@23389
   638
  have nSuc: "n = Suc m" by fact
paulson@15510
   639
  have mlessn: "m<n" by (simp add: nSuc)
paulson@15532
   640
  from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
nipkow@27165
   641
  let ?hm = "Fun.swap k m h"
paulson@15520
   642
  have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn 
huffman@35216
   643
    by (simp add: inj_on)
paulson@15510
   644
  show ?thesis
paulson@15520
   645
  proof (intro exI conjI)
paulson@15520
   646
    show "inj_on ?hm {i. i < m}" using inj_hm
paulson@15510
   647
      by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
paulson@15520
   648
    show "m<n" by (rule mlessn)
paulson@15520
   649
    show "A = ?hm ` {i. i < m}" 
paulson@15520
   650
    proof (rule insert_image_inj_on_eq)
nipkow@27165
   651
      show "inj_on (Fun.swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
paulson@15520
   652
      show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) 
paulson@15520
   653
      show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
wenzelm@32960
   654
        using aA hkeq nSuc klessn
wenzelm@32960
   655
        by (auto simp add: swap_def image_less_Suc fun_upd_image 
wenzelm@32960
   656
                           less_Suc_eq inj_on_image_set_diff [OF inj_on])
nipkow@15479
   657
    qed
nipkow@15479
   658
  qed
nipkow@15479
   659
qed
nipkow@15479
   660
nipkow@28853
   661
context fun_left_comm
haftmann@26041
   662
begin
haftmann@26041
   663
nipkow@28853
   664
lemma fold_graph_determ_aux:
nipkow@28853
   665
  "A = h`{i::nat. i<n} \<Longrightarrow> inj_on h {i. i<n}
nipkow@28853
   666
   \<Longrightarrow> fold_graph f z A x \<Longrightarrow> fold_graph f z A x'
nipkow@15392
   667
   \<Longrightarrow> x' = x"
nipkow@28853
   668
proof (induct n arbitrary: A x x' h rule: less_induct)
paulson@15510
   669
  case (less n)
nipkow@28853
   670
  have IH: "\<And>m h A x x'. m < n \<Longrightarrow> A = h ` {i. i<m}
nipkow@28853
   671
      \<Longrightarrow> inj_on h {i. i<m} \<Longrightarrow> fold_graph f z A x
nipkow@28853
   672
      \<Longrightarrow> fold_graph f z A x' \<Longrightarrow> x' = x" by fact
nipkow@28853
   673
  have Afoldx: "fold_graph f z A x" and Afoldx': "fold_graph f z A x'"
nipkow@28853
   674
    and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" by fact+
nipkow@28853
   675
  show ?case
nipkow@28853
   676
  proof (rule fold_graph.cases [OF Afoldx])
nipkow@28853
   677
    assume "A = {}" and "x = z"
nipkow@28853
   678
    with Afoldx' show "x' = x" by auto
nipkow@28853
   679
  next
nipkow@28853
   680
    fix B b u
nipkow@28853
   681
    assume AbB: "A = insert b B" and x: "x = f b u"
nipkow@28853
   682
      and notinB: "b \<notin> B" and Bu: "fold_graph f z B u"
nipkow@28853
   683
    show "x'=x" 
nipkow@28853
   684
    proof (rule fold_graph.cases [OF Afoldx'])
nipkow@28853
   685
      assume "A = {}" and "x' = z"
nipkow@28853
   686
      with AbB show "x' = x" by blast
nipkow@15392
   687
    next
nipkow@28853
   688
      fix C c v
nipkow@28853
   689
      assume AcC: "A = insert c C" and x': "x' = f c v"
nipkow@28853
   690
        and notinC: "c \<notin> C" and Cv: "fold_graph f z C v"
nipkow@28853
   691
      from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
nipkow@28853
   692
      from insert_inj_onE [OF Beq notinB injh]
nipkow@28853
   693
      obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" 
nipkow@28853
   694
        and Beq: "B = hB ` {i. i < mB}" and lessB: "mB < n" by auto 
nipkow@28853
   695
      from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
nipkow@28853
   696
      from insert_inj_onE [OF Ceq notinC injh]
nipkow@28853
   697
      obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
nipkow@28853
   698
        and Ceq: "C = hC ` {i. i < mC}" and lessC: "mC < n" by auto 
nipkow@28853
   699
      show "x'=x"
nipkow@28853
   700
      proof cases
nipkow@28853
   701
        assume "b=c"
wenzelm@32960
   702
        then moreover have "B = C" using AbB AcC notinB notinC by auto
wenzelm@32960
   703
        ultimately show ?thesis  using Bu Cv x x' IH [OF lessC Ceq inj_onC]
nipkow@28853
   704
          by auto
nipkow@15392
   705
      next
wenzelm@32960
   706
        assume diff: "b \<noteq> c"
wenzelm@32960
   707
        let ?D = "B - {c}"
wenzelm@32960
   708
        have B: "B = insert c ?D" and C: "C = insert b ?D"
wenzelm@32960
   709
          using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
wenzelm@32960
   710
        have "finite A" by(rule fold_graph_imp_finite [OF Afoldx])
wenzelm@32960
   711
        with AbB have "finite ?D" by simp
wenzelm@32960
   712
        then obtain d where Dfoldd: "fold_graph f z ?D d"
wenzelm@32960
   713
          using finite_imp_fold_graph by iprover
wenzelm@32960
   714
        moreover have cinB: "c \<in> B" using B by auto
wenzelm@32960
   715
        ultimately have "fold_graph f z B (f c d)" by(rule Diff1_fold_graph)
wenzelm@32960
   716
        hence "f c d = u" by (rule IH [OF lessB Beq inj_onB Bu]) 
nipkow@28853
   717
        moreover have "f b d = v"
wenzelm@32960
   718
        proof (rule IH[OF lessC Ceq inj_onC Cv])
wenzelm@32960
   719
          show "fold_graph f z C (f b d)" using C notinB Dfoldd by fastsimp
wenzelm@32960
   720
        qed
wenzelm@32960
   721
        ultimately show ?thesis
nipkow@28853
   722
          using fun_left_comm [of c b] x x' by (auto simp add: o_def)
nipkow@15392
   723
      qed
nipkow@15392
   724
    qed
nipkow@15392
   725
  qed
nipkow@28853
   726
qed
nipkow@28853
   727
nipkow@28853
   728
lemma fold_graph_determ:
nipkow@28853
   729
  "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
nipkow@28853
   730
apply (frule fold_graph_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) 
nipkow@28853
   731
apply (blast intro: fold_graph_determ_aux [rule_format])
nipkow@15392
   732
done
nipkow@15392
   733
nipkow@28853
   734
lemma fold_equality:
nipkow@28853
   735
  "fold_graph f z A y \<Longrightarrow> fold f z A = y"
nipkow@28853
   736
by (unfold fold_def) (blast intro: fold_graph_determ)
nipkow@15392
   737
nipkow@15392
   738
text{* The base case for @{text fold}: *}
nipkow@15392
   739
nipkow@28853
   740
lemma (in -) fold_empty [simp]: "fold f z {} = z"
nipkow@28853
   741
by (unfold fold_def) blast
nipkow@28853
   742
nipkow@28853
   743
text{* The various recursion equations for @{const fold}: *}
nipkow@28853
   744
nipkow@28853
   745
lemma fold_insert_aux: "x \<notin> A
nipkow@28853
   746
  \<Longrightarrow> fold_graph f z (insert x A) v \<longleftrightarrow>
nipkow@28853
   747
      (\<exists>y. fold_graph f z A y \<and> v = f x y)"
nipkow@28853
   748
apply auto
nipkow@28853
   749
apply (rule_tac A1 = A and f1 = f in finite_imp_fold_graph [THEN exE])
nipkow@28853
   750
 apply (fastsimp dest: fold_graph_imp_finite)
nipkow@28853
   751
apply (blast intro: fold_graph_determ)
nipkow@28853
   752
done
nipkow@15392
   753
haftmann@26041
   754
lemma fold_insert [simp]:
nipkow@28853
   755
  "finite A ==> x \<notin> A ==> fold f z (insert x A) = f x (fold f z A)"
nipkow@28853
   756
apply (simp add: fold_def fold_insert_aux)
nipkow@28853
   757
apply (rule the_equality)
nipkow@28853
   758
 apply (auto intro: finite_imp_fold_graph
nipkow@28853
   759
        cong add: conj_cong simp add: fold_def[symmetric] fold_equality)
nipkow@28853
   760
done
nipkow@28853
   761
nipkow@28853
   762
lemma fold_fun_comm:
nipkow@28853
   763
  "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
nipkow@28853
   764
proof (induct rule: finite_induct)
nipkow@28853
   765
  case empty then show ?case by simp
nipkow@28853
   766
next
nipkow@28853
   767
  case (insert y A) then show ?case
nipkow@28853
   768
    by (simp add: fun_left_comm[of x])
nipkow@28853
   769
qed
nipkow@28853
   770
nipkow@28853
   771
lemma fold_insert2:
nipkow@28853
   772
  "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
huffman@35216
   773
by (simp add: fold_fun_comm)
nipkow@15392
   774
haftmann@26041
   775
lemma fold_rec:
nipkow@28853
   776
assumes "finite A" and "x \<in> A"
nipkow@28853
   777
shows "fold f z A = f x (fold f z (A - {x}))"
nipkow@28853
   778
proof -
nipkow@28853
   779
  have A: "A = insert x (A - {x})" using `x \<in> A` by blast
nipkow@28853
   780
  then have "fold f z A = fold f z (insert x (A - {x}))" by simp
nipkow@28853
   781
  also have "\<dots> = f x (fold f z (A - {x}))"
nipkow@28853
   782
    by (rule fold_insert) (simp add: `finite A`)+
nipkow@15535
   783
  finally show ?thesis .
nipkow@15535
   784
qed
nipkow@15535
   785
nipkow@28853
   786
lemma fold_insert_remove:
nipkow@28853
   787
  assumes "finite A"
nipkow@28853
   788
  shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
nipkow@28853
   789
proof -
nipkow@28853
   790
  from `finite A` have "finite (insert x A)" by auto
nipkow@28853
   791
  moreover have "x \<in> insert x A" by auto
nipkow@28853
   792
  ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
nipkow@28853
   793
    by (rule fold_rec)
nipkow@28853
   794
  then show ?thesis by simp
nipkow@28853
   795
qed
nipkow@28853
   796
haftmann@26041
   797
end
nipkow@15392
   798
nipkow@15480
   799
text{* A simplified version for idempotent functions: *}
nipkow@15480
   800
nipkow@28853
   801
locale fun_left_comm_idem = fun_left_comm +
nipkow@28853
   802
  assumes fun_left_idem: "f x (f x z) = f x z"
haftmann@26041
   803
begin
haftmann@26041
   804
nipkow@28853
   805
text{* The nice version: *}
nipkow@28853
   806
lemma fun_comp_idem : "f x o f x = f x"
nipkow@28853
   807
by (simp add: fun_left_idem expand_fun_eq)
nipkow@28853
   808
haftmann@26041
   809
lemma fold_insert_idem:
nipkow@28853
   810
  assumes fin: "finite A"
nipkow@28853
   811
  shows "fold f z (insert x A) = f x (fold f z A)"
nipkow@15480
   812
proof cases
nipkow@28853
   813
  assume "x \<in> A"
nipkow@28853
   814
  then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
nipkow@28853
   815
  then show ?thesis using assms by (simp add:fun_left_idem)
nipkow@15480
   816
next
nipkow@28853
   817
  assume "x \<notin> A" then show ?thesis using assms by simp
nipkow@15480
   818
qed
nipkow@15480
   819
nipkow@28853
   820
declare fold_insert[simp del] fold_insert_idem[simp]
nipkow@28853
   821
nipkow@28853
   822
lemma fold_insert_idem2:
nipkow@28853
   823
  "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
nipkow@28853
   824
by(simp add:fold_fun_comm)
nipkow@15484
   825
haftmann@26041
   826
end
haftmann@26041
   827
haftmann@35817
   828
haftmann@35817
   829
subsubsection {* Expressing set operations via @{const fold} *}
haftmann@35817
   830
haftmann@35817
   831
lemma (in fun_left_comm) fun_left_comm_apply:
haftmann@35817
   832
  "fun_left_comm (\<lambda>x. f (g x))"
haftmann@35817
   833
proof
haftmann@35817
   834
qed (simp_all add: fun_left_comm)
haftmann@35817
   835
haftmann@35817
   836
lemma (in fun_left_comm_idem) fun_left_comm_idem_apply:
haftmann@35817
   837
  "fun_left_comm_idem (\<lambda>x. f (g x))"
haftmann@35817
   838
  by (rule fun_left_comm_idem.intro, rule fun_left_comm_apply, unfold_locales)
haftmann@35817
   839
    (simp_all add: fun_left_idem)
haftmann@35817
   840
haftmann@35817
   841
lemma fun_left_comm_idem_insert:
haftmann@35817
   842
  "fun_left_comm_idem insert"
haftmann@35817
   843
proof
haftmann@35817
   844
qed auto
haftmann@35817
   845
haftmann@35817
   846
lemma fun_left_comm_idem_remove:
haftmann@35817
   847
  "fun_left_comm_idem (\<lambda>x A. A - {x})"
haftmann@35817
   848
proof
haftmann@35817
   849
qed auto
nipkow@31992
   850
haftmann@35817
   851
lemma (in semilattice_inf) fun_left_comm_idem_inf:
haftmann@35817
   852
  "fun_left_comm_idem inf"
haftmann@35817
   853
proof
haftmann@35817
   854
qed (auto simp add: inf_left_commute)
haftmann@35817
   855
haftmann@35817
   856
lemma (in semilattice_sup) fun_left_comm_idem_sup:
haftmann@35817
   857
  "fun_left_comm_idem sup"
haftmann@35817
   858
proof
haftmann@35817
   859
qed (auto simp add: sup_left_commute)
nipkow@31992
   860
haftmann@35817
   861
lemma union_fold_insert:
haftmann@35817
   862
  assumes "finite A"
haftmann@35817
   863
  shows "A \<union> B = fold insert B A"
haftmann@35817
   864
proof -
haftmann@35817
   865
  interpret fun_left_comm_idem insert by (fact fun_left_comm_idem_insert)
haftmann@35817
   866
  from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
haftmann@35817
   867
qed
nipkow@31992
   868
haftmann@35817
   869
lemma minus_fold_remove:
haftmann@35817
   870
  assumes "finite A"
haftmann@35817
   871
  shows "B - A = fold (\<lambda>x A. A - {x}) B A"
haftmann@35817
   872
proof -
haftmann@35817
   873
  interpret fun_left_comm_idem "\<lambda>x A. A - {x}" by (fact fun_left_comm_idem_remove)
haftmann@35817
   874
  from `finite A` show ?thesis by (induct A arbitrary: B) auto
haftmann@35817
   875
qed
haftmann@35817
   876
haftmann@35817
   877
context complete_lattice
nipkow@31992
   878
begin
nipkow@31992
   879
haftmann@35817
   880
lemma inf_Inf_fold_inf:
haftmann@35817
   881
  assumes "finite A"
haftmann@35817
   882
  shows "inf B (Inf A) = fold inf B A"
haftmann@35817
   883
proof -
haftmann@35817
   884
  interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf)
haftmann@35817
   885
  from `finite A` show ?thesis by (induct A arbitrary: B)
haftmann@35817
   886
    (simp_all add: Inf_empty Inf_insert inf_commute fold_fun_comm)
haftmann@35817
   887
qed
nipkow@31992
   888
haftmann@35817
   889
lemma sup_Sup_fold_sup:
haftmann@35817
   890
  assumes "finite A"
haftmann@35817
   891
  shows "sup B (Sup A) = fold sup B A"
haftmann@35817
   892
proof -
haftmann@35817
   893
  interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)
haftmann@35817
   894
  from `finite A` show ?thesis by (induct A arbitrary: B)
haftmann@35817
   895
    (simp_all add: Sup_empty Sup_insert sup_commute fold_fun_comm)
nipkow@31992
   896
qed
nipkow@31992
   897
haftmann@35817
   898
lemma Inf_fold_inf:
haftmann@35817
   899
  assumes "finite A"
haftmann@35817
   900
  shows "Inf A = fold inf top A"
haftmann@35817
   901
  using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
haftmann@35817
   902
haftmann@35817
   903
lemma Sup_fold_sup:
haftmann@35817
   904
  assumes "finite A"
haftmann@35817
   905
  shows "Sup A = fold sup bot A"
haftmann@35817
   906
  using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
nipkow@31992
   907
haftmann@35817
   908
lemma inf_INFI_fold_inf:
haftmann@35817
   909
  assumes "finite A"
haftmann@35817
   910
  shows "inf B (INFI A f) = fold (\<lambda>A. inf (f A)) B A" (is "?inf = ?fold") 
haftmann@35817
   911
proof (rule sym)
haftmann@35817
   912
  interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf)
haftmann@35817
   913
  interpret fun_left_comm_idem "\<lambda>A. inf (f A)" by (fact fun_left_comm_idem_apply)
haftmann@35817
   914
  from `finite A` show "?fold = ?inf"
haftmann@35817
   915
  by (induct A arbitrary: B)
haftmann@35817
   916
    (simp_all add: INFI_def Inf_empty Inf_insert inf_left_commute)
haftmann@35817
   917
qed
nipkow@31992
   918
haftmann@35817
   919
lemma sup_SUPR_fold_sup:
haftmann@35817
   920
  assumes "finite A"
haftmann@35817
   921
  shows "sup B (SUPR A f) = fold (\<lambda>A. sup (f A)) B A" (is "?sup = ?fold") 
haftmann@35817
   922
proof (rule sym)
haftmann@35817
   923
  interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)
haftmann@35817
   924
  interpret fun_left_comm_idem "\<lambda>A. sup (f A)" by (fact fun_left_comm_idem_apply)
haftmann@35817
   925
  from `finite A` show "?fold = ?sup"
haftmann@35817
   926
  by (induct A arbitrary: B)
haftmann@35817
   927
    (simp_all add: SUPR_def Sup_empty Sup_insert sup_left_commute)
haftmann@35817
   928
qed
nipkow@31992
   929
haftmann@35817
   930
lemma INFI_fold_inf:
haftmann@35817
   931
  assumes "finite A"
haftmann@35817
   932
  shows "INFI A f = fold (\<lambda>A. inf (f A)) top A"
haftmann@35817
   933
  using assms inf_INFI_fold_inf [of A top] by simp
nipkow@31992
   934
haftmann@35817
   935
lemma SUPR_fold_sup:
haftmann@35817
   936
  assumes "finite A"
haftmann@35817
   937
  shows "SUPR A f = fold (\<lambda>A. sup (f A)) bot A"
haftmann@35817
   938
  using assms sup_SUPR_fold_sup [of A bot] by simp
nipkow@31992
   939
nipkow@31992
   940
end
nipkow@31992
   941
nipkow@31992
   942
haftmann@35817
   943
subsection {* The derived combinator @{text fold_image} *}
nipkow@28853
   944
nipkow@28853
   945
definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
nipkow@28853
   946
where "fold_image f g = fold (%x y. f (g x) y)"
nipkow@28853
   947
nipkow@28853
   948
lemma fold_image_empty[simp]: "fold_image f g z {} = z"
nipkow@28853
   949
by(simp add:fold_image_def)
nipkow@15392
   950
haftmann@26041
   951
context ab_semigroup_mult
haftmann@26041
   952
begin
haftmann@26041
   953
nipkow@28853
   954
lemma fold_image_insert[simp]:
nipkow@28853
   955
assumes "finite A" and "a \<notin> A"
nipkow@28853
   956
shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
nipkow@28853
   957
proof -
ballarin@29223
   958
  interpret I: fun_left_comm "%x y. (g x) * y"
nipkow@28853
   959
    by unfold_locales (simp add: mult_ac)
nipkow@31992
   960
  show ?thesis using assms by(simp add:fold_image_def)
nipkow@28853
   961
qed
nipkow@28853
   962
nipkow@28853
   963
(*
haftmann@26041
   964
lemma fold_commute:
haftmann@26041
   965
  "finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)"
berghofe@22262
   966
  apply (induct set: finite)
wenzelm@21575
   967
   apply simp
haftmann@26041
   968
  apply (simp add: mult_left_commute [of x])
nipkow@15392
   969
  done
nipkow@15392
   970
haftmann@26041
   971
lemma fold_nest_Un_Int:
nipkow@15392
   972
  "finite A ==> finite B
haftmann@26041
   973
    ==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)"
berghofe@22262
   974
  apply (induct set: finite)
wenzelm@21575
   975
   apply simp
nipkow@15392
   976
  apply (simp add: fold_commute Int_insert_left insert_absorb)
nipkow@15392
   977
  done
nipkow@15392
   978
haftmann@26041
   979
lemma fold_nest_Un_disjoint:
nipkow@15392
   980
  "finite A ==> finite B ==> A Int B = {}
haftmann@26041
   981
    ==> fold times g z (A Un B) = fold times g (fold times g z B) A"
nipkow@15392
   982
  by (simp add: fold_nest_Un_Int)
nipkow@28853
   983
*)
nipkow@28853
   984
nipkow@28853
   985
lemma fold_image_reindex:
paulson@15487
   986
assumes fin: "finite A"
nipkow@28853
   987
shows "inj_on h A \<Longrightarrow> fold_image times g z (h`A) = fold_image times (g\<circ>h) z A"
nipkow@31992
   988
using fin by induct auto
nipkow@15392
   989
nipkow@28853
   990
(*
haftmann@26041
   991
text{*
haftmann@26041
   992
  Fusion theorem, as described in Graham Hutton's paper,
haftmann@26041
   993
  A Tutorial on the Universality and Expressiveness of Fold,
haftmann@26041
   994
  JFP 9:4 (355-372), 1999.
haftmann@26041
   995
*}
haftmann@26041
   996
haftmann@26041
   997
lemma fold_fusion:
ballarin@27611
   998
  assumes "ab_semigroup_mult g"
haftmann@26041
   999
  assumes fin: "finite A"
haftmann@26041
  1000
    and hyp: "\<And>x y. h (g x y) = times x (h y)"
haftmann@26041
  1001
  shows "h (fold g j w A) = fold times j (h w) A"
ballarin@27611
  1002
proof -
ballarin@29223
  1003
  class_interpret ab_semigroup_mult [g] by fact
ballarin@27611
  1004
  show ?thesis using fin hyp by (induct set: finite) simp_all
ballarin@27611
  1005
qed
nipkow@28853
  1006
*)
nipkow@28853
  1007
nipkow@28853
  1008
lemma fold_image_cong:
nipkow@28853
  1009
  "finite A \<Longrightarrow>
nipkow@28853
  1010
  (!!x. x:A ==> g x = h x) ==> fold_image times g z A = fold_image times h z A"
nipkow@28853
  1011
apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C")
nipkow@28853
  1012
 apply simp
nipkow@28853
  1013
apply (erule finite_induct, simp)
nipkow@28853
  1014
apply (simp add: subset_insert_iff, clarify)
nipkow@28853
  1015
apply (subgoal_tac "finite C")
nipkow@28853
  1016
 prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
nipkow@28853
  1017
apply (subgoal_tac "C = insert x (C - {x})")
nipkow@28853
  1018
 prefer 2 apply blast
nipkow@28853
  1019
apply (erule ssubst)
nipkow@28853
  1020
apply (drule spec)
nipkow@28853
  1021
apply (erule (1) notE impE)
nipkow@28853
  1022
apply (simp add: Ball_def del: insert_Diff_single)
nipkow@28853
  1023
done
nipkow@15392
  1024
haftmann@26041
  1025
end
haftmann@26041
  1026
haftmann@26041
  1027
context comm_monoid_mult
haftmann@26041
  1028
begin
haftmann@26041
  1029
haftmann@35817
  1030
lemma fold_image_1:
haftmann@35817
  1031
  "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
haftmann@35817
  1032
  apply (induct set: finite)
haftmann@35817
  1033
  apply simp by auto
haftmann@35817
  1034
nipkow@28853
  1035
lemma fold_image_Un_Int:
haftmann@26041
  1036
  "finite A ==> finite B ==>
nipkow@28853
  1037
    fold_image times g 1 A * fold_image times g 1 B =
nipkow@28853
  1038
    fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
nipkow@28853
  1039
by (induct set: finite) 
nipkow@28853
  1040
   (auto simp add: mult_ac insert_absorb Int_insert_left)
haftmann@26041
  1041
haftmann@35817
  1042
lemma fold_image_Un_one:
haftmann@35817
  1043
  assumes fS: "finite S" and fT: "finite T"
haftmann@35817
  1044
  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
haftmann@35817
  1045
  shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
haftmann@35817
  1046
proof-
haftmann@35817
  1047
  have "fold_image op * f 1 (S \<inter> T) = 1" 
haftmann@35817
  1048
    apply (rule fold_image_1)
haftmann@35817
  1049
    using fS fT I0 by auto 
haftmann@35817
  1050
  with fold_image_Un_Int[OF fS fT] show ?thesis by simp
haftmann@35817
  1051
qed
haftmann@35817
  1052
haftmann@26041
  1053
corollary fold_Un_disjoint:
haftmann@26041
  1054
  "finite A ==> finite B ==> A Int B = {} ==>
nipkow@28853
  1055
   fold_image times g 1 (A Un B) =
nipkow@28853
  1056
   fold_image times g 1 A * fold_image times g 1 B"
nipkow@28853
  1057
by (simp add: fold_image_Un_Int)
nipkow@28853
  1058
nipkow@28853
  1059
lemma fold_image_UN_disjoint:
haftmann@26041
  1060
  "\<lbrakk> finite I; ALL i:I. finite (A i);
haftmann@26041
  1061
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
nipkow@28853
  1062
   \<Longrightarrow> fold_image times g 1 (UNION I A) =
nipkow@28853
  1063
       fold_image times (%i. fold_image times g 1 (A i)) 1 I"
nipkow@28853
  1064
apply (induct set: finite, simp, atomize)
nipkow@28853
  1065
apply (subgoal_tac "ALL i:F. x \<noteq> i")
nipkow@28853
  1066
 prefer 2 apply blast
nipkow@28853
  1067
apply (subgoal_tac "A x Int UNION F A = {}")
nipkow@28853
  1068
 prefer 2 apply blast
nipkow@28853
  1069
apply (simp add: fold_Un_disjoint)
nipkow@28853
  1070
done
nipkow@28853
  1071
nipkow@28853
  1072
lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@28853
  1073
  fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
nipkow@28853
  1074
  fold_image times (split g) 1 (SIGMA x:A. B x)"
nipkow@15392
  1075
apply (subst Sigma_def)
nipkow@28853
  1076
apply (subst fold_image_UN_disjoint, assumption, simp)
nipkow@15392
  1077
 apply blast
nipkow@28853
  1078
apply (erule fold_image_cong)
nipkow@28853
  1079
apply (subst fold_image_UN_disjoint, simp, simp)
nipkow@15392
  1080
 apply blast
paulson@15506
  1081
apply simp
nipkow@15392
  1082
done
nipkow@15392
  1083
nipkow@28853
  1084
lemma fold_image_distrib: "finite A \<Longrightarrow>
nipkow@28853
  1085
   fold_image times (%x. g x * h x) 1 A =
nipkow@28853
  1086
   fold_image times g 1 A *  fold_image times h 1 A"
nipkow@28853
  1087
by (erule finite_induct) (simp_all add: mult_ac)
haftmann@26041
  1088
chaieb@30260
  1089
lemma fold_image_related: 
chaieb@30260
  1090
  assumes Re: "R e e" 
chaieb@30260
  1091
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
chaieb@30260
  1092
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
chaieb@30260
  1093
  shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
chaieb@30260
  1094
  using fS by (rule finite_subset_induct) (insert assms, auto)
chaieb@30260
  1095
chaieb@30260
  1096
lemma  fold_image_eq_general:
chaieb@30260
  1097
  assumes fS: "finite S"
chaieb@30260
  1098
  and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 
chaieb@30260
  1099
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
chaieb@30260
  1100
  shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
chaieb@30260
  1101
proof-
chaieb@30260
  1102
  from h f12 have hS: "h ` S = S'" by auto
chaieb@30260
  1103
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
chaieb@30260
  1104
    from f12 h H  have "x = y" by auto }
chaieb@30260
  1105
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
chaieb@30260
  1106
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
chaieb@30260
  1107
  from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
chaieb@30260
  1108
  also have "\<dots> = fold_image (op *) (f2 o h) e S" 
chaieb@30260
  1109
    using fold_image_reindex[OF fS hinj, of f2 e] .
chaieb@30260
  1110
  also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
chaieb@30260
  1111
    by blast
chaieb@30260
  1112
  finally show ?thesis ..
chaieb@30260
  1113
qed
chaieb@30260
  1114
chaieb@30260
  1115
lemma fold_image_eq_general_inverses:
chaieb@30260
  1116
  assumes fS: "finite S" 
chaieb@30260
  1117
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
chaieb@30260
  1118
  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
  1119
  shows "fold_image (op *) f e S = fold_image (op *) g e T"
chaieb@30260
  1120
  (* metis solves it, but not yet available here *)
chaieb@30260
  1121
  apply (rule fold_image_eq_general[OF fS, of T h g f e])
chaieb@30260
  1122
  apply (rule ballI)
chaieb@30260
  1123
  apply (frule kh)
chaieb@30260
  1124
  apply (rule ex1I[])
chaieb@30260
  1125
  apply blast
chaieb@30260
  1126
  apply clarsimp
chaieb@30260
  1127
  apply (drule hk) apply simp
chaieb@30260
  1128
  apply (rule sym)
chaieb@30260
  1129
  apply (erule conjunct1[OF conjunct2[OF hk]])
chaieb@30260
  1130
  apply (rule ballI)
chaieb@30260
  1131
  apply (drule  hk)
chaieb@30260
  1132
  apply blast
chaieb@30260
  1133
  done
chaieb@30260
  1134
haftmann@26041
  1135
end
haftmann@22917
  1136
nipkow@25162
  1137
haftmann@35817
  1138
subsection {* A fold functional for non-empty sets *}
nipkow@15392
  1139
nipkow@15392
  1140
text{* Does not require start value. *}
wenzelm@12396
  1141
berghofe@23736
  1142
inductive
berghofe@22262
  1143
  fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
berghofe@22262
  1144
  for f :: "'a => 'a => 'a"
berghofe@22262
  1145
where
paulson@15506
  1146
  fold1Set_insertI [intro]:
nipkow@28853
  1147
   "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
wenzelm@12396
  1148
haftmann@35416
  1149
definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
berghofe@22262
  1150
  "fold1 f A == THE x. fold1Set f A x"
paulson@15506
  1151
paulson@15506
  1152
lemma fold1Set_nonempty:
haftmann@22917
  1153
  "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
nipkow@28853
  1154
by(erule fold1Set.cases, simp_all)
nipkow@15392
  1155
berghofe@23736
  1156
inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
berghofe@23736
  1157
berghofe@23736
  1158
inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
berghofe@22262
  1159
berghofe@22262
  1160
berghofe@22262
  1161
lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
huffman@35216
  1162
by (blast elim: fold_graph.cases)
nipkow@15392
  1163
haftmann@22917
  1164
lemma fold1_singleton [simp]: "fold1 f {a} = a"
nipkow@28853
  1165
by (unfold fold1_def) blast
wenzelm@12396
  1166
paulson@15508
  1167
lemma finite_nonempty_imp_fold1Set:
berghofe@22262
  1168
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
paulson@15508
  1169
apply (induct A rule: finite_induct)
nipkow@28853
  1170
apply (auto dest: finite_imp_fold_graph [of _ f])
paulson@15508
  1171
done
paulson@15506
  1172
nipkow@28853
  1173
text{*First, some lemmas about @{const fold_graph}.*}
nipkow@15392
  1174
haftmann@26041
  1175
context ab_semigroup_mult
haftmann@26041
  1176
begin
haftmann@26041
  1177
nipkow@28853
  1178
lemma fun_left_comm: "fun_left_comm(op *)"
nipkow@28853
  1179
by unfold_locales (simp add: mult_ac)
nipkow@28853
  1180
nipkow@28853
  1181
lemma fold_graph_insert_swap:
nipkow@28853
  1182
assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
nipkow@28853
  1183
shows "fold_graph times z (insert b A) (z * y)"
nipkow@28853
  1184
proof -
ballarin@29223
  1185
  interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
nipkow@28853
  1186
from assms show ?thesis
nipkow@28853
  1187
proof (induct rule: fold_graph.induct)
haftmann@26041
  1188
  case emptyI thus ?case by (force simp add: fold_insert_aux mult_commute)
paulson@15508
  1189
next
berghofe@22262
  1190
  case (insertI x A y)
nipkow@28853
  1191
    have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
paulson@15521
  1192
      using insertI by force  --{*how does @{term id} get unfolded?*}
haftmann@26041
  1193
    thus ?case by (simp add: insert_commute mult_ac)
paulson@15508
  1194
qed
nipkow@28853
  1195
qed
nipkow@28853
  1196
nipkow@28853
  1197
lemma fold_graph_permute_diff:
nipkow@28853
  1198
assumes fold: "fold_graph times b A x"
nipkow@28853
  1199
shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
paulson@15508
  1200
using fold
nipkow@28853
  1201
proof (induct rule: fold_graph.induct)
paulson@15508
  1202
  case emptyI thus ?case by simp
paulson@15508
  1203
next
berghofe@22262
  1204
  case (insertI x A y)
paulson@15521
  1205
  have "a = x \<or> a \<in> A" using insertI by simp
paulson@15521
  1206
  thus ?case
paulson@15521
  1207
  proof
paulson@15521
  1208
    assume "a = x"
paulson@15521
  1209
    with insertI show ?thesis
nipkow@28853
  1210
      by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
paulson@15521
  1211
  next
paulson@15521
  1212
    assume ainA: "a \<in> A"
nipkow@28853
  1213
    hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
nipkow@28853
  1214
      using insertI by force
paulson@15521
  1215
    moreover
paulson@15521
  1216
    have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
paulson@15521
  1217
      using ainA insertI by blast
nipkow@28853
  1218
    ultimately show ?thesis by simp
paulson@15508
  1219
  qed
paulson@15508
  1220
qed
paulson@15508
  1221
haftmann@26041
  1222
lemma fold1_eq_fold:
nipkow@28853
  1223
assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
nipkow@28853
  1224
proof -
ballarin@29223
  1225
  interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
nipkow@28853
  1226
  from assms show ?thesis
nipkow@28853
  1227
apply (simp add: fold1_def fold_def)
paulson@15508
  1228
apply (rule the_equality)
nipkow@28853
  1229
apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
paulson@15508
  1230
apply (rule sym, clarify)
paulson@15508
  1231
apply (case_tac "Aa=A")
huffman@35216
  1232
 apply (best intro: fold_graph_determ)
nipkow@28853
  1233
apply (subgoal_tac "fold_graph times a A x")
huffman@35216
  1234
 apply (best intro: fold_graph_determ)
nipkow@28853
  1235
apply (subgoal_tac "insert aa (Aa - {a}) = A")
nipkow@28853
  1236
 prefer 2 apply (blast elim: equalityE)
nipkow@28853
  1237
apply (auto dest: fold_graph_permute_diff [where a=a])
paulson@15508
  1238
done
nipkow@28853
  1239
qed
paulson@15508
  1240
paulson@15521
  1241
lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
paulson@15521
  1242
apply safe
nipkow@28853
  1243
 apply simp
nipkow@28853
  1244
 apply (drule_tac x=x in spec)
nipkow@28853
  1245
 apply (drule_tac x="A-{x}" in spec, auto)
paulson@15508
  1246
done
paulson@15508
  1247
haftmann@26041
  1248
lemma fold1_insert:
paulson@15521
  1249
  assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
haftmann@26041
  1250
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1251
proof -
ballarin@29223
  1252
  interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
nipkow@28853
  1253
  from nonempty obtain a A' where "A = insert a A' & a ~: A'"
paulson@15521
  1254
    by (auto simp add: nonempty_iff)
paulson@15521
  1255
  with A show ?thesis
nipkow@28853
  1256
    by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
paulson@15521
  1257
qed
paulson@15521
  1258
haftmann@26041
  1259
end
haftmann@26041
  1260
haftmann@26041
  1261
context ab_semigroup_idem_mult
haftmann@26041
  1262
begin
haftmann@26041
  1263
haftmann@35817
  1264
lemma fun_left_comm_idem: "fun_left_comm_idem(op *)"
haftmann@35817
  1265
apply unfold_locales
haftmann@35817
  1266
 apply (rule mult_left_commute)
haftmann@35817
  1267
apply (rule mult_left_idem)
haftmann@35817
  1268
done
haftmann@35817
  1269
haftmann@26041
  1270
lemma fold1_insert_idem [simp]:
paulson@15521
  1271
  assumes nonempty: "A \<noteq> {}" and A: "finite A" 
haftmann@26041
  1272
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1273
proof -
ballarin@29223
  1274
  interpret fun_left_comm_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
nipkow@28853
  1275
    by (rule fun_left_comm_idem)
nipkow@28853
  1276
  from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
paulson@15521
  1277
    by (auto simp add: nonempty_iff)
paulson@15521
  1278
  show ?thesis
paulson@15521
  1279
  proof cases
paulson@15521
  1280
    assume "a = x"
nipkow@28853
  1281
    thus ?thesis
paulson@15521
  1282
    proof cases
paulson@15521
  1283
      assume "A' = {}"
huffman@35216
  1284
      with prems show ?thesis by simp
paulson@15521
  1285
    next
paulson@15521
  1286
      assume "A' \<noteq> {}"
paulson@15521
  1287
      with prems show ?thesis
huffman@35216
  1288
        by (simp add: fold1_insert mult_assoc [symmetric])
paulson@15521
  1289
    qed
paulson@15521
  1290
  next
paulson@15521
  1291
    assume "a \<noteq> x"
paulson@15521
  1292
    with prems show ?thesis
huffman@35216
  1293
      by (simp add: insert_commute fold1_eq_fold)
paulson@15521
  1294
  qed
paulson@15521
  1295
qed
paulson@15506
  1296
haftmann@26041
  1297
lemma hom_fold1_commute:
haftmann@26041
  1298
assumes hom: "!!x y. h (x * y) = h x * h y"
haftmann@26041
  1299
and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
haftmann@22917
  1300
using N proof (induct rule: finite_ne_induct)
haftmann@22917
  1301
  case singleton thus ?case by simp
haftmann@22917
  1302
next
haftmann@22917
  1303
  case (insert n N)
haftmann@26041
  1304
  then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
haftmann@26041
  1305
  also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
haftmann@26041
  1306
  also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
haftmann@26041
  1307
  also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
haftmann@22917
  1308
    using insert by(simp)
haftmann@22917
  1309
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@22917
  1310
  finally show ?case .
haftmann@22917
  1311
qed
haftmann@22917
  1312
haftmann@32679
  1313
lemma fold1_eq_fold_idem:
haftmann@32679
  1314
  assumes "finite A"
haftmann@32679
  1315
  shows "fold1 times (insert a A) = fold times a A"
haftmann@32679
  1316
proof (cases "a \<in> A")
haftmann@32679
  1317
  case False
haftmann@32679
  1318
  with assms show ?thesis by (simp add: fold1_eq_fold)
haftmann@32679
  1319
next
haftmann@32679
  1320
  interpret fun_left_comm_idem times by (fact fun_left_comm_idem)
haftmann@32679
  1321
  case True then obtain b B
haftmann@32679
  1322
    where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
haftmann@32679
  1323
  with assms have "finite B" by auto
haftmann@32679
  1324
  then have "fold times a (insert a B) = fold times (a * a) B"
haftmann@32679
  1325
    using `a \<notin> B` by (rule fold_insert2)
haftmann@32679
  1326
  then show ?thesis
haftmann@32679
  1327
    using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
haftmann@32679
  1328
qed
haftmann@32679
  1329
haftmann@26041
  1330
end
haftmann@26041
  1331
paulson@15506
  1332
paulson@15508
  1333
text{* Now the recursion rules for definitions: *}
paulson@15508
  1334
haftmann@22917
  1335
lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
huffman@35216
  1336
by simp
paulson@15508
  1337
haftmann@26041
  1338
lemma (in ab_semigroup_mult) fold1_insert_def:
haftmann@26041
  1339
  "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1340
by (simp add:fold1_insert)
haftmann@26041
  1341
haftmann@26041
  1342
lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
haftmann@26041
  1343
  "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1344
by simp
paulson@15508
  1345
paulson@15508
  1346
subsubsection{* Determinacy for @{term fold1Set} *}
paulson@15508
  1347
nipkow@28853
  1348
(*Not actually used!!*)
nipkow@28853
  1349
(*
haftmann@26041
  1350
context ab_semigroup_mult
haftmann@26041
  1351
begin
haftmann@26041
  1352
nipkow@28853
  1353
lemma fold_graph_permute:
nipkow@28853
  1354
  "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
nipkow@28853
  1355
   ==> fold_graph times id a (insert b A) x"
haftmann@26041
  1356
apply (cases "a=b") 
nipkow@28853
  1357
apply (auto dest: fold_graph_permute_diff) 
paulson@15506
  1358
done
nipkow@15376
  1359
haftmann@26041
  1360
lemma fold1Set_determ:
haftmann@26041
  1361
  "fold1Set times A x ==> fold1Set times A y ==> y = x"
paulson@15506
  1362
proof (clarify elim!: fold1Set.cases)
paulson@15506
  1363
  fix A x B y a b
nipkow@28853
  1364
  assume Ax: "fold_graph times id a A x"
nipkow@28853
  1365
  assume By: "fold_graph times id b B y"
paulson@15506
  1366
  assume anotA:  "a \<notin> A"
paulson@15506
  1367
  assume bnotB:  "b \<notin> B"
paulson@15506
  1368
  assume eq: "insert a A = insert b B"
paulson@15506
  1369
  show "y=x"
paulson@15506
  1370
  proof cases
paulson@15506
  1371
    assume same: "a=b"
paulson@15506
  1372
    hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
nipkow@28853
  1373
    thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
nipkow@15392
  1374
  next
paulson@15506
  1375
    assume diff: "a\<noteq>b"
paulson@15506
  1376
    let ?D = "B - {a}"
paulson@15506
  1377
    have B: "B = insert a ?D" and A: "A = insert b ?D"
paulson@15506
  1378
     and aB: "a \<in> B" and bA: "b \<in> A"
paulson@15506
  1379
      using eq anotA bnotB diff by (blast elim!:equalityE)+
paulson@15506
  1380
    with aB bnotB By
nipkow@28853
  1381
    have "fold_graph times id a (insert b ?D) y" 
nipkow@28853
  1382
      by (auto intro: fold_graph_permute simp add: insert_absorb)
paulson@15506
  1383
    moreover
nipkow@28853
  1384
    have "fold_graph times id a (insert b ?D) x"
paulson@15506
  1385
      by (simp add: A [symmetric] Ax) 
nipkow@28853
  1386
    ultimately show ?thesis by (blast intro: fold_graph_determ) 
nipkow@15392
  1387
  qed
wenzelm@12396
  1388
qed
wenzelm@12396
  1389
haftmann@26041
  1390
lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
paulson@15506
  1391
  by (unfold fold1_def) (blast intro: fold1Set_determ)
paulson@15506
  1392
haftmann@26041
  1393
end
nipkow@28853
  1394
*)
haftmann@26041
  1395
paulson@15506
  1396
declare
nipkow@28853
  1397
  empty_fold_graphE [rule del]  fold_graph.intros [rule del]
paulson@15506
  1398
  empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
ballarin@19931
  1399
  -- {* No more proofs involve these relations. *}
nipkow@15376
  1400
haftmann@26041
  1401
subsubsection {* Lemmas about @{text fold1} *}
haftmann@26041
  1402
haftmann@26041
  1403
context ab_semigroup_mult
haftmann@22917
  1404
begin
haftmann@22917
  1405
haftmann@26041
  1406
lemma fold1_Un:
nipkow@15484
  1407
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  1408
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
haftmann@26041
  1409
       fold1 times (A Un B) = fold1 times A * fold1 times B"
haftmann@26041
  1410
using A by (induct rule: finite_ne_induct)
haftmann@26041
  1411
  (simp_all add: fold1_insert mult_assoc)
haftmann@26041
  1412
haftmann@26041
  1413
lemma fold1_in:
haftmann@26041
  1414
  assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
haftmann@26041
  1415
  shows "fold1 times A \<in> A"
nipkow@15484
  1416
using A
nipkow@15484
  1417
proof (induct rule:finite_ne_induct)
paulson@15506
  1418
  case singleton thus ?case by simp
nipkow@15484
  1419
next
nipkow@15484
  1420
  case insert thus ?case using elem by (force simp add:fold1_insert)
nipkow@15484
  1421
qed
nipkow@15484
  1422
haftmann@26041
  1423
end
haftmann@26041
  1424
haftmann@26041
  1425
lemma (in ab_semigroup_idem_mult) fold1_Un2:
nipkow@15497
  1426
assumes A: "finite A" "A \<noteq> {}"
haftmann@26041
  1427
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
haftmann@26041
  1428
       fold1 times (A Un B) = fold1 times A * fold1 times B"
nipkow@15497
  1429
using A
haftmann@26041
  1430
proof(induct rule:finite_ne_induct)
nipkow@15497
  1431
  case singleton thus ?case by simp
nipkow@15484
  1432
next
haftmann@26041
  1433
  case insert thus ?case by (simp add: mult_assoc)
nipkow@18423
  1434
qed
nipkow@18423
  1435
nipkow@18423
  1436
haftmann@35817
  1437
subsection {* Locales as mini-packages for fold operations *}
haftmann@34007
  1438
haftmann@35817
  1439
subsubsection {* The natural case *}
haftmann@35719
  1440
haftmann@35719
  1441
locale folding =
haftmann@35719
  1442
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@35719
  1443
  fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@35817
  1444
  assumes commute_comp: "f y \<circ> f x = f x \<circ> f y"
haftmann@35722
  1445
  assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
haftmann@35719
  1446
begin
haftmann@35719
  1447
haftmann@35719
  1448
lemma empty [simp]:
haftmann@35719
  1449
  "F {} = id"
haftmann@35719
  1450
  by (simp add: eq_fold expand_fun_eq)
haftmann@35719
  1451
haftmann@35719
  1452
lemma insert [simp]:
haftmann@35719
  1453
  assumes "finite A" and "x \<notin> A"
haftmann@35719
  1454
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35719
  1455
proof -
haftmann@35817
  1456
  interpret fun_left_comm f proof
haftmann@35817
  1457
  qed (insert commute_comp, simp add: expand_fun_eq)
haftmann@35719
  1458
  from fold_insert2 assms
haftmann@35722
  1459
  have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
haftmann@35722
  1460
  with `finite A` show ?thesis by (simp add: eq_fold expand_fun_eq)
haftmann@35719
  1461
qed
haftmann@35719
  1462
haftmann@35719
  1463
lemma remove:
haftmann@35719
  1464
  assumes "finite A" and "x \<in> A"
haftmann@35719
  1465
  shows "F A = F (A - {x}) \<circ> f x"
haftmann@35719
  1466
proof -
haftmann@35719
  1467
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35719
  1468
    by (auto dest: mk_disjoint_insert)
haftmann@35719
  1469
  moreover from `finite A` this have "finite B" by simp
haftmann@35719
  1470
  ultimately show ?thesis by simp
haftmann@35719
  1471
qed
haftmann@35719
  1472
haftmann@35719
  1473
lemma insert_remove:
haftmann@35719
  1474
  assumes "finite A"
haftmann@35719
  1475
  shows "F (insert x A) = F (A - {x}) \<circ> f x"
haftmann@35722
  1476
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35719
  1477
haftmann@35817
  1478
lemma commute_left_comp:
haftmann@35817
  1479
  "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
haftmann@35817
  1480
  by (simp add: o_assoc commute_comp)
haftmann@35817
  1481
haftmann@35719
  1482
lemma commute_comp':
haftmann@35719
  1483
  assumes "finite A"
haftmann@35719
  1484
  shows "f x \<circ> F A = F A \<circ> f x"
haftmann@35817
  1485
  using assms by (induct A)
haftmann@35817
  1486
    (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: o_assoc [symmetric] commute_comp)
haftmann@35817
  1487
haftmann@35817
  1488
lemma commute_left_comp':
haftmann@35817
  1489
  assumes "finite A"
haftmann@35817
  1490
  shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
haftmann@35817
  1491
  using assms by (simp add: o_assoc commute_comp')
haftmann@35817
  1492
haftmann@35817
  1493
lemma commute_comp'':
haftmann@35817
  1494
  assumes "finite A" and "finite B"
haftmann@35817
  1495
  shows "F B \<circ> F A = F A \<circ> F B"
haftmann@35817
  1496
  using assms by (induct A)
haftmann@35817
  1497
    (simp_all add: o_assoc, simp add: o_assoc [symmetric] commute_comp')
haftmann@35719
  1498
haftmann@35817
  1499
lemma commute_left_comp'':
haftmann@35817
  1500
  assumes "finite A" and "finite B"
haftmann@35817
  1501
  shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
haftmann@35817
  1502
  using assms by (simp add: o_assoc commute_comp'')
haftmann@35817
  1503
haftmann@35817
  1504
lemmas commute_comps = o_assoc [symmetric] commute_comp commute_left_comp
haftmann@35817
  1505
  commute_comp' commute_left_comp' commute_comp'' commute_left_comp''
haftmann@35817
  1506
haftmann@35817
  1507
lemma union_inter:
haftmann@35817
  1508
  assumes "finite A" and "finite B"
haftmann@35817
  1509
  shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
haftmann@35817
  1510
  using assms by (induct A)
haftmann@35817
  1511
    (simp_all del: o_apply add: insert_absorb Int_insert_left commute_comps,
haftmann@35817
  1512
      simp add: o_assoc)
haftmann@35719
  1513
haftmann@35719
  1514
lemma union:
haftmann@35719
  1515
  assumes "finite A" and "finite B"
haftmann@35719
  1516
  and "A \<inter> B = {}"
haftmann@35719
  1517
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1518
proof -
haftmann@35817
  1519
  from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
haftmann@35817
  1520
  with `A \<inter> B = {}` show ?thesis by simp
haftmann@35719
  1521
qed
haftmann@35719
  1522
haftmann@34007
  1523
end
haftmann@35719
  1524
haftmann@35817
  1525
haftmann@35817
  1526
subsubsection {* The natural case with idempotency *}
haftmann@35817
  1527
haftmann@35719
  1528
locale folding_idem = folding +
haftmann@35719
  1529
  assumes idem_comp: "f x \<circ> f x = f x"
haftmann@35719
  1530
begin
haftmann@35719
  1531
haftmann@35817
  1532
lemma idem_left_comp:
haftmann@35817
  1533
  "f x \<circ> (f x \<circ> g) = f x \<circ> g"
haftmann@35817
  1534
  by (simp add: o_assoc idem_comp)
haftmann@35817
  1535
haftmann@35817
  1536
lemma in_comp_idem:
haftmann@35817
  1537
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1538
  shows "F A \<circ> f x = F A"
haftmann@35817
  1539
using assms by (induct A)
haftmann@35817
  1540
  (auto simp add: commute_comps idem_comp, simp add: commute_left_comp' [symmetric] commute_comp')
haftmann@35719
  1541
haftmann@35817
  1542
lemma subset_comp_idem:
haftmann@35817
  1543
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1544
  shows "F A \<circ> F B = F A"
haftmann@35817
  1545
proof -
haftmann@35817
  1546
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1547
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1548
    (simp_all add: o_assoc in_comp_idem `finite A`)
haftmann@35817
  1549
qed
haftmann@35719
  1550
haftmann@35817
  1551
declare insert [simp del]
haftmann@35719
  1552
haftmann@35719
  1553
lemma insert_idem [simp]:
haftmann@35719
  1554
  assumes "finite A"
haftmann@35719
  1555
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35817
  1556
  using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
haftmann@35719
  1557
haftmann@35719
  1558
lemma union_idem:
haftmann@35719
  1559
  assumes "finite A" and "finite B"
haftmann@35719
  1560
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1561
proof -
haftmann@35817
  1562
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1563
  then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
haftmann@35817
  1564
  with assms show ?thesis by (simp add: union_inter)
haftmann@35719
  1565
qed
haftmann@35719
  1566
haftmann@35719
  1567
end
haftmann@35719
  1568
haftmann@35817
  1569
haftmann@35817
  1570
subsubsection {* The image case with fixed function *}
haftmann@35817
  1571
haftmann@35796
  1572
no_notation times (infixl "*" 70)
haftmann@35796
  1573
no_notation Groups.one ("1")
haftmann@35722
  1574
haftmann@35722
  1575
locale folding_image_simple = comm_monoid +
haftmann@35722
  1576
  fixes g :: "('b \<Rightarrow> 'a)"
haftmann@35722
  1577
  fixes F :: "'b set \<Rightarrow> 'a"
haftmann@35817
  1578
  assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
haftmann@35722
  1579
begin
haftmann@35722
  1580
haftmann@35722
  1581
lemma empty [simp]:
haftmann@35722
  1582
  "F {} = 1"
haftmann@35817
  1583
  by (simp add: eq_fold_g)
haftmann@35722
  1584
haftmann@35722
  1585
lemma insert [simp]:
haftmann@35722
  1586
  assumes "finite A" and "x \<notin> A"
haftmann@35722
  1587
  shows "F (insert x A) = g x * F A"
haftmann@35722
  1588
proof -
haftmann@35722
  1589
  interpret fun_left_comm "%x y. (g x) * y" proof
haftmann@35722
  1590
  qed (simp add: ac_simps)
haftmann@35722
  1591
  with assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
haftmann@35722
  1592
    by (simp add: fold_image_def)
haftmann@35817
  1593
  with `finite A` show ?thesis by (simp add: eq_fold_g)
haftmann@35722
  1594
qed
haftmann@35722
  1595
haftmann@35722
  1596
lemma remove:
haftmann@35722
  1597
  assumes "finite A" and "x \<in> A"
haftmann@35722
  1598
  shows "F A = g x * F (A - {x})"
haftmann@35722
  1599
proof -
haftmann@35722
  1600
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35722
  1601
    by (auto dest: mk_disjoint_insert)
haftmann@35722
  1602
  moreover from `finite A` this have "finite B" by simp
haftmann@35722
  1603
  ultimately show ?thesis by simp
haftmann@35722
  1604
qed
haftmann@35722
  1605
haftmann@35722
  1606
lemma insert_remove:
haftmann@35722
  1607
  assumes "finite A"
haftmann@35722
  1608
  shows "F (insert x A) = g x * F (A - {x})"
haftmann@35722
  1609
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35722
  1610
haftmann@35817
  1611
lemma neutral:
haftmann@35817
  1612
  assumes "finite A" and "\<forall>x\<in>A. g x = 1"
haftmann@35817
  1613
  shows "F A = 1"
haftmann@35817
  1614
  using assms by (induct A) simp_all
haftmann@35817
  1615
haftmann@35722
  1616
lemma union_inter:
haftmann@35722
  1617
  assumes "finite A" and "finite B"
haftmann@35817
  1618
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35722
  1619
using assms proof (induct A)
haftmann@35722
  1620
  case empty then show ?case by simp
haftmann@35722
  1621
next
haftmann@35722
  1622
  case (insert x A) then show ?case
haftmann@35722
  1623
    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
haftmann@35722
  1624
qed
haftmann@35722
  1625
haftmann@35817
  1626
corollary union_inter_neutral:
haftmann@35817
  1627
  assumes "finite A" and "finite B"
haftmann@35817
  1628
  and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
haftmann@35817
  1629
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1630
  using assms by (simp add: union_inter [symmetric] neutral)
haftmann@35817
  1631
haftmann@35722
  1632
corollary union_disjoint:
haftmann@35722
  1633
  assumes "finite A" and "finite B"
haftmann@35722
  1634
  assumes "A \<inter> B = {}"
haftmann@35722
  1635
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1636
  using assms by (simp add: union_inter_neutral)
haftmann@35722
  1637
haftmann@35719
  1638
end
haftmann@35722
  1639
haftmann@35817
  1640
haftmann@35817
  1641
subsubsection {* The image case with flexible function *}
haftmann@35817
  1642
haftmann@35722
  1643
locale folding_image = comm_monoid +
haftmann@35722
  1644
  fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@35722
  1645
  assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
haftmann@35722
  1646
haftmann@35722
  1647
sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
haftmann@35722
  1648
qed (fact eq_fold)
haftmann@35722
  1649
haftmann@35722
  1650
context folding_image
haftmann@35722
  1651
begin
haftmann@35722
  1652
haftmann@35817
  1653
lemma reindex: (* FIXME polymorhism *)
haftmann@35722
  1654
  assumes "finite A" and "inj_on h A"
haftmann@35722
  1655
  shows "F g (h ` A) = F (g \<circ> h) A"
haftmann@35722
  1656
  using assms by (induct A) auto
haftmann@35722
  1657
haftmann@35722
  1658
lemma cong:
haftmann@35722
  1659
  assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
haftmann@35722
  1660
  shows "F g A = F h A"
haftmann@35722
  1661
proof -
haftmann@35722
  1662
  from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
haftmann@35722
  1663
  apply - apply (erule finite_induct) apply simp
haftmann@35722
  1664
  apply (simp add: subset_insert_iff, clarify)
haftmann@35722
  1665
  apply (subgoal_tac "finite C")
haftmann@35722
  1666
  prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
haftmann@35722
  1667
  apply (subgoal_tac "C = insert x (C - {x})")
haftmann@35722
  1668
  prefer 2 apply blast
haftmann@35722
  1669
  apply (erule ssubst)
haftmann@35722
  1670
  apply (drule spec)
haftmann@35722
  1671
  apply (erule (1) notE impE)
haftmann@35722
  1672
  apply (simp add: Ball_def del: insert_Diff_single)
haftmann@35722
  1673
  done
haftmann@35722
  1674
  with assms show ?thesis by simp
haftmann@35722
  1675
qed
haftmann@35722
  1676
haftmann@35722
  1677
lemma UNION_disjoint:
haftmann@35722
  1678
  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
haftmann@35722
  1679
  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
haftmann@35722
  1680
  shows "F g (UNION I A) = F (F g \<circ> A) I"
haftmann@35722
  1681
apply (insert assms)
haftmann@35722
  1682
apply (induct set: finite, simp, atomize)
haftmann@35722
  1683
apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
haftmann@35722
  1684
 prefer 2 apply blast
haftmann@35722
  1685
apply (subgoal_tac "A x Int UNION Fa A = {}")
haftmann@35722
  1686
 prefer 2 apply blast
haftmann@35722
  1687
apply (simp add: union_disjoint)
haftmann@35722
  1688
done
haftmann@35722
  1689
haftmann@35722
  1690
lemma distrib:
haftmann@35722
  1691
  assumes "finite A"
haftmann@35722
  1692
  shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
haftmann@35722
  1693
  using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
haftmann@35722
  1694
haftmann@35722
  1695
lemma related: 
haftmann@35722
  1696
  assumes Re: "R 1 1" 
haftmann@35722
  1697
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
haftmann@35722
  1698
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
haftmann@35722
  1699
  shows "R (F h S) (F g S)"
haftmann@35722
  1700
  using fS by (rule finite_subset_induct) (insert assms, auto)
haftmann@35722
  1701
haftmann@35722
  1702
lemma eq_general:
haftmann@35722
  1703
  assumes fS: "finite S"
haftmann@35722
  1704
  and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 
haftmann@35722
  1705
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
haftmann@35722
  1706
  shows "F f1 S = F f2 S'"
haftmann@35722
  1707
proof-
haftmann@35722
  1708
  from h f12 have hS: "h ` S = S'" by blast
haftmann@35722
  1709
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
haftmann@35722
  1710
    from f12 h H  have "x = y" by auto }
haftmann@35722
  1711
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
haftmann@35722
  1712
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
haftmann@35722
  1713
  from hS have "F f2 S' = F f2 (h ` S)" by simp
haftmann@35722
  1714
  also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
haftmann@35722
  1715
  also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
haftmann@35722
  1716
    by blast
haftmann@35722
  1717
  finally show ?thesis ..
haftmann@35722
  1718
qed
haftmann@35722
  1719
haftmann@35722
  1720
lemma eq_general_inverses:
haftmann@35722
  1721
  assumes fS: "finite S" 
haftmann@35722
  1722
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
haftmann@35722
  1723
  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
  1724
  shows "F j S = F g T"
haftmann@35722
  1725
  (* metis solves it, but not yet available here *)
haftmann@35722
  1726
  apply (rule eq_general [OF fS, of T h g j])
haftmann@35722
  1727
  apply (rule ballI)
haftmann@35722
  1728
  apply (frule kh)
haftmann@35722
  1729
  apply (rule ex1I[])
haftmann@35722
  1730
  apply blast
haftmann@35722
  1731
  apply clarsimp
haftmann@35722
  1732
  apply (drule hk) apply simp
haftmann@35722
  1733
  apply (rule sym)
haftmann@35722
  1734
  apply (erule conjunct1[OF conjunct2[OF hk]])
haftmann@35722
  1735
  apply (rule ballI)
haftmann@35722
  1736
  apply (drule hk)
haftmann@35722
  1737
  apply blast
haftmann@35722
  1738
  done
haftmann@35722
  1739
haftmann@35722
  1740
end
haftmann@35722
  1741
haftmann@35817
  1742
haftmann@35817
  1743
subsubsection {* The image case with fixed function and idempotency *}
haftmann@35817
  1744
haftmann@35817
  1745
locale folding_image_simple_idem = folding_image_simple +
haftmann@35817
  1746
  assumes idem: "x * x = x"
haftmann@35817
  1747
haftmann@35817
  1748
sublocale folding_image_simple_idem < semilattice proof
haftmann@35817
  1749
qed (fact idem)
haftmann@35817
  1750
haftmann@35817
  1751
context folding_image_simple_idem
haftmann@35817
  1752
begin
haftmann@35817
  1753
haftmann@35817
  1754
lemma in_idem:
haftmann@35817
  1755
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1756
  shows "g x * F A = F A"
haftmann@35817
  1757
  using assms by (induct A) (auto simp add: left_commute)
haftmann@35817
  1758
haftmann@35817
  1759
lemma subset_idem:
haftmann@35817
  1760
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1761
  shows "F B * F A = F A"
haftmann@35817
  1762
proof -
haftmann@35817
  1763
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1764
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1765
    (auto simp add: assoc in_idem `finite A`)
haftmann@35817
  1766
qed
haftmann@35817
  1767
haftmann@35817
  1768
declare insert [simp del]
haftmann@35817
  1769
haftmann@35817
  1770
lemma insert_idem [simp]:
haftmann@35817
  1771
  assumes "finite A"
haftmann@35817
  1772
  shows "F (insert x A) = g x * F A"
haftmann@35817
  1773
  using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
haftmann@35817
  1774
haftmann@35817
  1775
lemma union_idem:
haftmann@35817
  1776
  assumes "finite A" and "finite B"
haftmann@35817
  1777
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1778
proof -
haftmann@35817
  1779
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1780
  then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
haftmann@35817
  1781
  with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1782
qed
haftmann@35817
  1783
haftmann@35817
  1784
end
haftmann@35817
  1785
haftmann@35817
  1786
haftmann@35817
  1787
subsubsection {* The image case with flexible function and idempotency *}
haftmann@35817
  1788
haftmann@35817
  1789
locale folding_image_idem = folding_image +
haftmann@35817
  1790
  assumes idem: "x * x = x"
haftmann@35817
  1791
haftmann@35817
  1792
sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
haftmann@35817
  1793
qed (fact idem)
haftmann@35817
  1794
haftmann@35817
  1795
haftmann@35817
  1796
subsubsection {* The neutral-less case *}
haftmann@35817
  1797
haftmann@35817
  1798
locale folding_one = abel_semigroup +
haftmann@35817
  1799
  fixes F :: "'a set \<Rightarrow> 'a"
haftmann@35817
  1800
  assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
haftmann@35817
  1801
begin
haftmann@35817
  1802
haftmann@35817
  1803
lemma singleton [simp]:
haftmann@35817
  1804
  "F {x} = x"
haftmann@35817
  1805
  by (simp add: eq_fold)
haftmann@35817
  1806
haftmann@35817
  1807
lemma eq_fold':
haftmann@35817
  1808
  assumes "finite A" and "x \<notin> A"
haftmann@35817
  1809
  shows "F (insert x A) = fold (op *) x A"
haftmann@35817
  1810
proof -
haftmann@35817
  1811
  interpret ab_semigroup_mult "op *" proof qed (simp_all add: ac_simps)
haftmann@35817
  1812
  with assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
haftmann@35817
  1813
qed
haftmann@35817
  1814
haftmann@35817
  1815
lemma insert [simp]:
haftmann@35817
  1816
  assumes "finite A" and "x \<notin> A"
haftmann@35817
  1817
  shows "F (insert x A) = (if A = {} then x else x * F A)"
haftmann@35817
  1818
proof (cases "A = {}")
haftmann@35817
  1819
  case True then show ?thesis by simp
haftmann@35817
  1820
next
haftmann@35817
  1821
  case False then obtain b where "b \<in> A" by blast
haftmann@35817
  1822
  then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1823
  with `finite A` have "finite B" by simp
haftmann@35817
  1824
  interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
haftmann@35817
  1825
  qed (simp_all add: expand_fun_eq ac_simps)
haftmann@35817
  1826
  thm fold.commute_comp' [of B b, simplified expand_fun_eq, simplified]
haftmann@35817
  1827
  from `finite B` fold.commute_comp' [of B x]
haftmann@35817
  1828
    have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
haftmann@35817
  1829
  then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: expand_fun_eq commute)
haftmann@35817
  1830
  from `finite B` * fold.insert [of B b]
haftmann@35817
  1831
    have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
haftmann@35817
  1832
  then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: expand_fun_eq)
haftmann@35817
  1833
  from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
haftmann@35817
  1834
qed
haftmann@35817
  1835
haftmann@35817
  1836
lemma remove:
haftmann@35817
  1837
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1838
  shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1839
proof -
haftmann@35817
  1840
  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1841
  with assms show ?thesis by simp
haftmann@35817
  1842
qed
haftmann@35817
  1843
haftmann@35817
  1844
lemma insert_remove:
haftmann@35817
  1845
  assumes "finite A"
haftmann@35817
  1846
  shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1847
  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
haftmann@35817
  1848
haftmann@35817
  1849
lemma union_disjoint:
haftmann@35817
  1850
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
haftmann@35817
  1851
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1852
  using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
haftmann@35817
  1853
haftmann@35817
  1854
lemma union_inter:
haftmann@35817
  1855
  assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
haftmann@35817
  1856
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35817
  1857
proof -
haftmann@35817
  1858
  from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
haftmann@35817
  1859
  from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
haftmann@35817
  1860
    case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
haftmann@35817
  1861
  next
haftmann@35817
  1862
    case (insert x A) show ?case proof (cases "x \<in> B")
haftmann@35817
  1863
      case True then have "B \<noteq> {}" by auto
haftmann@35817
  1864
      with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
haftmann@35817
  1865
        (simp_all add: insert_absorb ac_simps union_disjoint)
haftmann@35817
  1866
    next
haftmann@35817
  1867
      case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
haftmann@35817
  1868
      moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
haftmann@35817
  1869
        by auto
haftmann@35817
  1870
      ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
haftmann@35817
  1871
    qed
haftmann@35817
  1872
  qed
haftmann@35817
  1873
qed
haftmann@35817
  1874
haftmann@35817
  1875
lemma closed:
haftmann@35817
  1876
  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
haftmann@35817
  1877
  shows "F A \<in> A"
haftmann@35817
  1878
using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
haftmann@35817
  1879
  case singleton then show ?case by simp
haftmann@35817
  1880
next
haftmann@35817
  1881
  case insert with elem show ?case by force
haftmann@35817
  1882
qed
haftmann@35817
  1883
haftmann@35817
  1884
end
haftmann@35817
  1885
haftmann@35817
  1886
haftmann@35817
  1887
subsubsection {* The neutral-less case with idempotency *}
haftmann@35817
  1888
haftmann@35817
  1889
locale folding_one_idem = folding_one +
haftmann@35817
  1890
  assumes idem: "x * x = x"
haftmann@35817
  1891
haftmann@35817
  1892
sublocale folding_one_idem < semilattice proof
haftmann@35817
  1893
qed (fact idem)
haftmann@35817
  1894
haftmann@35817
  1895
context folding_one_idem
haftmann@35817
  1896
begin
haftmann@35817
  1897
haftmann@35817
  1898
lemma in_idem:
haftmann@35817
  1899
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1900
  shows "x * F A = F A"
haftmann@35817
  1901
proof -
haftmann@35817
  1902
  from assms have "A \<noteq> {}" by auto
haftmann@35817
  1903
  with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
haftmann@35817
  1904
qed
haftmann@35817
  1905
haftmann@35817
  1906
lemma subset_idem:
haftmann@35817
  1907
  assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
haftmann@35817
  1908
  shows "F B * F A = F A"
haftmann@35817
  1909
proof -
haftmann@35817
  1910
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1911
  then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
haftmann@35817
  1912
    (simp_all add: assoc in_idem `finite A`)
haftmann@35817
  1913
qed
haftmann@35817
  1914
haftmann@35817
  1915
declare insert [simp del]
haftmann@35817
  1916
haftmann@35817
  1917
lemma eq_fold_idem':
haftmann@35817
  1918
  assumes "finite A"
haftmann@35817
  1919
  shows "F (insert a A) = fold (op *) a A"
haftmann@35817
  1920
proof -
haftmann@35817
  1921
  interpret ab_semigroup_idem_mult "op *" proof qed (simp_all add: ac_simps)
haftmann@35817
  1922
  with assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
haftmann@35817
  1923
qed
haftmann@35817
  1924
haftmann@35817
  1925
lemma insert_idem [simp]:
haftmann@35817
  1926
  assumes "finite A"
haftmann@35817
  1927
  shows "F (insert x A) = (if A = {} then x else x * F A)"
haftmann@35817
  1928
proof (cases "x \<in> A")
haftmann@35817
  1929
  case False with `finite A` show ?thesis by (rule insert)
haftmann@35817
  1930
next
haftmann@35817
  1931
  case True then have "A \<noteq> {}" by auto
haftmann@35817
  1932
  with `finite A` show ?thesis by (simp add: in_idem insert_absorb True)
haftmann@35817
  1933
qed
haftmann@35817
  1934
  
haftmann@35817
  1935
lemma union_idem:
haftmann@35817
  1936
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
haftmann@35817
  1937
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1938
proof (cases "A \<inter> B = {}")
haftmann@35817
  1939
  case True with assms show ?thesis by (simp add: union_disjoint)
haftmann@35817
  1940
next
haftmann@35817
  1941
  case False
haftmann@35817
  1942
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1943
  with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
haftmann@35817
  1944
  with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1945
qed
haftmann@35817
  1946
haftmann@35817
  1947
lemma hom_commute:
haftmann@35817
  1948
  assumes hom: "\<And>x y. h (x * y) = h x * h y"
haftmann@35817
  1949
  and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
haftmann@35817
  1950
using N proof (induct rule: finite_ne_induct)
haftmann@35817
  1951
  case singleton thus ?case by simp
haftmann@35817
  1952
next
haftmann@35817
  1953
  case (insert n N)
haftmann@35817
  1954
  then have "h (F (insert n N)) = h (n * F N)" by simp
haftmann@35817
  1955
  also have "\<dots> = h n * h (F N)" by (rule hom)
haftmann@35817
  1956
  also have "h (F N) = F (h ` N)" by(rule insert)
haftmann@35817
  1957
  also have "h n * \<dots> = F (insert (h n) (h ` N))"
haftmann@35817
  1958
    using insert by(simp)
haftmann@35817
  1959
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@35817
  1960
  finally show ?case .
haftmann@35817
  1961
qed
haftmann@35817
  1962
haftmann@35817
  1963
end
haftmann@35817
  1964
haftmann@35796
  1965
notation times (infixl "*" 70)
haftmann@35796
  1966
notation Groups.one ("1")
haftmann@35722
  1967
haftmann@35722
  1968
haftmann@35722
  1969
subsection {* Finite cardinality *}
haftmann@35722
  1970
haftmann@35722
  1971
text {* This definition, although traditional, is ugly to work with:
haftmann@35722
  1972
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
haftmann@35722
  1973
But now that we have @{text fold_image} things are easy:
haftmann@35722
  1974
*}
haftmann@35722
  1975
haftmann@35722
  1976
definition card :: "'a set \<Rightarrow> nat" where
haftmann@35722
  1977
  "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
haftmann@35722
  1978
haftmann@35722
  1979
interpretation card!: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
haftmann@35722
  1980
qed (simp add: card_def)
haftmann@35722
  1981
haftmann@35722
  1982
lemma card_infinite [simp]:
haftmann@35722
  1983
  "\<not> finite A \<Longrightarrow> card A = 0"
haftmann@35722
  1984
  by (simp add: card_def)
haftmann@35722
  1985
haftmann@35722
  1986
lemma card_empty:
haftmann@35722
  1987
  "card {} = 0"
haftmann@35722
  1988
  by (fact card.empty)
haftmann@35722
  1989
haftmann@35722
  1990
lemma card_insert_disjoint:
haftmann@35722
  1991
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
haftmann@35722
  1992
  by simp
haftmann@35722
  1993
haftmann@35722
  1994
lemma card_insert_if:
haftmann@35722
  1995
  "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
haftmann@35722
  1996
  by auto (simp add: card.insert_remove card.remove)
haftmann@35722
  1997
haftmann@35722
  1998
lemma card_ge_0_finite:
haftmann@35722
  1999
  "card A > 0 \<Longrightarrow> finite A"
haftmann@35722
  2000
  by (rule ccontr) simp
haftmann@35722
  2001
haftmann@35722
  2002
lemma card_0_eq [simp, noatp]:
haftmann@35722
  2003
  "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
haftmann@35722
  2004
  by (auto dest: mk_disjoint_insert)
haftmann@35722
  2005
haftmann@35722
  2006
lemma finite_UNIV_card_ge_0:
haftmann@35722
  2007
  "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
haftmann@35722
  2008
  by (rule ccontr) simp
haftmann@35722
  2009
haftmann@35722
  2010
lemma card_eq_0_iff:
haftmann@35722
  2011
  "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
haftmann@35722
  2012
  by auto
haftmann@35722
  2013
haftmann@35722
  2014
lemma card_gt_0_iff:
haftmann@35722
  2015
  "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
haftmann@35722
  2016
  by (simp add: neq0_conv [symmetric] card_eq_0_iff) 
haftmann@35722
  2017
haftmann@35722
  2018
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
haftmann@35722
  2019
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
haftmann@35722
  2020
apply(simp del:insert_Diff_single)
haftmann@35722
  2021
done
haftmann@35722
  2022
haftmann@35722
  2023
lemma card_Diff_singleton:
haftmann@35722
  2024
  "finite A ==> x: A ==> card (A - {x}) = card A - 1"
haftmann@35722
  2025
by (simp add: card_Suc_Diff1 [symmetric])
haftmann@35722
  2026
haftmann@35722
  2027
lemma card_Diff_singleton_if:
haftmann@35722
  2028
  "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
haftmann@35722
  2029
by (simp add: card_Diff_singleton)
haftmann@35722
  2030
haftmann@35722
  2031
lemma card_Diff_insert[simp]:
haftmann@35722
  2032
assumes "finite A" and "a:A" and "a ~: B"
haftmann@35722
  2033
shows "card(A - insert a B) = card(A - B) - 1"
haftmann@35722
  2034
proof -
haftmann@35722
  2035
  have "A - insert a B = (A - B) - {a}" using assms by blast
haftmann@35722
  2036
  then show ?thesis using assms by(simp add:card_Diff_singleton)
haftmann@35722
  2037
qed
haftmann@35722
  2038
haftmann@35722
  2039
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
haftmann@35722
  2040
by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  2041
haftmann@35722
  2042
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
haftmann@35722
  2043
by (simp add: card_insert_if)
haftmann@35722
  2044
haftmann@35722
  2045
lemma card_mono:
haftmann@35722
  2046
  assumes "finite B" and "A \<subseteq> B"
haftmann@35722
  2047
  shows "card A \<le> card B"
haftmann@35722
  2048
proof -
haftmann@35722
  2049
  from assms have "finite A" by (auto intro: finite_subset)
haftmann@35722
  2050
  then show ?thesis using assms proof (induct A arbitrary: B)
haftmann@35722
  2051
    case empty then show ?case by simp
haftmann@35722
  2052
  next
haftmann@35722
  2053
    case (insert x A)
haftmann@35722
  2054
    then have "x \<in> B" by simp
haftmann@35722
  2055
    from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
haftmann@35722
  2056
    with insert.hyps have "card A \<le> card (B - {x})" by auto
haftmann@35722
  2057
    with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
haftmann@35722
  2058
  qed
haftmann@35722
  2059
qed
haftmann@35722
  2060
haftmann@35722
  2061
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
haftmann@35722
  2062
apply (induct set: finite, simp, clarify)
haftmann@35722
  2063
apply (subgoal_tac "finite A & A - {x} <= F")
haftmann@35722
  2064
 prefer 2 apply (blast intro: finite_subset, atomize)
haftmann@35722
  2065
apply (drule_tac x = "A - {x}" in spec)
haftmann@35722
  2066
apply (simp add: card_Diff_singleton_if split add: split_if_asm)
haftmann@35722
  2067
apply (case_tac "card A", auto)
haftmann@35722
  2068
done
haftmann@35722
  2069
haftmann@35722
  2070
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
haftmann@35722
  2071
apply (simp add: psubset_eq linorder_not_le [symmetric])
haftmann@35722
  2072
apply (blast dest: card_seteq)
haftmann@35722
  2073
done
haftmann@35722
  2074
haftmann@35722
  2075
lemma card_Un_Int: "finite A ==> finite B
haftmann@35722
  2076
    ==> card A + card B = card (A Un B) + card (A Int B)"
haftmann@35817
  2077
  by (fact card.union_inter [symmetric])
haftmann@35722
  2078
haftmann@35722
  2079
lemma card_Un_disjoint: "finite A ==> finite B
haftmann@35722
  2080
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
haftmann@35722
  2081
  by (fact card.union_disjoint)
haftmann@35722
  2082
haftmann@35722
  2083
lemma card_Diff_subset:
haftmann@35722
  2084
  assumes "finite B" and "B \<subseteq> A"
haftmann@35722
  2085
  shows "card (A - B) = card A - card B"
haftmann@35722
  2086
proof (cases "finite A")
haftmann@35722
  2087
  case False with assms show ?thesis by simp
haftmann@35722
  2088
next
haftmann@35722
  2089
  case True with assms show ?thesis by (induct B arbitrary: A) simp_all
haftmann@35722
  2090
qed
haftmann@35722
  2091
haftmann@35722
  2092
lemma card_Diff_subset_Int:
haftmann@35722
  2093
  assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
haftmann@35722
  2094
proof -
haftmann@35722
  2095
  have "A - B = A - A \<inter> B" by auto
haftmann@35722
  2096
  thus ?thesis
haftmann@35722
  2097
    by (simp add: card_Diff_subset AB) 
haftmann@35722
  2098
qed
haftmann@35722
  2099
haftmann@35722
  2100
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
haftmann@35722
  2101
apply (rule Suc_less_SucD)
haftmann@35722
  2102
apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  2103
done
haftmann@35722
  2104
haftmann@35722
  2105
lemma card_Diff2_less:
haftmann@35722
  2106
  "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
haftmann@35722
  2107
apply (case_tac "x = y")
haftmann@35722
  2108
 apply (simp add: card_Diff1_less del:card_Diff_insert)
haftmann@35722
  2109
apply (rule less_trans)
haftmann@35722
  2110
 prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
haftmann@35722
  2111
done
haftmann@35722
  2112
haftmann@35722
  2113
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
haftmann@35722
  2114
apply (case_tac "x : A")
haftmann@35722
  2115
 apply (simp_all add: card_Diff1_less less_imp_le)
haftmann@35722
  2116
done
haftmann@35722
  2117
haftmann@35722
  2118
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
haftmann@35722
  2119
by (erule psubsetI, blast)
haftmann@35722
  2120
haftmann@35722
  2121
lemma insert_partition:
haftmann@35722
  2122
  "\<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
  2123
  \<Longrightarrow> x \<inter> \<Union> F = {}"
haftmann@35722
  2124
by auto
haftmann@35722
  2125
haftmann@35722
  2126
lemma finite_psubset_induct[consumes 1, case_names psubset]:
haftmann@35722
  2127
  assumes "finite A" and "!!A. finite A \<Longrightarrow> (!!B. finite B \<Longrightarrow> B \<subset> A \<Longrightarrow> P(B)) \<Longrightarrow> P(A)" shows "P A"
haftmann@35722
  2128
using assms(1)
haftmann@35722
  2129
proof (induct A rule: measure_induct_rule[where f=card])
haftmann@35722
  2130
  case (less A)
haftmann@35722
  2131
  show ?case
haftmann@35722
  2132
  proof(rule assms(2)[OF less(2)])
haftmann@35722
  2133
    fix B assume "finite B" "B \<subset> A"
haftmann@35722
  2134
    show "P B" by(rule less(1)[OF psubset_card_mono[OF less(2) `B \<subset> A`] `finite B`])
haftmann@35722
  2135
  qed
haftmann@35722
  2136
qed
haftmann@35722
  2137
haftmann@35722
  2138
text{* main cardinality theorem *}
haftmann@35722
  2139
lemma card_partition [rule_format]:
haftmann@35722
  2140
  "finite C ==>
haftmann@35722
  2141
     finite (\<Union> C) -->
haftmann@35722
  2142
     (\<forall>c\<in>C. card c = k) -->
haftmann@35722
  2143
     (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
haftmann@35722
  2144
     k * card(C) = card (\<Union> C)"
haftmann@35722
  2145
apply (erule finite_induct, simp)
haftmann@35722
  2146
apply (simp add: card_Un_disjoint insert_partition 
haftmann@35722
  2147
       finite_subset [of _ "\<Union> (insert x F)"])
haftmann@35722
  2148
done
haftmann@35722
  2149
haftmann@35722
  2150
lemma card_eq_UNIV_imp_eq_UNIV:
haftmann@35722
  2151
  assumes fin: "finite (UNIV :: 'a set)"
haftmann@35722
  2152
  and card: "card A = card (UNIV :: 'a set)"
haftmann@35722
  2153
  shows "A = (UNIV :: 'a set)"
haftmann@35722
  2154
proof
haftmann@35722
  2155
  show "A \<subseteq> UNIV" by simp
haftmann@35722
  2156
  show "UNIV \<subseteq> A"
haftmann@35722
  2157
  proof
haftmann@35722
  2158
    fix x
haftmann@35722
  2159
    show "x \<in> A"
haftmann@35722
  2160
    proof (rule ccontr)
haftmann@35722
  2161
      assume "x \<notin> A"
haftmann@35722
  2162
      then have "A \<subset> UNIV" by auto
haftmann@35722
  2163
      with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
haftmann@35722
  2164
      with card show False by simp
haftmann@35722
  2165
    qed
haftmann@35722
  2166
  qed
haftmann@35722
  2167
qed
haftmann@35722
  2168
haftmann@35722
  2169
text{*The form of a finite set of given cardinality*}
haftmann@35722
  2170
haftmann@35722
  2171
lemma card_eq_SucD:
haftmann@35722
  2172
assumes "card A = Suc k"
haftmann@35722
  2173
shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
haftmann@35722
  2174
proof -
haftmann@35722
  2175
  have fin: "finite A" using assms by (auto intro: ccontr)
haftmann@35722
  2176
  moreover have "card A \<noteq> 0" using assms by auto
haftmann@35722
  2177
  ultimately obtain b where b: "b \<in> A" by auto
haftmann@35722
  2178
  show ?thesis
haftmann@35722
  2179
  proof (intro exI conjI)
haftmann@35722
  2180
    show "A = insert b (A-{b})" using b by blast
haftmann@35722
  2181
    show "b \<notin> A - {b}" by blast
haftmann@35722
  2182
    show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
haftmann@35722
  2183
      using assms b fin by(fastsimp dest:mk_disjoint_insert)+
haftmann@35722
  2184
  qed
haftmann@35722
  2185
qed
haftmann@35722
  2186
haftmann@35722
  2187
lemma card_Suc_eq:
haftmann@35722
  2188
  "(card A = Suc k) =
haftmann@35722
  2189
   (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
haftmann@35722
  2190
apply(rule iffI)
haftmann@35722
  2191
 apply(erule card_eq_SucD)
haftmann@35722
  2192
apply(auto)
haftmann@35722
  2193
apply(subst card_insert)
haftmann@35722
  2194
 apply(auto intro:ccontr)
haftmann@35722
  2195
done
haftmann@35722
  2196
haftmann@35722
  2197
lemma finite_fun_UNIVD2:
haftmann@35722
  2198
  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
haftmann@35722
  2199
  shows "finite (UNIV :: 'b set)"
haftmann@35722
  2200
proof -
haftmann@35722
  2201
  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
haftmann@35722
  2202
    by(rule finite_imageI)
haftmann@35722
  2203
  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
haftmann@35722
  2204
    by(rule UNIV_eq_I) auto
haftmann@35722
  2205
  ultimately show "finite (UNIV :: 'b set)" by simp
haftmann@35722
  2206
qed
haftmann@35722
  2207
haftmann@35722
  2208
lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
haftmann@35722
  2209
  unfolding UNIV_unit by simp
haftmann@35722
  2210
haftmann@35722
  2211
haftmann@35722
  2212
subsubsection {* Cardinality of image *}
haftmann@35722
  2213
haftmann@35722
  2214
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
haftmann@35722
  2215
apply (induct set: finite)
haftmann@35722
  2216
 apply simp
haftmann@35722
  2217
apply (simp add: le_SucI card_insert_if)
haftmann@35722
  2218
done
haftmann@35722
  2219
haftmann@35722
  2220
lemma card_image:
haftmann@35722
  2221
  assumes "inj_on f A"
haftmann@35722
  2222
  shows "card (f ` A) = card A"
haftmann@35722
  2223
proof (cases "finite A")
haftmann@35722
  2224
  case True then show ?thesis using assms by (induct A) simp_all
haftmann@35722
  2225
next
haftmann@35722
  2226
  case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
haftmann@35722
  2227
  with False show ?thesis by simp
haftmann@35722
  2228
qed
haftmann@35722
  2229
haftmann@35722
  2230
lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
haftmann@35722
  2231
by(auto simp: card_image bij_betw_def)
haftmann@35722
  2232
haftmann@35722
  2233
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
haftmann@35722
  2234
by (simp add: card_seteq card_image)
haftmann@35722
  2235
haftmann@35722
  2236
lemma eq_card_imp_inj_on:
haftmann@35722
  2237
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
haftmann@35722
  2238
apply (induct rule:finite_induct)
haftmann@35722
  2239
apply simp
haftmann@35722
  2240
apply(frule card_image_le[where f = f])
haftmann@35722
  2241
apply(simp add:card_insert_if split:if_splits)
haftmann@35722
  2242
done
haftmann@35722
  2243
haftmann@35722
  2244
lemma inj_on_iff_eq_card:
haftmann@35722
  2245
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
haftmann@35722
  2246
by(blast intro: card_image eq_card_imp_inj_on)
haftmann@35722
  2247
haftmann@35722
  2248
haftmann@35722
  2249
lemma card_inj_on_le:
haftmann@35722
  2250
  "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
haftmann@35722
  2251
apply (subgoal_tac "finite A") 
haftmann@35722
  2252
 apply (force intro: card_mono simp add: card_image [symmetric])
haftmann@35722
  2253
apply (blast intro: finite_imageD dest: finite_subset) 
haftmann@35722
  2254
done
haftmann@35722
  2255
haftmann@35722
  2256
lemma card_bij_eq:
haftmann@35722
  2257
  "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
haftmann@35722
  2258
     finite A; finite B |] ==> card A = card B"
haftmann@35722
  2259
by (auto intro: le_antisym card_inj_on_le)
haftmann@35722
  2260
haftmann@35722
  2261
haftmann@35722
  2262
subsubsection {* Cardinality of sums *}
haftmann@35722
  2263
haftmann@35722
  2264
lemma card_Plus:
haftmann@35722
  2265
  assumes "finite A" and "finite B"
haftmann@35722
  2266
  shows "card (A <+> B) = card A + card B"
haftmann@35722
  2267
proof -
haftmann@35722
  2268
  have "Inl`A \<inter> Inr`B = {}" by fast
haftmann@35722
  2269
  with assms show ?thesis
haftmann@35722
  2270
    unfolding Plus_def
haftmann@35722
  2271
    by (simp add: card_Un_disjoint card_image)
haftmann@35722
  2272
qed
haftmann@35722
  2273
haftmann@35722
  2274
lemma card_Plus_conv_if:
haftmann@35722
  2275
  "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
haftmann@35722
  2276
  by (auto simp add: card_Plus)
haftmann@35722
  2277
haftmann@35722
  2278
haftmann@35722
  2279
subsubsection {* Cardinality of the Powerset *}
haftmann@35722
  2280
haftmann@35722
  2281
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
haftmann@35722
  2282
apply (induct set: finite)
haftmann@35722
  2283
 apply (simp_all add: Pow_insert)
haftmann@35722
  2284
apply (subst card_Un_disjoint, blast)
haftmann@35722
  2285
  apply (blast intro: finite_imageI, blast)
haftmann@35722
  2286
apply (subgoal_tac "inj_on (insert x) (Pow F)")
haftmann@35722
  2287
 apply (simp add: card_image Pow_insert)
haftmann@35722
  2288
apply (unfold inj_on_def)
haftmann@35722
  2289
apply (blast elim!: equalityE)
haftmann@35722
  2290
done
haftmann@35722
  2291
haftmann@35722
  2292
text {* Relates to equivalence classes.  Based on a theorem of F. Kammüller.  *}
haftmann@35722
  2293
haftmann@35722
  2294
lemma dvd_partition:
haftmann@35722
  2295
  "finite (Union C) ==>
haftmann@35722
  2296
    ALL c : C. k dvd card c ==>
haftmann@35722
  2297
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
haftmann@35722
  2298
  k dvd card (Union C)"
haftmann@35722
  2299
apply(frule finite_UnionD)
haftmann@35722
  2300
apply(rotate_tac -1)
haftmann@35722
  2301
apply (induct set: finite, simp_all, clarify)
haftmann@35722
  2302
apply (subst card_Un_disjoint)
haftmann@35722
  2303
   apply (auto simp add: disjoint_eq_subset_Compl)
haftmann@35722
  2304
done
haftmann@35722
  2305
haftmann@35722
  2306
haftmann@35722
  2307
subsubsection {* Relating injectivity and surjectivity *}
haftmann@35722
  2308
haftmann@35722
  2309
lemma finite_surj_inj: "finite(A) \<Longrightarrow> A <= f`A \<Longrightarrow> inj_on f A"
haftmann@35722
  2310
apply(rule eq_card_imp_inj_on, assumption)
haftmann@35722
  2311
apply(frule finite_imageI)
haftmann@35722
  2312
apply(drule (1) card_seteq)
haftmann@35722
  2313
 apply(erule card_image_le)
haftmann@35722
  2314
apply simp
haftmann@35722
  2315
done
haftmann@35722
  2316
haftmann@35722
  2317
lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2318
shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
haftmann@35722
  2319
by (blast intro: finite_surj_inj subset_UNIV dest:surj_range)
haftmann@35722
  2320
haftmann@35722
  2321
lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2322
shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
haftmann@35722
  2323
by(fastsimp simp:surj_def dest!: endo_inj_surj)
haftmann@35722
  2324
haftmann@35722
  2325
corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
haftmann@35722
  2326
proof
haftmann@35722
  2327
  assume "finite(UNIV::nat set)"
haftmann@35722
  2328
  with finite_UNIV_inj_surj[of Suc]
haftmann@35722
  2329
  show False by simp (blast dest: Suc_neq_Zero surjD)
haftmann@35722
  2330
qed
haftmann@35722
  2331
haftmann@35722
  2332
(* Often leads to bogus ATP proofs because of reduced type information, hence noatp *)
haftmann@35722
  2333
lemma infinite_UNIV_char_0[noatp]:
haftmann@35722
  2334
  "\<not> finite (UNIV::'a::semiring_char_0 set)"
haftmann@35722
  2335
proof
haftmann@35722
  2336
  assume "finite (UNIV::'a set)"
haftmann@35722
  2337
  with subset_UNIV have "finite (range of_nat::'a set)"
haftmann@35722
  2338
    by (rule finite_subset)
haftmann@35722
  2339
  moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
haftmann@35722
  2340
    by (simp add: inj_on_def)
haftmann@35722
  2341
  ultimately have "finite (UNIV::nat set)"
haftmann@35722
  2342
    by (rule finite_imageD)
haftmann@35722
  2343
  then show "False"
haftmann@35722
  2344
    by simp
haftmann@35722
  2345
qed
haftmann@35722
  2346
haftmann@35722
  2347
end