src/HOL/Finite_Set.thy
author hoelzl
Tue Nov 23 14:14:17 2010 +0100 (2010-11-23)
changeset 40703 d1fc454d6735
parent 40702 cf26dd7395e4
child 40716 a92d744bca5f
permissions -rw-r--r--
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
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(*  Title:      HOL/Finite_Set.thy
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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                with contributions by Jeremy Avigad
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*)
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header {* Finite sets *}
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theory Finite_Set
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imports Option Power
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begin
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subsection {* Predicate for finite sets *}
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inductive finite :: "'a set => 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})")
wenzelm@12396
   333
     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@29797
   512
class finite =
haftmann@26041
   513
  assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
huffman@27430
   514
begin
huffman@27430
   515
huffman@27430
   516
lemma finite [simp]: "finite (A \<Colon> 'a set)"
haftmann@26441
   517
  by (rule subset_UNIV finite_UNIV finite_subset)+
haftmann@26041
   518
huffman@27430
   519
end
huffman@27430
   520
blanchet@35828
   521
lemma UNIV_unit [no_atp]:
haftmann@26041
   522
  "UNIV = {()}" by auto
haftmann@26041
   523
haftmann@35719
   524
instance unit :: finite proof
haftmann@35719
   525
qed (simp add: UNIV_unit)
haftmann@26146
   526
blanchet@35828
   527
lemma UNIV_bool [no_atp]:
haftmann@26041
   528
  "UNIV = {False, True}" by auto
haftmann@26041
   529
haftmann@35719
   530
instance bool :: finite proof
haftmann@35719
   531
qed (simp add: UNIV_bool)
haftmann@35719
   532
haftmann@37678
   533
instance prod :: (finite, finite) finite proof
haftmann@35719
   534
qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
haftmann@26146
   535
haftmann@35719
   536
lemma finite_option_UNIV [simp]:
haftmann@35719
   537
  "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
haftmann@35719
   538
  by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
haftmann@35719
   539
haftmann@35719
   540
instance option :: (finite) finite proof
haftmann@35719
   541
qed (simp add: UNIV_option_conv)
haftmann@26146
   542
haftmann@26041
   543
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
nipkow@39302
   544
  by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
haftmann@26041
   545
haftmann@26146
   546
instance "fun" :: (finite, finite) finite
haftmann@26146
   547
proof
haftmann@26041
   548
  show "finite (UNIV :: ('a => 'b) set)"
haftmann@26041
   549
  proof (rule finite_imageD)
haftmann@26041
   550
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
berghofe@26792
   551
    have "range ?graph \<subseteq> Pow UNIV" by simp
berghofe@26792
   552
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
berghofe@26792
   553
      by (simp only: finite_Pow_iff finite)
berghofe@26792
   554
    ultimately show "finite (range ?graph)"
berghofe@26792
   555
      by (rule finite_subset)
haftmann@26041
   556
    show "inj ?graph" by (rule inj_graph)
haftmann@26041
   557
  qed
haftmann@26041
   558
qed
haftmann@26041
   559
haftmann@37678
   560
instance sum :: (finite, finite) finite proof
haftmann@35719
   561
qed (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
haftmann@27981
   562
haftmann@26041
   563
haftmann@35817
   564
subsection {* A basic fold functional for finite sets *}
nipkow@15392
   565
nipkow@15392
   566
text {* The intended behaviour is
wenzelm@31916
   567
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
nipkow@28853
   568
if @{text f} is ``left-commutative'':
nipkow@15392
   569
*}
nipkow@15392
   570
nipkow@28853
   571
locale fun_left_comm =
nipkow@28853
   572
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@28853
   573
  assumes fun_left_comm: "f x (f y z) = f y (f x z)"
nipkow@28853
   574
begin
nipkow@28853
   575
nipkow@28853
   576
text{* On a functional level it looks much nicer: *}
nipkow@28853
   577
nipkow@28853
   578
lemma fun_comp_comm:  "f x \<circ> f y = f y \<circ> f x"
nipkow@39302
   579
by (simp add: fun_left_comm fun_eq_iff)
nipkow@28853
   580
nipkow@28853
   581
end
nipkow@28853
   582
nipkow@28853
   583
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
nipkow@28853
   584
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
nipkow@28853
   585
  emptyI [intro]: "fold_graph f z {} z" |
nipkow@28853
   586
  insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
nipkow@28853
   587
      \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
nipkow@28853
   588
nipkow@28853
   589
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
nipkow@28853
   590
nipkow@28853
   591
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
haftmann@37767
   592
  "fold f z A = (THE y. fold_graph f z A y)"
nipkow@15392
   593
paulson@15498
   594
text{*A tempting alternative for the definiens is
nipkow@28853
   595
@{term "if finite A then THE y. fold_graph f z A y else e"}.
paulson@15498
   596
It allows the removal of finiteness assumptions from the theorems
nipkow@28853
   597
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
nipkow@28853
   598
The proofs become ugly. It is not worth the effort. (???) *}
nipkow@28853
   599
nipkow@28853
   600
nipkow@28853
   601
lemma Diff1_fold_graph:
nipkow@28853
   602
  "fold_graph f z (A - {x}) y \<Longrightarrow> x \<in> A \<Longrightarrow> fold_graph f z A (f x y)"
nipkow@28853
   603
by (erule insert_Diff [THEN subst], rule fold_graph.intros, auto)
nipkow@28853
   604
nipkow@28853
   605
lemma fold_graph_imp_finite: "fold_graph f z A x \<Longrightarrow> finite A"
nipkow@28853
   606
by (induct set: fold_graph) auto
nipkow@28853
   607
nipkow@28853
   608
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
nipkow@28853
   609
by (induct set: finite) auto
nipkow@28853
   610
nipkow@28853
   611
nipkow@28853
   612
subsubsection{*From @{const fold_graph} to @{term fold}*}
nipkow@15392
   613
nipkow@28853
   614
context fun_left_comm
haftmann@26041
   615
begin
haftmann@26041
   616
huffman@36045
   617
lemma fold_graph_insertE_aux:
huffman@36045
   618
  "fold_graph f z A y \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'"
huffman@36045
   619
proof (induct set: fold_graph)
huffman@36045
   620
  case (insertI x A y) show ?case
huffman@36045
   621
  proof (cases "x = a")
huffman@36045
   622
    assume "x = a" with insertI show ?case by auto
nipkow@28853
   623
  next
huffman@36045
   624
    assume "x \<noteq> a"
huffman@36045
   625
    then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
huffman@36045
   626
      using insertI by auto
huffman@36045
   627
    have 1: "f x y = f a (f x y')"
huffman@36045
   628
      unfolding y by (rule fun_left_comm)
huffman@36045
   629
    have 2: "fold_graph f z (insert x A - {a}) (f x y')"
huffman@36045
   630
      using y' and `x \<noteq> a` and `x \<notin> A`
huffman@36045
   631
      by (simp add: insert_Diff_if fold_graph.insertI)
huffman@36045
   632
    from 1 2 show ?case by fast
nipkow@15392
   633
  qed
huffman@36045
   634
qed simp
huffman@36045
   635
huffman@36045
   636
lemma fold_graph_insertE:
huffman@36045
   637
  assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
huffman@36045
   638
  obtains y where "v = f x y" and "fold_graph f z A y"
huffman@36045
   639
using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
nipkow@28853
   640
nipkow@28853
   641
lemma fold_graph_determ:
nipkow@28853
   642
  "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
huffman@36045
   643
proof (induct arbitrary: y set: fold_graph)
huffman@36045
   644
  case (insertI x A y v)
huffman@36045
   645
  from `fold_graph f z (insert x A) v` and `x \<notin> A`
huffman@36045
   646
  obtain y' where "v = f x y'" and "fold_graph f z A y'"
huffman@36045
   647
    by (rule fold_graph_insertE)
huffman@36045
   648
  from `fold_graph f z A y'` have "y' = y" by (rule insertI)
huffman@36045
   649
  with `v = f x y'` show "v = f x y" by simp
huffman@36045
   650
qed fast
nipkow@15392
   651
nipkow@28853
   652
lemma fold_equality:
nipkow@28853
   653
  "fold_graph f z A y \<Longrightarrow> fold f z A = y"
nipkow@28853
   654
by (unfold fold_def) (blast intro: fold_graph_determ)
nipkow@15392
   655
huffman@36045
   656
lemma fold_graph_fold: "finite A \<Longrightarrow> fold_graph f z A (fold f z A)"
huffman@36045
   657
unfolding fold_def
huffman@36045
   658
apply (rule theI')
huffman@36045
   659
apply (rule ex_ex1I)
huffman@36045
   660
apply (erule finite_imp_fold_graph)
huffman@36045
   661
apply (erule (1) fold_graph_determ)
huffman@36045
   662
done
huffman@36045
   663
nipkow@15392
   664
text{* The base case for @{text fold}: *}
nipkow@15392
   665
nipkow@28853
   666
lemma (in -) fold_empty [simp]: "fold f z {} = z"
nipkow@28853
   667
by (unfold fold_def) blast
nipkow@28853
   668
nipkow@28853
   669
text{* The various recursion equations for @{const fold}: *}
nipkow@28853
   670
haftmann@26041
   671
lemma fold_insert [simp]:
nipkow@28853
   672
  "finite A ==> x \<notin> A ==> fold f z (insert x A) = f x (fold f z A)"
huffman@36045
   673
apply (rule fold_equality)
huffman@36045
   674
apply (erule fold_graph.insertI)
huffman@36045
   675
apply (erule fold_graph_fold)
nipkow@28853
   676
done
nipkow@28853
   677
nipkow@28853
   678
lemma fold_fun_comm:
nipkow@28853
   679
  "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
nipkow@28853
   680
proof (induct rule: finite_induct)
nipkow@28853
   681
  case empty then show ?case by simp
nipkow@28853
   682
next
nipkow@28853
   683
  case (insert y A) then show ?case
nipkow@28853
   684
    by (simp add: fun_left_comm[of x])
nipkow@28853
   685
qed
nipkow@28853
   686
nipkow@28853
   687
lemma fold_insert2:
nipkow@28853
   688
  "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
huffman@35216
   689
by (simp add: fold_fun_comm)
nipkow@15392
   690
haftmann@26041
   691
lemma fold_rec:
nipkow@28853
   692
assumes "finite A" and "x \<in> A"
nipkow@28853
   693
shows "fold f z A = f x (fold f z (A - {x}))"
nipkow@28853
   694
proof -
nipkow@28853
   695
  have A: "A = insert x (A - {x})" using `x \<in> A` by blast
nipkow@28853
   696
  then have "fold f z A = fold f z (insert x (A - {x}))" by simp
nipkow@28853
   697
  also have "\<dots> = f x (fold f z (A - {x}))"
nipkow@28853
   698
    by (rule fold_insert) (simp add: `finite A`)+
nipkow@15535
   699
  finally show ?thesis .
nipkow@15535
   700
qed
nipkow@15535
   701
nipkow@28853
   702
lemma fold_insert_remove:
nipkow@28853
   703
  assumes "finite A"
nipkow@28853
   704
  shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
nipkow@28853
   705
proof -
nipkow@28853
   706
  from `finite A` have "finite (insert x A)" by auto
nipkow@28853
   707
  moreover have "x \<in> insert x A" by auto
nipkow@28853
   708
  ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
nipkow@28853
   709
    by (rule fold_rec)
nipkow@28853
   710
  then show ?thesis by simp
nipkow@28853
   711
qed
nipkow@28853
   712
haftmann@26041
   713
end
nipkow@15392
   714
nipkow@15480
   715
text{* A simplified version for idempotent functions: *}
nipkow@15480
   716
nipkow@28853
   717
locale fun_left_comm_idem = fun_left_comm +
nipkow@28853
   718
  assumes fun_left_idem: "f x (f x z) = f x z"
haftmann@26041
   719
begin
haftmann@26041
   720
nipkow@28853
   721
text{* The nice version: *}
nipkow@28853
   722
lemma fun_comp_idem : "f x o f x = f x"
nipkow@39302
   723
by (simp add: fun_left_idem fun_eq_iff)
nipkow@28853
   724
haftmann@26041
   725
lemma fold_insert_idem:
nipkow@28853
   726
  assumes fin: "finite A"
nipkow@28853
   727
  shows "fold f z (insert x A) = f x (fold f z A)"
nipkow@15480
   728
proof cases
nipkow@28853
   729
  assume "x \<in> A"
nipkow@28853
   730
  then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
nipkow@28853
   731
  then show ?thesis using assms by (simp add:fun_left_idem)
nipkow@15480
   732
next
nipkow@28853
   733
  assume "x \<notin> A" then show ?thesis using assms by simp
nipkow@15480
   734
qed
nipkow@15480
   735
nipkow@28853
   736
declare fold_insert[simp del] fold_insert_idem[simp]
nipkow@28853
   737
nipkow@28853
   738
lemma fold_insert_idem2:
nipkow@28853
   739
  "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
nipkow@28853
   740
by(simp add:fold_fun_comm)
nipkow@15484
   741
haftmann@26041
   742
end
haftmann@26041
   743
haftmann@35817
   744
haftmann@35817
   745
subsubsection {* Expressing set operations via @{const fold} *}
haftmann@35817
   746
haftmann@35817
   747
lemma (in fun_left_comm) fun_left_comm_apply:
haftmann@35817
   748
  "fun_left_comm (\<lambda>x. f (g x))"
haftmann@35817
   749
proof
haftmann@35817
   750
qed (simp_all add: fun_left_comm)
haftmann@35817
   751
haftmann@35817
   752
lemma (in fun_left_comm_idem) fun_left_comm_idem_apply:
haftmann@35817
   753
  "fun_left_comm_idem (\<lambda>x. f (g x))"
haftmann@35817
   754
  by (rule fun_left_comm_idem.intro, rule fun_left_comm_apply, unfold_locales)
haftmann@35817
   755
    (simp_all add: fun_left_idem)
haftmann@35817
   756
haftmann@35817
   757
lemma fun_left_comm_idem_insert:
haftmann@35817
   758
  "fun_left_comm_idem insert"
haftmann@35817
   759
proof
haftmann@35817
   760
qed auto
haftmann@35817
   761
haftmann@35817
   762
lemma fun_left_comm_idem_remove:
haftmann@35817
   763
  "fun_left_comm_idem (\<lambda>x A. A - {x})"
haftmann@35817
   764
proof
haftmann@35817
   765
qed auto
nipkow@31992
   766
haftmann@35817
   767
lemma (in semilattice_inf) fun_left_comm_idem_inf:
haftmann@35817
   768
  "fun_left_comm_idem inf"
haftmann@35817
   769
proof
haftmann@35817
   770
qed (auto simp add: inf_left_commute)
haftmann@35817
   771
haftmann@35817
   772
lemma (in semilattice_sup) fun_left_comm_idem_sup:
haftmann@35817
   773
  "fun_left_comm_idem sup"
haftmann@35817
   774
proof
haftmann@35817
   775
qed (auto simp add: sup_left_commute)
nipkow@31992
   776
haftmann@35817
   777
lemma union_fold_insert:
haftmann@35817
   778
  assumes "finite A"
haftmann@35817
   779
  shows "A \<union> B = fold insert B A"
haftmann@35817
   780
proof -
haftmann@35817
   781
  interpret fun_left_comm_idem insert by (fact fun_left_comm_idem_insert)
haftmann@35817
   782
  from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
haftmann@35817
   783
qed
nipkow@31992
   784
haftmann@35817
   785
lemma minus_fold_remove:
haftmann@35817
   786
  assumes "finite A"
haftmann@35817
   787
  shows "B - A = fold (\<lambda>x A. A - {x}) B A"
haftmann@35817
   788
proof -
haftmann@35817
   789
  interpret fun_left_comm_idem "\<lambda>x A. A - {x}" by (fact fun_left_comm_idem_remove)
haftmann@35817
   790
  from `finite A` show ?thesis by (induct A arbitrary: B) auto
haftmann@35817
   791
qed
haftmann@35817
   792
haftmann@35817
   793
context complete_lattice
nipkow@31992
   794
begin
nipkow@31992
   795
haftmann@35817
   796
lemma inf_Inf_fold_inf:
haftmann@35817
   797
  assumes "finite A"
haftmann@35817
   798
  shows "inf B (Inf A) = fold inf B A"
haftmann@35817
   799
proof -
haftmann@35817
   800
  interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf)
haftmann@35817
   801
  from `finite A` show ?thesis by (induct A arbitrary: B)
haftmann@35817
   802
    (simp_all add: Inf_empty Inf_insert inf_commute fold_fun_comm)
haftmann@35817
   803
qed
nipkow@31992
   804
haftmann@35817
   805
lemma sup_Sup_fold_sup:
haftmann@35817
   806
  assumes "finite A"
haftmann@35817
   807
  shows "sup B (Sup A) = fold sup B A"
haftmann@35817
   808
proof -
haftmann@35817
   809
  interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)
haftmann@35817
   810
  from `finite A` show ?thesis by (induct A arbitrary: B)
haftmann@35817
   811
    (simp_all add: Sup_empty Sup_insert sup_commute fold_fun_comm)
nipkow@31992
   812
qed
nipkow@31992
   813
haftmann@35817
   814
lemma Inf_fold_inf:
haftmann@35817
   815
  assumes "finite A"
haftmann@35817
   816
  shows "Inf A = fold inf top A"
haftmann@35817
   817
  using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
haftmann@35817
   818
haftmann@35817
   819
lemma Sup_fold_sup:
haftmann@35817
   820
  assumes "finite A"
haftmann@35817
   821
  shows "Sup A = fold sup bot A"
haftmann@35817
   822
  using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
nipkow@31992
   823
haftmann@35817
   824
lemma inf_INFI_fold_inf:
haftmann@35817
   825
  assumes "finite A"
haftmann@35817
   826
  shows "inf B (INFI A f) = fold (\<lambda>A. inf (f A)) B A" (is "?inf = ?fold") 
haftmann@35817
   827
proof (rule sym)
haftmann@35817
   828
  interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf)
haftmann@35817
   829
  interpret fun_left_comm_idem "\<lambda>A. inf (f A)" by (fact fun_left_comm_idem_apply)
haftmann@35817
   830
  from `finite A` show "?fold = ?inf"
haftmann@35817
   831
  by (induct A arbitrary: B)
haftmann@35817
   832
    (simp_all add: INFI_def Inf_empty Inf_insert inf_left_commute)
haftmann@35817
   833
qed
nipkow@31992
   834
haftmann@35817
   835
lemma sup_SUPR_fold_sup:
haftmann@35817
   836
  assumes "finite A"
haftmann@35817
   837
  shows "sup B (SUPR A f) = fold (\<lambda>A. sup (f A)) B A" (is "?sup = ?fold") 
haftmann@35817
   838
proof (rule sym)
haftmann@35817
   839
  interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)
haftmann@35817
   840
  interpret fun_left_comm_idem "\<lambda>A. sup (f A)" by (fact fun_left_comm_idem_apply)
haftmann@35817
   841
  from `finite A` show "?fold = ?sup"
haftmann@35817
   842
  by (induct A arbitrary: B)
haftmann@35817
   843
    (simp_all add: SUPR_def Sup_empty Sup_insert sup_left_commute)
haftmann@35817
   844
qed
nipkow@31992
   845
haftmann@35817
   846
lemma INFI_fold_inf:
haftmann@35817
   847
  assumes "finite A"
haftmann@35817
   848
  shows "INFI A f = fold (\<lambda>A. inf (f A)) top A"
haftmann@35817
   849
  using assms inf_INFI_fold_inf [of A top] by simp
nipkow@31992
   850
haftmann@35817
   851
lemma SUPR_fold_sup:
haftmann@35817
   852
  assumes "finite A"
haftmann@35817
   853
  shows "SUPR A f = fold (\<lambda>A. sup (f A)) bot A"
haftmann@35817
   854
  using assms sup_SUPR_fold_sup [of A bot] by simp
nipkow@31992
   855
nipkow@31992
   856
end
nipkow@31992
   857
nipkow@31992
   858
haftmann@35817
   859
subsection {* The derived combinator @{text fold_image} *}
nipkow@28853
   860
nipkow@28853
   861
definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
nipkow@28853
   862
where "fold_image f g = fold (%x y. f (g x) y)"
nipkow@28853
   863
nipkow@28853
   864
lemma fold_image_empty[simp]: "fold_image f g z {} = z"
nipkow@28853
   865
by(simp add:fold_image_def)
nipkow@15392
   866
haftmann@26041
   867
context ab_semigroup_mult
haftmann@26041
   868
begin
haftmann@26041
   869
nipkow@28853
   870
lemma fold_image_insert[simp]:
nipkow@28853
   871
assumes "finite A" and "a \<notin> A"
nipkow@28853
   872
shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
nipkow@28853
   873
proof -
ballarin@29223
   874
  interpret I: fun_left_comm "%x y. (g x) * y"
nipkow@28853
   875
    by unfold_locales (simp add: mult_ac)
nipkow@31992
   876
  show ?thesis using assms by(simp add:fold_image_def)
nipkow@28853
   877
qed
nipkow@28853
   878
nipkow@28853
   879
(*
haftmann@26041
   880
lemma fold_commute:
haftmann@26041
   881
  "finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)"
berghofe@22262
   882
  apply (induct set: finite)
wenzelm@21575
   883
   apply simp
haftmann@26041
   884
  apply (simp add: mult_left_commute [of x])
nipkow@15392
   885
  done
nipkow@15392
   886
haftmann@26041
   887
lemma fold_nest_Un_Int:
nipkow@15392
   888
  "finite A ==> finite B
haftmann@26041
   889
    ==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)"
berghofe@22262
   890
  apply (induct set: finite)
wenzelm@21575
   891
   apply simp
nipkow@15392
   892
  apply (simp add: fold_commute Int_insert_left insert_absorb)
nipkow@15392
   893
  done
nipkow@15392
   894
haftmann@26041
   895
lemma fold_nest_Un_disjoint:
nipkow@15392
   896
  "finite A ==> finite B ==> A Int B = {}
haftmann@26041
   897
    ==> fold times g z (A Un B) = fold times g (fold times g z B) A"
nipkow@15392
   898
  by (simp add: fold_nest_Un_Int)
nipkow@28853
   899
*)
nipkow@28853
   900
nipkow@28853
   901
lemma fold_image_reindex:
paulson@15487
   902
assumes fin: "finite A"
nipkow@28853
   903
shows "inj_on h A \<Longrightarrow> fold_image times g z (h`A) = fold_image times (g\<circ>h) z A"
nipkow@31992
   904
using fin by induct auto
nipkow@15392
   905
nipkow@28853
   906
(*
haftmann@26041
   907
text{*
haftmann@26041
   908
  Fusion theorem, as described in Graham Hutton's paper,
haftmann@26041
   909
  A Tutorial on the Universality and Expressiveness of Fold,
haftmann@26041
   910
  JFP 9:4 (355-372), 1999.
haftmann@26041
   911
*}
haftmann@26041
   912
haftmann@26041
   913
lemma fold_fusion:
ballarin@27611
   914
  assumes "ab_semigroup_mult g"
haftmann@26041
   915
  assumes fin: "finite A"
haftmann@26041
   916
    and hyp: "\<And>x y. h (g x y) = times x (h y)"
haftmann@26041
   917
  shows "h (fold g j w A) = fold times j (h w) A"
ballarin@27611
   918
proof -
ballarin@29223
   919
  class_interpret ab_semigroup_mult [g] by fact
ballarin@27611
   920
  show ?thesis using fin hyp by (induct set: finite) simp_all
ballarin@27611
   921
qed
nipkow@28853
   922
*)
nipkow@28853
   923
nipkow@28853
   924
lemma fold_image_cong:
nipkow@28853
   925
  "finite A \<Longrightarrow>
nipkow@28853
   926
  (!!x. x:A ==> g x = h x) ==> fold_image times g z A = fold_image times h z A"
nipkow@28853
   927
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
   928
 apply simp
nipkow@28853
   929
apply (erule finite_induct, simp)
nipkow@28853
   930
apply (simp add: subset_insert_iff, clarify)
nipkow@28853
   931
apply (subgoal_tac "finite C")
nipkow@28853
   932
 prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
nipkow@28853
   933
apply (subgoal_tac "C = insert x (C - {x})")
nipkow@28853
   934
 prefer 2 apply blast
nipkow@28853
   935
apply (erule ssubst)
nipkow@28853
   936
apply (drule spec)
nipkow@28853
   937
apply (erule (1) notE impE)
nipkow@28853
   938
apply (simp add: Ball_def del: insert_Diff_single)
nipkow@28853
   939
done
nipkow@15392
   940
haftmann@26041
   941
end
haftmann@26041
   942
haftmann@26041
   943
context comm_monoid_mult
haftmann@26041
   944
begin
haftmann@26041
   945
haftmann@35817
   946
lemma fold_image_1:
haftmann@35817
   947
  "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
haftmann@35817
   948
  apply (induct set: finite)
haftmann@35817
   949
  apply simp by auto
haftmann@35817
   950
nipkow@28853
   951
lemma fold_image_Un_Int:
haftmann@26041
   952
  "finite A ==> finite B ==>
nipkow@28853
   953
    fold_image times g 1 A * fold_image times g 1 B =
nipkow@28853
   954
    fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
nipkow@28853
   955
by (induct set: finite) 
nipkow@28853
   956
   (auto simp add: mult_ac insert_absorb Int_insert_left)
haftmann@26041
   957
haftmann@35817
   958
lemma fold_image_Un_one:
haftmann@35817
   959
  assumes fS: "finite S" and fT: "finite T"
haftmann@35817
   960
  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
haftmann@35817
   961
  shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
haftmann@35817
   962
proof-
haftmann@35817
   963
  have "fold_image op * f 1 (S \<inter> T) = 1" 
haftmann@35817
   964
    apply (rule fold_image_1)
haftmann@35817
   965
    using fS fT I0 by auto 
haftmann@35817
   966
  with fold_image_Un_Int[OF fS fT] show ?thesis by simp
haftmann@35817
   967
qed
haftmann@35817
   968
haftmann@26041
   969
corollary fold_Un_disjoint:
haftmann@26041
   970
  "finite A ==> finite B ==> A Int B = {} ==>
nipkow@28853
   971
   fold_image times g 1 (A Un B) =
nipkow@28853
   972
   fold_image times g 1 A * fold_image times g 1 B"
nipkow@28853
   973
by (simp add: fold_image_Un_Int)
nipkow@28853
   974
nipkow@28853
   975
lemma fold_image_UN_disjoint:
haftmann@26041
   976
  "\<lbrakk> finite I; ALL i:I. finite (A i);
haftmann@26041
   977
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
nipkow@28853
   978
   \<Longrightarrow> fold_image times g 1 (UNION I A) =
nipkow@28853
   979
       fold_image times (%i. fold_image times g 1 (A i)) 1 I"
nipkow@28853
   980
apply (induct set: finite, simp, atomize)
nipkow@28853
   981
apply (subgoal_tac "ALL i:F. x \<noteq> i")
nipkow@28853
   982
 prefer 2 apply blast
nipkow@28853
   983
apply (subgoal_tac "A x Int UNION F A = {}")
nipkow@28853
   984
 prefer 2 apply blast
nipkow@28853
   985
apply (simp add: fold_Un_disjoint)
nipkow@28853
   986
done
nipkow@28853
   987
nipkow@28853
   988
lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@28853
   989
  fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
nipkow@28853
   990
  fold_image times (split g) 1 (SIGMA x:A. B x)"
nipkow@15392
   991
apply (subst Sigma_def)
nipkow@28853
   992
apply (subst fold_image_UN_disjoint, assumption, simp)
nipkow@15392
   993
 apply blast
nipkow@28853
   994
apply (erule fold_image_cong)
nipkow@28853
   995
apply (subst fold_image_UN_disjoint, simp, simp)
nipkow@15392
   996
 apply blast
paulson@15506
   997
apply simp
nipkow@15392
   998
done
nipkow@15392
   999
nipkow@28853
  1000
lemma fold_image_distrib: "finite A \<Longrightarrow>
nipkow@28853
  1001
   fold_image times (%x. g x * h x) 1 A =
nipkow@28853
  1002
   fold_image times g 1 A *  fold_image times h 1 A"
nipkow@28853
  1003
by (erule finite_induct) (simp_all add: mult_ac)
haftmann@26041
  1004
chaieb@30260
  1005
lemma fold_image_related: 
chaieb@30260
  1006
  assumes Re: "R e e" 
chaieb@30260
  1007
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
chaieb@30260
  1008
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
chaieb@30260
  1009
  shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
chaieb@30260
  1010
  using fS by (rule finite_subset_induct) (insert assms, auto)
chaieb@30260
  1011
chaieb@30260
  1012
lemma  fold_image_eq_general:
chaieb@30260
  1013
  assumes fS: "finite S"
chaieb@30260
  1014
  and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 
chaieb@30260
  1015
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
chaieb@30260
  1016
  shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
chaieb@30260
  1017
proof-
chaieb@30260
  1018
  from h f12 have hS: "h ` S = S'" by auto
chaieb@30260
  1019
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
chaieb@30260
  1020
    from f12 h H  have "x = y" by auto }
chaieb@30260
  1021
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
chaieb@30260
  1022
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
chaieb@30260
  1023
  from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
chaieb@30260
  1024
  also have "\<dots> = fold_image (op *) (f2 o h) e S" 
chaieb@30260
  1025
    using fold_image_reindex[OF fS hinj, of f2 e] .
chaieb@30260
  1026
  also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
chaieb@30260
  1027
    by blast
chaieb@30260
  1028
  finally show ?thesis ..
chaieb@30260
  1029
qed
chaieb@30260
  1030
chaieb@30260
  1031
lemma fold_image_eq_general_inverses:
chaieb@30260
  1032
  assumes fS: "finite S" 
chaieb@30260
  1033
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
chaieb@30260
  1034
  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
  1035
  shows "fold_image (op *) f e S = fold_image (op *) g e T"
chaieb@30260
  1036
  (* metis solves it, but not yet available here *)
chaieb@30260
  1037
  apply (rule fold_image_eq_general[OF fS, of T h g f e])
chaieb@30260
  1038
  apply (rule ballI)
chaieb@30260
  1039
  apply (frule kh)
chaieb@30260
  1040
  apply (rule ex1I[])
chaieb@30260
  1041
  apply blast
chaieb@30260
  1042
  apply clarsimp
chaieb@30260
  1043
  apply (drule hk) apply simp
chaieb@30260
  1044
  apply (rule sym)
chaieb@30260
  1045
  apply (erule conjunct1[OF conjunct2[OF hk]])
chaieb@30260
  1046
  apply (rule ballI)
chaieb@30260
  1047
  apply (drule  hk)
chaieb@30260
  1048
  apply blast
chaieb@30260
  1049
  done
chaieb@30260
  1050
haftmann@26041
  1051
end
haftmann@22917
  1052
nipkow@25162
  1053
haftmann@35817
  1054
subsection {* A fold functional for non-empty sets *}
nipkow@15392
  1055
nipkow@15392
  1056
text{* Does not require start value. *}
wenzelm@12396
  1057
berghofe@23736
  1058
inductive
berghofe@22262
  1059
  fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
berghofe@22262
  1060
  for f :: "'a => 'a => 'a"
berghofe@22262
  1061
where
paulson@15506
  1062
  fold1Set_insertI [intro]:
nipkow@28853
  1063
   "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
wenzelm@12396
  1064
haftmann@35416
  1065
definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
berghofe@22262
  1066
  "fold1 f A == THE x. fold1Set f A x"
paulson@15506
  1067
paulson@15506
  1068
lemma fold1Set_nonempty:
haftmann@22917
  1069
  "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
nipkow@28853
  1070
by(erule fold1Set.cases, simp_all)
nipkow@15392
  1071
berghofe@23736
  1072
inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
berghofe@23736
  1073
berghofe@23736
  1074
inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
berghofe@22262
  1075
berghofe@22262
  1076
berghofe@22262
  1077
lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
huffman@35216
  1078
by (blast elim: fold_graph.cases)
nipkow@15392
  1079
haftmann@22917
  1080
lemma fold1_singleton [simp]: "fold1 f {a} = a"
nipkow@28853
  1081
by (unfold fold1_def) blast
wenzelm@12396
  1082
paulson@15508
  1083
lemma finite_nonempty_imp_fold1Set:
berghofe@22262
  1084
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
paulson@15508
  1085
apply (induct A rule: finite_induct)
nipkow@28853
  1086
apply (auto dest: finite_imp_fold_graph [of _ f])
paulson@15508
  1087
done
paulson@15506
  1088
nipkow@28853
  1089
text{*First, some lemmas about @{const fold_graph}.*}
nipkow@15392
  1090
haftmann@26041
  1091
context ab_semigroup_mult
haftmann@26041
  1092
begin
haftmann@26041
  1093
nipkow@28853
  1094
lemma fun_left_comm: "fun_left_comm(op *)"
nipkow@28853
  1095
by unfold_locales (simp add: mult_ac)
nipkow@28853
  1096
nipkow@28853
  1097
lemma fold_graph_insert_swap:
nipkow@28853
  1098
assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
nipkow@28853
  1099
shows "fold_graph times z (insert b A) (z * y)"
nipkow@28853
  1100
proof -
ballarin@29223
  1101
  interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
nipkow@28853
  1102
from assms show ?thesis
nipkow@28853
  1103
proof (induct rule: fold_graph.induct)
huffman@36045
  1104
  case emptyI show ?case by (subst mult_commute [of z b], fast)
paulson@15508
  1105
next
berghofe@22262
  1106
  case (insertI x A y)
nipkow@28853
  1107
    have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
paulson@15521
  1108
      using insertI by force  --{*how does @{term id} get unfolded?*}
haftmann@26041
  1109
    thus ?case by (simp add: insert_commute mult_ac)
paulson@15508
  1110
qed
nipkow@28853
  1111
qed
nipkow@28853
  1112
nipkow@28853
  1113
lemma fold_graph_permute_diff:
nipkow@28853
  1114
assumes fold: "fold_graph times b A x"
nipkow@28853
  1115
shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
paulson@15508
  1116
using fold
nipkow@28853
  1117
proof (induct rule: fold_graph.induct)
paulson@15508
  1118
  case emptyI thus ?case by simp
paulson@15508
  1119
next
berghofe@22262
  1120
  case (insertI x A y)
paulson@15521
  1121
  have "a = x \<or> a \<in> A" using insertI by simp
paulson@15521
  1122
  thus ?case
paulson@15521
  1123
  proof
paulson@15521
  1124
    assume "a = x"
paulson@15521
  1125
    with insertI show ?thesis
nipkow@28853
  1126
      by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
paulson@15521
  1127
  next
paulson@15521
  1128
    assume ainA: "a \<in> A"
nipkow@28853
  1129
    hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
nipkow@28853
  1130
      using insertI by force
paulson@15521
  1131
    moreover
paulson@15521
  1132
    have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
paulson@15521
  1133
      using ainA insertI by blast
nipkow@28853
  1134
    ultimately show ?thesis by simp
paulson@15508
  1135
  qed
paulson@15508
  1136
qed
paulson@15508
  1137
haftmann@26041
  1138
lemma fold1_eq_fold:
nipkow@28853
  1139
assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
nipkow@28853
  1140
proof -
ballarin@29223
  1141
  interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
nipkow@28853
  1142
  from assms show ?thesis
nipkow@28853
  1143
apply (simp add: fold1_def fold_def)
paulson@15508
  1144
apply (rule the_equality)
nipkow@28853
  1145
apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
paulson@15508
  1146
apply (rule sym, clarify)
paulson@15508
  1147
apply (case_tac "Aa=A")
huffman@35216
  1148
 apply (best intro: fold_graph_determ)
nipkow@28853
  1149
apply (subgoal_tac "fold_graph times a A x")
huffman@35216
  1150
 apply (best intro: fold_graph_determ)
nipkow@28853
  1151
apply (subgoal_tac "insert aa (Aa - {a}) = A")
nipkow@28853
  1152
 prefer 2 apply (blast elim: equalityE)
nipkow@28853
  1153
apply (auto dest: fold_graph_permute_diff [where a=a])
paulson@15508
  1154
done
nipkow@28853
  1155
qed
paulson@15508
  1156
paulson@15521
  1157
lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
paulson@15521
  1158
apply safe
nipkow@28853
  1159
 apply simp
nipkow@28853
  1160
 apply (drule_tac x=x in spec)
nipkow@28853
  1161
 apply (drule_tac x="A-{x}" in spec, auto)
paulson@15508
  1162
done
paulson@15508
  1163
haftmann@26041
  1164
lemma fold1_insert:
paulson@15521
  1165
  assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
haftmann@26041
  1166
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1167
proof -
ballarin@29223
  1168
  interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
nipkow@28853
  1169
  from nonempty obtain a A' where "A = insert a A' & a ~: A'"
paulson@15521
  1170
    by (auto simp add: nonempty_iff)
paulson@15521
  1171
  with A show ?thesis
nipkow@28853
  1172
    by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
paulson@15521
  1173
qed
paulson@15521
  1174
haftmann@26041
  1175
end
haftmann@26041
  1176
haftmann@26041
  1177
context ab_semigroup_idem_mult
haftmann@26041
  1178
begin
haftmann@26041
  1179
haftmann@35817
  1180
lemma fun_left_comm_idem: "fun_left_comm_idem(op *)"
haftmann@35817
  1181
apply unfold_locales
haftmann@35817
  1182
 apply (rule mult_left_commute)
haftmann@35817
  1183
apply (rule mult_left_idem)
haftmann@35817
  1184
done
haftmann@35817
  1185
haftmann@26041
  1186
lemma fold1_insert_idem [simp]:
paulson@15521
  1187
  assumes nonempty: "A \<noteq> {}" and A: "finite A" 
haftmann@26041
  1188
  shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521
  1189
proof -
ballarin@29223
  1190
  interpret fun_left_comm_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
nipkow@28853
  1191
    by (rule fun_left_comm_idem)
nipkow@28853
  1192
  from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
paulson@15521
  1193
    by (auto simp add: nonempty_iff)
paulson@15521
  1194
  show ?thesis
paulson@15521
  1195
  proof cases
paulson@15521
  1196
    assume "a = x"
nipkow@28853
  1197
    thus ?thesis
paulson@15521
  1198
    proof cases
paulson@15521
  1199
      assume "A' = {}"
huffman@35216
  1200
      with prems show ?thesis by simp
paulson@15521
  1201
    next
paulson@15521
  1202
      assume "A' \<noteq> {}"
paulson@15521
  1203
      with prems show ?thesis
huffman@35216
  1204
        by (simp add: fold1_insert mult_assoc [symmetric])
paulson@15521
  1205
    qed
paulson@15521
  1206
  next
paulson@15521
  1207
    assume "a \<noteq> x"
paulson@15521
  1208
    with prems show ?thesis
huffman@35216
  1209
      by (simp add: insert_commute fold1_eq_fold)
paulson@15521
  1210
  qed
paulson@15521
  1211
qed
paulson@15506
  1212
haftmann@26041
  1213
lemma hom_fold1_commute:
haftmann@26041
  1214
assumes hom: "!!x y. h (x * y) = h x * h y"
haftmann@26041
  1215
and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
haftmann@22917
  1216
using N proof (induct rule: finite_ne_induct)
haftmann@22917
  1217
  case singleton thus ?case by simp
haftmann@22917
  1218
next
haftmann@22917
  1219
  case (insert n N)
haftmann@26041
  1220
  then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
haftmann@26041
  1221
  also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
haftmann@26041
  1222
  also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
haftmann@26041
  1223
  also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
haftmann@22917
  1224
    using insert by(simp)
haftmann@22917
  1225
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@22917
  1226
  finally show ?case .
haftmann@22917
  1227
qed
haftmann@22917
  1228
haftmann@32679
  1229
lemma fold1_eq_fold_idem:
haftmann@32679
  1230
  assumes "finite A"
haftmann@32679
  1231
  shows "fold1 times (insert a A) = fold times a A"
haftmann@32679
  1232
proof (cases "a \<in> A")
haftmann@32679
  1233
  case False
haftmann@32679
  1234
  with assms show ?thesis by (simp add: fold1_eq_fold)
haftmann@32679
  1235
next
haftmann@32679
  1236
  interpret fun_left_comm_idem times by (fact fun_left_comm_idem)
haftmann@32679
  1237
  case True then obtain b B
haftmann@32679
  1238
    where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
haftmann@32679
  1239
  with assms have "finite B" by auto
haftmann@32679
  1240
  then have "fold times a (insert a B) = fold times (a * a) B"
haftmann@32679
  1241
    using `a \<notin> B` by (rule fold_insert2)
haftmann@32679
  1242
  then show ?thesis
haftmann@32679
  1243
    using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
haftmann@32679
  1244
qed
haftmann@32679
  1245
haftmann@26041
  1246
end
haftmann@26041
  1247
paulson@15506
  1248
paulson@15508
  1249
text{* Now the recursion rules for definitions: *}
paulson@15508
  1250
haftmann@22917
  1251
lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
huffman@35216
  1252
by simp
paulson@15508
  1253
haftmann@26041
  1254
lemma (in ab_semigroup_mult) fold1_insert_def:
haftmann@26041
  1255
  "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1256
by (simp add:fold1_insert)
haftmann@26041
  1257
haftmann@26041
  1258
lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
haftmann@26041
  1259
  "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041
  1260
by simp
paulson@15508
  1261
paulson@15508
  1262
subsubsection{* Determinacy for @{term fold1Set} *}
paulson@15508
  1263
nipkow@28853
  1264
(*Not actually used!!*)
nipkow@28853
  1265
(*
haftmann@26041
  1266
context ab_semigroup_mult
haftmann@26041
  1267
begin
haftmann@26041
  1268
nipkow@28853
  1269
lemma fold_graph_permute:
nipkow@28853
  1270
  "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
nipkow@28853
  1271
   ==> fold_graph times id a (insert b A) x"
haftmann@26041
  1272
apply (cases "a=b") 
nipkow@28853
  1273
apply (auto dest: fold_graph_permute_diff) 
paulson@15506
  1274
done
nipkow@15376
  1275
haftmann@26041
  1276
lemma fold1Set_determ:
haftmann@26041
  1277
  "fold1Set times A x ==> fold1Set times A y ==> y = x"
paulson@15506
  1278
proof (clarify elim!: fold1Set.cases)
paulson@15506
  1279
  fix A x B y a b
nipkow@28853
  1280
  assume Ax: "fold_graph times id a A x"
nipkow@28853
  1281
  assume By: "fold_graph times id b B y"
paulson@15506
  1282
  assume anotA:  "a \<notin> A"
paulson@15506
  1283
  assume bnotB:  "b \<notin> B"
paulson@15506
  1284
  assume eq: "insert a A = insert b B"
paulson@15506
  1285
  show "y=x"
paulson@15506
  1286
  proof cases
paulson@15506
  1287
    assume same: "a=b"
paulson@15506
  1288
    hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
nipkow@28853
  1289
    thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
nipkow@15392
  1290
  next
paulson@15506
  1291
    assume diff: "a\<noteq>b"
paulson@15506
  1292
    let ?D = "B - {a}"
paulson@15506
  1293
    have B: "B = insert a ?D" and A: "A = insert b ?D"
paulson@15506
  1294
     and aB: "a \<in> B" and bA: "b \<in> A"
paulson@15506
  1295
      using eq anotA bnotB diff by (blast elim!:equalityE)+
paulson@15506
  1296
    with aB bnotB By
nipkow@28853
  1297
    have "fold_graph times id a (insert b ?D) y" 
nipkow@28853
  1298
      by (auto intro: fold_graph_permute simp add: insert_absorb)
paulson@15506
  1299
    moreover
nipkow@28853
  1300
    have "fold_graph times id a (insert b ?D) x"
paulson@15506
  1301
      by (simp add: A [symmetric] Ax) 
nipkow@28853
  1302
    ultimately show ?thesis by (blast intro: fold_graph_determ) 
nipkow@15392
  1303
  qed
wenzelm@12396
  1304
qed
wenzelm@12396
  1305
haftmann@26041
  1306
lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
paulson@15506
  1307
  by (unfold fold1_def) (blast intro: fold1Set_determ)
paulson@15506
  1308
haftmann@26041
  1309
end
nipkow@28853
  1310
*)
haftmann@26041
  1311
paulson@15506
  1312
declare
nipkow@28853
  1313
  empty_fold_graphE [rule del]  fold_graph.intros [rule del]
paulson@15506
  1314
  empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
ballarin@19931
  1315
  -- {* No more proofs involve these relations. *}
nipkow@15376
  1316
haftmann@26041
  1317
subsubsection {* Lemmas about @{text fold1} *}
haftmann@26041
  1318
haftmann@26041
  1319
context ab_semigroup_mult
haftmann@22917
  1320
begin
haftmann@22917
  1321
haftmann@26041
  1322
lemma fold1_Un:
nipkow@15484
  1323
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  1324
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
haftmann@26041
  1325
       fold1 times (A Un B) = fold1 times A * fold1 times B"
haftmann@26041
  1326
using A by (induct rule: finite_ne_induct)
haftmann@26041
  1327
  (simp_all add: fold1_insert mult_assoc)
haftmann@26041
  1328
haftmann@26041
  1329
lemma fold1_in:
haftmann@26041
  1330
  assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
haftmann@26041
  1331
  shows "fold1 times A \<in> A"
nipkow@15484
  1332
using A
nipkow@15484
  1333
proof (induct rule:finite_ne_induct)
paulson@15506
  1334
  case singleton thus ?case by simp
nipkow@15484
  1335
next
nipkow@15484
  1336
  case insert thus ?case using elem by (force simp add:fold1_insert)
nipkow@15484
  1337
qed
nipkow@15484
  1338
haftmann@26041
  1339
end
haftmann@26041
  1340
haftmann@26041
  1341
lemma (in ab_semigroup_idem_mult) fold1_Un2:
nipkow@15497
  1342
assumes A: "finite A" "A \<noteq> {}"
haftmann@26041
  1343
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
haftmann@26041
  1344
       fold1 times (A Un B) = fold1 times A * fold1 times B"
nipkow@15497
  1345
using A
haftmann@26041
  1346
proof(induct rule:finite_ne_induct)
nipkow@15497
  1347
  case singleton thus ?case by simp
nipkow@15484
  1348
next
haftmann@26041
  1349
  case insert thus ?case by (simp add: mult_assoc)
nipkow@18423
  1350
qed
nipkow@18423
  1351
nipkow@18423
  1352
haftmann@35817
  1353
subsection {* Locales as mini-packages for fold operations *}
haftmann@34007
  1354
haftmann@35817
  1355
subsubsection {* The natural case *}
haftmann@35719
  1356
haftmann@35719
  1357
locale folding =
haftmann@35719
  1358
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@35719
  1359
  fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@35817
  1360
  assumes commute_comp: "f y \<circ> f x = f x \<circ> f y"
haftmann@35722
  1361
  assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
haftmann@35719
  1362
begin
haftmann@35719
  1363
haftmann@35719
  1364
lemma empty [simp]:
haftmann@35719
  1365
  "F {} = id"
nipkow@39302
  1366
  by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1367
haftmann@35719
  1368
lemma insert [simp]:
haftmann@35719
  1369
  assumes "finite A" and "x \<notin> A"
haftmann@35719
  1370
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35719
  1371
proof -
haftmann@35817
  1372
  interpret fun_left_comm f proof
nipkow@39302
  1373
  qed (insert commute_comp, simp add: fun_eq_iff)
haftmann@35719
  1374
  from fold_insert2 assms
haftmann@35722
  1375
  have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
nipkow@39302
  1376
  with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
haftmann@35719
  1377
qed
haftmann@35719
  1378
haftmann@35719
  1379
lemma remove:
haftmann@35719
  1380
  assumes "finite A" and "x \<in> A"
haftmann@35719
  1381
  shows "F A = F (A - {x}) \<circ> f x"
haftmann@35719
  1382
proof -
haftmann@35719
  1383
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35719
  1384
    by (auto dest: mk_disjoint_insert)
haftmann@35719
  1385
  moreover from `finite A` this have "finite B" by simp
haftmann@35719
  1386
  ultimately show ?thesis by simp
haftmann@35719
  1387
qed
haftmann@35719
  1388
haftmann@35719
  1389
lemma insert_remove:
haftmann@35719
  1390
  assumes "finite A"
haftmann@35719
  1391
  shows "F (insert x A) = F (A - {x}) \<circ> f x"
haftmann@35722
  1392
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35719
  1393
haftmann@35817
  1394
lemma commute_left_comp:
haftmann@35817
  1395
  "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
haftmann@35817
  1396
  by (simp add: o_assoc commute_comp)
haftmann@35817
  1397
haftmann@35719
  1398
lemma commute_comp':
haftmann@35719
  1399
  assumes "finite A"
haftmann@35719
  1400
  shows "f x \<circ> F A = F A \<circ> f x"
haftmann@35817
  1401
  using assms by (induct A)
haftmann@35817
  1402
    (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: o_assoc [symmetric] commute_comp)
haftmann@35817
  1403
haftmann@35817
  1404
lemma commute_left_comp':
haftmann@35817
  1405
  assumes "finite A"
haftmann@35817
  1406
  shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
haftmann@35817
  1407
  using assms by (simp add: o_assoc commute_comp')
haftmann@35817
  1408
haftmann@35817
  1409
lemma commute_comp'':
haftmann@35817
  1410
  assumes "finite A" and "finite B"
haftmann@35817
  1411
  shows "F B \<circ> F A = F A \<circ> F B"
haftmann@35817
  1412
  using assms by (induct A)
haftmann@35817
  1413
    (simp_all add: o_assoc, simp add: o_assoc [symmetric] commute_comp')
haftmann@35719
  1414
haftmann@35817
  1415
lemma commute_left_comp'':
haftmann@35817
  1416
  assumes "finite A" and "finite B"
haftmann@35817
  1417
  shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
haftmann@35817
  1418
  using assms by (simp add: o_assoc commute_comp'')
haftmann@35817
  1419
haftmann@35817
  1420
lemmas commute_comps = o_assoc [symmetric] commute_comp commute_left_comp
haftmann@35817
  1421
  commute_comp' commute_left_comp' commute_comp'' commute_left_comp''
haftmann@35817
  1422
haftmann@35817
  1423
lemma union_inter:
haftmann@35817
  1424
  assumes "finite A" and "finite B"
haftmann@35817
  1425
  shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
haftmann@35817
  1426
  using assms by (induct A)
haftmann@35817
  1427
    (simp_all del: o_apply add: insert_absorb Int_insert_left commute_comps,
haftmann@35817
  1428
      simp add: o_assoc)
haftmann@35719
  1429
haftmann@35719
  1430
lemma union:
haftmann@35719
  1431
  assumes "finite A" and "finite B"
haftmann@35719
  1432
  and "A \<inter> B = {}"
haftmann@35719
  1433
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1434
proof -
haftmann@35817
  1435
  from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
haftmann@35817
  1436
  with `A \<inter> B = {}` show ?thesis by simp
haftmann@35719
  1437
qed
haftmann@35719
  1438
haftmann@34007
  1439
end
haftmann@35719
  1440
haftmann@35817
  1441
haftmann@35817
  1442
subsubsection {* The natural case with idempotency *}
haftmann@35817
  1443
haftmann@35719
  1444
locale folding_idem = folding +
haftmann@35719
  1445
  assumes idem_comp: "f x \<circ> f x = f x"
haftmann@35719
  1446
begin
haftmann@35719
  1447
haftmann@35817
  1448
lemma idem_left_comp:
haftmann@35817
  1449
  "f x \<circ> (f x \<circ> g) = f x \<circ> g"
haftmann@35817
  1450
  by (simp add: o_assoc idem_comp)
haftmann@35817
  1451
haftmann@35817
  1452
lemma in_comp_idem:
haftmann@35817
  1453
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1454
  shows "F A \<circ> f x = F A"
haftmann@35817
  1455
using assms by (induct A)
haftmann@35817
  1456
  (auto simp add: commute_comps idem_comp, simp add: commute_left_comp' [symmetric] commute_comp')
haftmann@35719
  1457
haftmann@35817
  1458
lemma subset_comp_idem:
haftmann@35817
  1459
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1460
  shows "F A \<circ> F B = F A"
haftmann@35817
  1461
proof -
haftmann@35817
  1462
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1463
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1464
    (simp_all add: o_assoc in_comp_idem `finite A`)
haftmann@35817
  1465
qed
haftmann@35719
  1466
haftmann@35817
  1467
declare insert [simp del]
haftmann@35719
  1468
haftmann@35719
  1469
lemma insert_idem [simp]:
haftmann@35719
  1470
  assumes "finite A"
haftmann@35719
  1471
  shows "F (insert x A) = F A \<circ> f x"
haftmann@35817
  1472
  using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
haftmann@35719
  1473
haftmann@35719
  1474
lemma union_idem:
haftmann@35719
  1475
  assumes "finite A" and "finite B"
haftmann@35719
  1476
  shows "F (A \<union> B) = F A \<circ> F B"
haftmann@35817
  1477
proof -
haftmann@35817
  1478
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1479
  then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
haftmann@35817
  1480
  with assms show ?thesis by (simp add: union_inter)
haftmann@35719
  1481
qed
haftmann@35719
  1482
haftmann@35719
  1483
end
haftmann@35719
  1484
haftmann@35817
  1485
haftmann@35817
  1486
subsubsection {* The image case with fixed function *}
haftmann@35817
  1487
haftmann@35796
  1488
no_notation times (infixl "*" 70)
haftmann@35796
  1489
no_notation Groups.one ("1")
haftmann@35722
  1490
haftmann@35722
  1491
locale folding_image_simple = comm_monoid +
haftmann@35722
  1492
  fixes g :: "('b \<Rightarrow> 'a)"
haftmann@35722
  1493
  fixes F :: "'b set \<Rightarrow> 'a"
haftmann@35817
  1494
  assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
haftmann@35722
  1495
begin
haftmann@35722
  1496
haftmann@35722
  1497
lemma empty [simp]:
haftmann@35722
  1498
  "F {} = 1"
haftmann@35817
  1499
  by (simp add: eq_fold_g)
haftmann@35722
  1500
haftmann@35722
  1501
lemma insert [simp]:
haftmann@35722
  1502
  assumes "finite A" and "x \<notin> A"
haftmann@35722
  1503
  shows "F (insert x A) = g x * F A"
haftmann@35722
  1504
proof -
haftmann@35722
  1505
  interpret fun_left_comm "%x y. (g x) * y" proof
haftmann@35722
  1506
  qed (simp add: ac_simps)
haftmann@35722
  1507
  with assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
haftmann@35722
  1508
    by (simp add: fold_image_def)
haftmann@35817
  1509
  with `finite A` show ?thesis by (simp add: eq_fold_g)
haftmann@35722
  1510
qed
haftmann@35722
  1511
haftmann@35722
  1512
lemma remove:
haftmann@35722
  1513
  assumes "finite A" and "x \<in> A"
haftmann@35722
  1514
  shows "F A = g x * F (A - {x})"
haftmann@35722
  1515
proof -
haftmann@35722
  1516
  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@35722
  1517
    by (auto dest: mk_disjoint_insert)
haftmann@35722
  1518
  moreover from `finite A` this have "finite B" by simp
haftmann@35722
  1519
  ultimately show ?thesis by simp
haftmann@35722
  1520
qed
haftmann@35722
  1521
haftmann@35722
  1522
lemma insert_remove:
haftmann@35722
  1523
  assumes "finite A"
haftmann@35722
  1524
  shows "F (insert x A) = g x * F (A - {x})"
haftmann@35722
  1525
  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@35722
  1526
haftmann@35817
  1527
lemma neutral:
haftmann@35817
  1528
  assumes "finite A" and "\<forall>x\<in>A. g x = 1"
haftmann@35817
  1529
  shows "F A = 1"
haftmann@35817
  1530
  using assms by (induct A) simp_all
haftmann@35817
  1531
haftmann@35722
  1532
lemma union_inter:
haftmann@35722
  1533
  assumes "finite A" and "finite B"
haftmann@35817
  1534
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35722
  1535
using assms proof (induct A)
haftmann@35722
  1536
  case empty then show ?case by simp
haftmann@35722
  1537
next
haftmann@35722
  1538
  case (insert x A) then show ?case
haftmann@35722
  1539
    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
haftmann@35722
  1540
qed
haftmann@35722
  1541
haftmann@35817
  1542
corollary union_inter_neutral:
haftmann@35817
  1543
  assumes "finite A" and "finite B"
haftmann@35817
  1544
  and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
haftmann@35817
  1545
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1546
  using assms by (simp add: union_inter [symmetric] neutral)
haftmann@35817
  1547
haftmann@35722
  1548
corollary union_disjoint:
haftmann@35722
  1549
  assumes "finite A" and "finite B"
haftmann@35722
  1550
  assumes "A \<inter> B = {}"
haftmann@35722
  1551
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1552
  using assms by (simp add: union_inter_neutral)
haftmann@35722
  1553
haftmann@35719
  1554
end
haftmann@35722
  1555
haftmann@35817
  1556
haftmann@35817
  1557
subsubsection {* The image case with flexible function *}
haftmann@35817
  1558
haftmann@35722
  1559
locale folding_image = comm_monoid +
haftmann@35722
  1560
  fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@35722
  1561
  assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
haftmann@35722
  1562
haftmann@35722
  1563
sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
haftmann@35722
  1564
qed (fact eq_fold)
haftmann@35722
  1565
haftmann@35722
  1566
context folding_image
haftmann@35722
  1567
begin
haftmann@35722
  1568
haftmann@35817
  1569
lemma reindex: (* FIXME polymorhism *)
haftmann@35722
  1570
  assumes "finite A" and "inj_on h A"
haftmann@35722
  1571
  shows "F g (h ` A) = F (g \<circ> h) A"
haftmann@35722
  1572
  using assms by (induct A) auto
haftmann@35722
  1573
haftmann@35722
  1574
lemma cong:
haftmann@35722
  1575
  assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
haftmann@35722
  1576
  shows "F g A = F h A"
haftmann@35722
  1577
proof -
haftmann@35722
  1578
  from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
haftmann@35722
  1579
  apply - apply (erule finite_induct) apply simp
haftmann@35722
  1580
  apply (simp add: subset_insert_iff, clarify)
haftmann@35722
  1581
  apply (subgoal_tac "finite C")
haftmann@35722
  1582
  prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
haftmann@35722
  1583
  apply (subgoal_tac "C = insert x (C - {x})")
haftmann@35722
  1584
  prefer 2 apply blast
haftmann@35722
  1585
  apply (erule ssubst)
haftmann@35722
  1586
  apply (drule spec)
haftmann@35722
  1587
  apply (erule (1) notE impE)
haftmann@35722
  1588
  apply (simp add: Ball_def del: insert_Diff_single)
haftmann@35722
  1589
  done
haftmann@35722
  1590
  with assms show ?thesis by simp
haftmann@35722
  1591
qed
haftmann@35722
  1592
haftmann@35722
  1593
lemma UNION_disjoint:
haftmann@35722
  1594
  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
haftmann@35722
  1595
  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
haftmann@35722
  1596
  shows "F g (UNION I A) = F (F g \<circ> A) I"
haftmann@35722
  1597
apply (insert assms)
haftmann@35722
  1598
apply (induct set: finite, simp, atomize)
haftmann@35722
  1599
apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
haftmann@35722
  1600
 prefer 2 apply blast
haftmann@35722
  1601
apply (subgoal_tac "A x Int UNION Fa A = {}")
haftmann@35722
  1602
 prefer 2 apply blast
haftmann@35722
  1603
apply (simp add: union_disjoint)
haftmann@35722
  1604
done
haftmann@35722
  1605
haftmann@35722
  1606
lemma distrib:
haftmann@35722
  1607
  assumes "finite A"
haftmann@35722
  1608
  shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
haftmann@35722
  1609
  using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
haftmann@35722
  1610
haftmann@35722
  1611
lemma related: 
haftmann@35722
  1612
  assumes Re: "R 1 1" 
haftmann@35722
  1613
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
haftmann@35722
  1614
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
haftmann@35722
  1615
  shows "R (F h S) (F g S)"
haftmann@35722
  1616
  using fS by (rule finite_subset_induct) (insert assms, auto)
haftmann@35722
  1617
haftmann@35722
  1618
lemma eq_general:
haftmann@35722
  1619
  assumes fS: "finite S"
haftmann@35722
  1620
  and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 
haftmann@35722
  1621
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
haftmann@35722
  1622
  shows "F f1 S = F f2 S'"
haftmann@35722
  1623
proof-
haftmann@35722
  1624
  from h f12 have hS: "h ` S = S'" by blast
haftmann@35722
  1625
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
haftmann@35722
  1626
    from f12 h H  have "x = y" by auto }
haftmann@35722
  1627
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
haftmann@35722
  1628
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
haftmann@35722
  1629
  from hS have "F f2 S' = F f2 (h ` S)" by simp
haftmann@35722
  1630
  also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
haftmann@35722
  1631
  also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
haftmann@35722
  1632
    by blast
haftmann@35722
  1633
  finally show ?thesis ..
haftmann@35722
  1634
qed
haftmann@35722
  1635
haftmann@35722
  1636
lemma eq_general_inverses:
haftmann@35722
  1637
  assumes fS: "finite S" 
haftmann@35722
  1638
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
haftmann@35722
  1639
  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
  1640
  shows "F j S = F g T"
haftmann@35722
  1641
  (* metis solves it, but not yet available here *)
haftmann@35722
  1642
  apply (rule eq_general [OF fS, of T h g j])
haftmann@35722
  1643
  apply (rule ballI)
haftmann@35722
  1644
  apply (frule kh)
haftmann@35722
  1645
  apply (rule ex1I[])
haftmann@35722
  1646
  apply blast
haftmann@35722
  1647
  apply clarsimp
haftmann@35722
  1648
  apply (drule hk) apply simp
haftmann@35722
  1649
  apply (rule sym)
haftmann@35722
  1650
  apply (erule conjunct1[OF conjunct2[OF hk]])
haftmann@35722
  1651
  apply (rule ballI)
haftmann@35722
  1652
  apply (drule hk)
haftmann@35722
  1653
  apply blast
haftmann@35722
  1654
  done
haftmann@35722
  1655
haftmann@35722
  1656
end
haftmann@35722
  1657
haftmann@35817
  1658
haftmann@35817
  1659
subsubsection {* The image case with fixed function and idempotency *}
haftmann@35817
  1660
haftmann@35817
  1661
locale folding_image_simple_idem = folding_image_simple +
haftmann@35817
  1662
  assumes idem: "x * x = x"
haftmann@35817
  1663
haftmann@35817
  1664
sublocale folding_image_simple_idem < semilattice proof
haftmann@35817
  1665
qed (fact idem)
haftmann@35817
  1666
haftmann@35817
  1667
context folding_image_simple_idem
haftmann@35817
  1668
begin
haftmann@35817
  1669
haftmann@35817
  1670
lemma in_idem:
haftmann@35817
  1671
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1672
  shows "g x * F A = F A"
haftmann@35817
  1673
  using assms by (induct A) (auto simp add: left_commute)
haftmann@35817
  1674
haftmann@35817
  1675
lemma subset_idem:
haftmann@35817
  1676
  assumes "finite A" and "B \<subseteq> A"
haftmann@35817
  1677
  shows "F B * F A = F A"
haftmann@35817
  1678
proof -
haftmann@35817
  1679
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1680
  then show ?thesis using `B \<subseteq> A` by (induct B)
haftmann@35817
  1681
    (auto simp add: assoc in_idem `finite A`)
haftmann@35817
  1682
qed
haftmann@35817
  1683
haftmann@35817
  1684
declare insert [simp del]
haftmann@35817
  1685
haftmann@35817
  1686
lemma insert_idem [simp]:
haftmann@35817
  1687
  assumes "finite A"
haftmann@35817
  1688
  shows "F (insert x A) = g x * F A"
haftmann@35817
  1689
  using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
haftmann@35817
  1690
haftmann@35817
  1691
lemma union_idem:
haftmann@35817
  1692
  assumes "finite A" and "finite B"
haftmann@35817
  1693
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1694
proof -
haftmann@35817
  1695
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1696
  then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
haftmann@35817
  1697
  with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1698
qed
haftmann@35817
  1699
haftmann@35817
  1700
end
haftmann@35817
  1701
haftmann@35817
  1702
haftmann@35817
  1703
subsubsection {* The image case with flexible function and idempotency *}
haftmann@35817
  1704
haftmann@35817
  1705
locale folding_image_idem = folding_image +
haftmann@35817
  1706
  assumes idem: "x * x = x"
haftmann@35817
  1707
haftmann@35817
  1708
sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
haftmann@35817
  1709
qed (fact idem)
haftmann@35817
  1710
haftmann@35817
  1711
haftmann@35817
  1712
subsubsection {* The neutral-less case *}
haftmann@35817
  1713
haftmann@35817
  1714
locale folding_one = abel_semigroup +
haftmann@35817
  1715
  fixes F :: "'a set \<Rightarrow> 'a"
haftmann@35817
  1716
  assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
haftmann@35817
  1717
begin
haftmann@35817
  1718
haftmann@35817
  1719
lemma singleton [simp]:
haftmann@35817
  1720
  "F {x} = x"
haftmann@35817
  1721
  by (simp add: eq_fold)
haftmann@35817
  1722
haftmann@35817
  1723
lemma eq_fold':
haftmann@35817
  1724
  assumes "finite A" and "x \<notin> A"
haftmann@35817
  1725
  shows "F (insert x A) = fold (op *) x A"
haftmann@35817
  1726
proof -
haftmann@35817
  1727
  interpret ab_semigroup_mult "op *" proof qed (simp_all add: ac_simps)
haftmann@35817
  1728
  with assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
haftmann@35817
  1729
qed
haftmann@35817
  1730
haftmann@35817
  1731
lemma insert [simp]:
haftmann@36637
  1732
  assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
haftmann@36637
  1733
  shows "F (insert x A) = x * F A"
haftmann@36637
  1734
proof -
haftmann@36637
  1735
  from `A \<noteq> {}` obtain b where "b \<in> A" by blast
haftmann@35817
  1736
  then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1737
  with `finite A` have "finite B" by simp
haftmann@35817
  1738
  interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
nipkow@39302
  1739
  qed (simp_all add: fun_eq_iff ac_simps)
nipkow@39302
  1740
  thm fold.commute_comp' [of B b, simplified fun_eq_iff, simplified]
haftmann@35817
  1741
  from `finite B` fold.commute_comp' [of B x]
haftmann@35817
  1742
    have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
nipkow@39302
  1743
  then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)
haftmann@35817
  1744
  from `finite B` * fold.insert [of B b]
haftmann@35817
  1745
    have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
nipkow@39302
  1746
  then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)
haftmann@35817
  1747
  from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
haftmann@35817
  1748
qed
haftmann@35817
  1749
haftmann@35817
  1750
lemma remove:
haftmann@35817
  1751
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1752
  shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1753
proof -
haftmann@35817
  1754
  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@35817
  1755
  with assms show ?thesis by simp
haftmann@35817
  1756
qed
haftmann@35817
  1757
haftmann@35817
  1758
lemma insert_remove:
haftmann@35817
  1759
  assumes "finite A"
haftmann@35817
  1760
  shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@35817
  1761
  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
haftmann@35817
  1762
haftmann@35817
  1763
lemma union_disjoint:
haftmann@35817
  1764
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
haftmann@35817
  1765
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1766
  using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
haftmann@35817
  1767
haftmann@35817
  1768
lemma union_inter:
haftmann@35817
  1769
  assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
haftmann@35817
  1770
  shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
haftmann@35817
  1771
proof -
haftmann@35817
  1772
  from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
haftmann@35817
  1773
  from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
haftmann@35817
  1774
    case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
haftmann@35817
  1775
  next
haftmann@35817
  1776
    case (insert x A) show ?case proof (cases "x \<in> B")
haftmann@35817
  1777
      case True then have "B \<noteq> {}" by auto
haftmann@35817
  1778
      with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
haftmann@35817
  1779
        (simp_all add: insert_absorb ac_simps union_disjoint)
haftmann@35817
  1780
    next
haftmann@35817
  1781
      case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
haftmann@35817
  1782
      moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
haftmann@35817
  1783
        by auto
haftmann@35817
  1784
      ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
haftmann@35817
  1785
    qed
haftmann@35817
  1786
  qed
haftmann@35817
  1787
qed
haftmann@35817
  1788
haftmann@35817
  1789
lemma closed:
haftmann@35817
  1790
  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
haftmann@35817
  1791
  shows "F A \<in> A"
haftmann@35817
  1792
using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
haftmann@35817
  1793
  case singleton then show ?case by simp
haftmann@35817
  1794
next
haftmann@35817
  1795
  case insert with elem show ?case by force
haftmann@35817
  1796
qed
haftmann@35817
  1797
haftmann@35817
  1798
end
haftmann@35817
  1799
haftmann@35817
  1800
haftmann@35817
  1801
subsubsection {* The neutral-less case with idempotency *}
haftmann@35817
  1802
haftmann@35817
  1803
locale folding_one_idem = folding_one +
haftmann@35817
  1804
  assumes idem: "x * x = x"
haftmann@35817
  1805
haftmann@35817
  1806
sublocale folding_one_idem < semilattice proof
haftmann@35817
  1807
qed (fact idem)
haftmann@35817
  1808
haftmann@35817
  1809
context folding_one_idem
haftmann@35817
  1810
begin
haftmann@35817
  1811
haftmann@35817
  1812
lemma in_idem:
haftmann@35817
  1813
  assumes "finite A" and "x \<in> A"
haftmann@35817
  1814
  shows "x * F A = F A"
haftmann@35817
  1815
proof -
haftmann@35817
  1816
  from assms have "A \<noteq> {}" by auto
haftmann@35817
  1817
  with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
haftmann@35817
  1818
qed
haftmann@35817
  1819
haftmann@35817
  1820
lemma subset_idem:
haftmann@35817
  1821
  assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
haftmann@35817
  1822
  shows "F B * F A = F A"
haftmann@35817
  1823
proof -
haftmann@35817
  1824
  from assms have "finite B" by (blast dest: finite_subset)
haftmann@35817
  1825
  then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
haftmann@35817
  1826
    (simp_all add: assoc in_idem `finite A`)
haftmann@35817
  1827
qed
haftmann@35817
  1828
haftmann@35817
  1829
lemma eq_fold_idem':
haftmann@35817
  1830
  assumes "finite A"
haftmann@35817
  1831
  shows "F (insert a A) = fold (op *) a A"
haftmann@35817
  1832
proof -
haftmann@35817
  1833
  interpret ab_semigroup_idem_mult "op *" proof qed (simp_all add: ac_simps)
haftmann@35817
  1834
  with assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
haftmann@35817
  1835
qed
haftmann@35817
  1836
haftmann@35817
  1837
lemma insert_idem [simp]:
haftmann@36637
  1838
  assumes "finite A" and "A \<noteq> {}"
haftmann@36637
  1839
  shows "F (insert x A) = x * F A"
haftmann@35817
  1840
proof (cases "x \<in> A")
haftmann@36637
  1841
  case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
haftmann@35817
  1842
next
haftmann@36637
  1843
  case True
haftmann@36637
  1844
  from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
haftmann@35817
  1845
qed
haftmann@35817
  1846
  
haftmann@35817
  1847
lemma union_idem:
haftmann@35817
  1848
  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
haftmann@35817
  1849
  shows "F (A \<union> B) = F A * F B"
haftmann@35817
  1850
proof (cases "A \<inter> B = {}")
haftmann@35817
  1851
  case True with assms show ?thesis by (simp add: union_disjoint)
haftmann@35817
  1852
next
haftmann@35817
  1853
  case False
haftmann@35817
  1854
  from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
haftmann@35817
  1855
  with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
haftmann@35817
  1856
  with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
haftmann@35817
  1857
qed
haftmann@35817
  1858
haftmann@35817
  1859
lemma hom_commute:
haftmann@35817
  1860
  assumes hom: "\<And>x y. h (x * y) = h x * h y"
haftmann@35817
  1861
  and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
haftmann@35817
  1862
using N proof (induct rule: finite_ne_induct)
haftmann@35817
  1863
  case singleton thus ?case by simp
haftmann@35817
  1864
next
haftmann@35817
  1865
  case (insert n N)
haftmann@35817
  1866
  then have "h (F (insert n N)) = h (n * F N)" by simp
haftmann@35817
  1867
  also have "\<dots> = h n * h (F N)" by (rule hom)
haftmann@35817
  1868
  also have "h (F N) = F (h ` N)" by(rule insert)
haftmann@35817
  1869
  also have "h n * \<dots> = F (insert (h n) (h ` N))"
haftmann@35817
  1870
    using insert by(simp)
haftmann@35817
  1871
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@35817
  1872
  finally show ?case .
haftmann@35817
  1873
qed
haftmann@35817
  1874
haftmann@35817
  1875
end
haftmann@35817
  1876
haftmann@35796
  1877
notation times (infixl "*" 70)
haftmann@35796
  1878
notation Groups.one ("1")
haftmann@35722
  1879
haftmann@35722
  1880
haftmann@35722
  1881
subsection {* Finite cardinality *}
haftmann@35722
  1882
haftmann@35722
  1883
text {* This definition, although traditional, is ugly to work with:
haftmann@35722
  1884
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
haftmann@35722
  1885
But now that we have @{text fold_image} things are easy:
haftmann@35722
  1886
*}
haftmann@35722
  1887
haftmann@35722
  1888
definition card :: "'a set \<Rightarrow> nat" where
haftmann@35722
  1889
  "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
haftmann@35722
  1890
haftmann@37770
  1891
interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
haftmann@35722
  1892
qed (simp add: card_def)
haftmann@35722
  1893
haftmann@35722
  1894
lemma card_infinite [simp]:
haftmann@35722
  1895
  "\<not> finite A \<Longrightarrow> card A = 0"
haftmann@35722
  1896
  by (simp add: card_def)
haftmann@35722
  1897
haftmann@35722
  1898
lemma card_empty:
haftmann@35722
  1899
  "card {} = 0"
haftmann@35722
  1900
  by (fact card.empty)
haftmann@35722
  1901
haftmann@35722
  1902
lemma card_insert_disjoint:
haftmann@35722
  1903
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
haftmann@35722
  1904
  by simp
haftmann@35722
  1905
haftmann@35722
  1906
lemma card_insert_if:
haftmann@35722
  1907
  "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
haftmann@35722
  1908
  by auto (simp add: card.insert_remove card.remove)
haftmann@35722
  1909
haftmann@35722
  1910
lemma card_ge_0_finite:
haftmann@35722
  1911
  "card A > 0 \<Longrightarrow> finite A"
haftmann@35722
  1912
  by (rule ccontr) simp
haftmann@35722
  1913
blanchet@35828
  1914
lemma card_0_eq [simp, no_atp]:
haftmann@35722
  1915
  "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
haftmann@35722
  1916
  by (auto dest: mk_disjoint_insert)
haftmann@35722
  1917
haftmann@35722
  1918
lemma finite_UNIV_card_ge_0:
haftmann@35722
  1919
  "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
haftmann@35722
  1920
  by (rule ccontr) simp
haftmann@35722
  1921
haftmann@35722
  1922
lemma card_eq_0_iff:
haftmann@35722
  1923
  "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
haftmann@35722
  1924
  by auto
haftmann@35722
  1925
haftmann@35722
  1926
lemma card_gt_0_iff:
haftmann@35722
  1927
  "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
haftmann@35722
  1928
  by (simp add: neq0_conv [symmetric] card_eq_0_iff) 
haftmann@35722
  1929
haftmann@35722
  1930
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
haftmann@35722
  1931
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
haftmann@35722
  1932
apply(simp del:insert_Diff_single)
haftmann@35722
  1933
done
haftmann@35722
  1934
haftmann@35722
  1935
lemma card_Diff_singleton:
haftmann@35722
  1936
  "finite A ==> x: A ==> card (A - {x}) = card A - 1"
haftmann@35722
  1937
by (simp add: card_Suc_Diff1 [symmetric])
haftmann@35722
  1938
haftmann@35722
  1939
lemma card_Diff_singleton_if:
haftmann@35722
  1940
  "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
haftmann@35722
  1941
by (simp add: card_Diff_singleton)
haftmann@35722
  1942
haftmann@35722
  1943
lemma card_Diff_insert[simp]:
haftmann@35722
  1944
assumes "finite A" and "a:A" and "a ~: B"
haftmann@35722
  1945
shows "card(A - insert a B) = card(A - B) - 1"
haftmann@35722
  1946
proof -
haftmann@35722
  1947
  have "A - insert a B = (A - B) - {a}" using assms by blast
haftmann@35722
  1948
  then show ?thesis using assms by(simp add:card_Diff_singleton)
haftmann@35722
  1949
qed
haftmann@35722
  1950
haftmann@35722
  1951
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
haftmann@35722
  1952
by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  1953
haftmann@35722
  1954
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
haftmann@35722
  1955
by (simp add: card_insert_if)
haftmann@35722
  1956
haftmann@35722
  1957
lemma card_mono:
haftmann@35722
  1958
  assumes "finite B" and "A \<subseteq> B"
haftmann@35722
  1959
  shows "card A \<le> card B"
haftmann@35722
  1960
proof -
haftmann@35722
  1961
  from assms have "finite A" by (auto intro: finite_subset)
haftmann@35722
  1962
  then show ?thesis using assms proof (induct A arbitrary: B)
haftmann@35722
  1963
    case empty then show ?case by simp
haftmann@35722
  1964
  next
haftmann@35722
  1965
    case (insert x A)
haftmann@35722
  1966
    then have "x \<in> B" by simp
haftmann@35722
  1967
    from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
haftmann@35722
  1968
    with insert.hyps have "card A \<le> card (B - {x})" by auto
haftmann@35722
  1969
    with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)
haftmann@35722
  1970
  qed
haftmann@35722
  1971
qed
haftmann@35722
  1972
haftmann@35722
  1973
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
haftmann@35722
  1974
apply (induct set: finite, simp, clarify)
haftmann@35722
  1975
apply (subgoal_tac "finite A & A - {x} <= F")
haftmann@35722
  1976
 prefer 2 apply (blast intro: finite_subset, atomize)
haftmann@35722
  1977
apply (drule_tac x = "A - {x}" in spec)
haftmann@35722
  1978
apply (simp add: card_Diff_singleton_if split add: split_if_asm)
haftmann@35722
  1979
apply (case_tac "card A", auto)
haftmann@35722
  1980
done
haftmann@35722
  1981
haftmann@35722
  1982
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
haftmann@35722
  1983
apply (simp add: psubset_eq linorder_not_le [symmetric])
haftmann@35722
  1984
apply (blast dest: card_seteq)
haftmann@35722
  1985
done
haftmann@35722
  1986
haftmann@35722
  1987
lemma card_Un_Int: "finite A ==> finite B
haftmann@35722
  1988
    ==> card A + card B = card (A Un B) + card (A Int B)"
haftmann@35817
  1989
  by (fact card.union_inter [symmetric])
haftmann@35722
  1990
haftmann@35722
  1991
lemma card_Un_disjoint: "finite A ==> finite B
haftmann@35722
  1992
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
haftmann@35722
  1993
  by (fact card.union_disjoint)
haftmann@35722
  1994
haftmann@35722
  1995
lemma card_Diff_subset:
haftmann@35722
  1996
  assumes "finite B" and "B \<subseteq> A"
haftmann@35722
  1997
  shows "card (A - B) = card A - card B"
haftmann@35722
  1998
proof (cases "finite A")
haftmann@35722
  1999
  case False with assms show ?thesis by simp
haftmann@35722
  2000
next
haftmann@35722
  2001
  case True with assms show ?thesis by (induct B arbitrary: A) simp_all
haftmann@35722
  2002
qed
haftmann@35722
  2003
haftmann@35722
  2004
lemma card_Diff_subset_Int:
haftmann@35722
  2005
  assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
haftmann@35722
  2006
proof -
haftmann@35722
  2007
  have "A - B = A - A \<inter> B" by auto
haftmann@35722
  2008
  thus ?thesis
haftmann@35722
  2009
    by (simp add: card_Diff_subset AB) 
haftmann@35722
  2010
qed
haftmann@35722
  2011
haftmann@35722
  2012
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
haftmann@35722
  2013
apply (rule Suc_less_SucD)
haftmann@35722
  2014
apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
haftmann@35722
  2015
done
haftmann@35722
  2016
haftmann@35722
  2017
lemma card_Diff2_less:
haftmann@35722
  2018
  "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
haftmann@35722
  2019
apply (case_tac "x = y")
haftmann@35722
  2020
 apply (simp add: card_Diff1_less del:card_Diff_insert)
haftmann@35722
  2021
apply (rule less_trans)
haftmann@35722
  2022
 prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
haftmann@35722
  2023
done
haftmann@35722
  2024
haftmann@35722
  2025
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
haftmann@35722
  2026
apply (case_tac "x : A")
haftmann@35722
  2027
 apply (simp_all add: card_Diff1_less less_imp_le)
haftmann@35722
  2028
done
haftmann@35722
  2029
haftmann@35722
  2030
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
haftmann@35722
  2031
by (erule psubsetI, blast)
haftmann@35722
  2032
haftmann@35722
  2033
lemma insert_partition:
haftmann@35722
  2034
  "\<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
  2035
  \<Longrightarrow> x \<inter> \<Union> F = {}"
haftmann@35722
  2036
by auto
haftmann@35722
  2037
haftmann@35722
  2038
lemma finite_psubset_induct[consumes 1, case_names psubset]:
urbanc@36079
  2039
  assumes fin: "finite A" 
urbanc@36079
  2040
  and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A" 
urbanc@36079
  2041
  shows "P A"
urbanc@36079
  2042
using fin
urbanc@36079
  2043
proof (induct A taking: card rule: measure_induct_rule)
haftmann@35722
  2044
  case (less A)
urbanc@36079
  2045
  have fin: "finite A" by fact
urbanc@36079
  2046
  have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
urbanc@36079
  2047
  { fix B 
urbanc@36079
  2048
    assume asm: "B \<subset> A"
urbanc@36079
  2049
    from asm have "card B < card A" using psubset_card_mono fin by blast
urbanc@36079
  2050
    moreover
urbanc@36079
  2051
    from asm have "B \<subseteq> A" by auto
urbanc@36079
  2052
    then have "finite B" using fin finite_subset by blast
urbanc@36079
  2053
    ultimately 
urbanc@36079
  2054
    have "P B" using ih by simp
urbanc@36079
  2055
  }
urbanc@36079
  2056
  with fin show "P A" using major by blast
haftmann@35722
  2057
qed
haftmann@35722
  2058
haftmann@35722
  2059
text{* main cardinality theorem *}
haftmann@35722
  2060
lemma card_partition [rule_format]:
haftmann@35722
  2061
  "finite C ==>
haftmann@35722
  2062
     finite (\<Union> C) -->
haftmann@35722
  2063
     (\<forall>c\<in>C. card c = k) -->
haftmann@35722
  2064
     (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
haftmann@35722
  2065
     k * card(C) = card (\<Union> C)"
haftmann@35722
  2066
apply (erule finite_induct, simp)
haftmann@35722
  2067
apply (simp add: card_Un_disjoint insert_partition 
haftmann@35722
  2068
       finite_subset [of _ "\<Union> (insert x F)"])
haftmann@35722
  2069
done
haftmann@35722
  2070
haftmann@35722
  2071
lemma card_eq_UNIV_imp_eq_UNIV:
haftmann@35722
  2072
  assumes fin: "finite (UNIV :: 'a set)"
haftmann@35722
  2073
  and card: "card A = card (UNIV :: 'a set)"
haftmann@35722
  2074
  shows "A = (UNIV :: 'a set)"
haftmann@35722
  2075
proof
haftmann@35722
  2076
  show "A \<subseteq> UNIV" by simp
haftmann@35722
  2077
  show "UNIV \<subseteq> A"
haftmann@35722
  2078
  proof
haftmann@35722
  2079
    fix x
haftmann@35722
  2080
    show "x \<in> A"
haftmann@35722
  2081
    proof (rule ccontr)
haftmann@35722
  2082
      assume "x \<notin> A"
haftmann@35722
  2083
      then have "A \<subset> UNIV" by auto
haftmann@35722
  2084
      with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
haftmann@35722
  2085
      with card show False by simp
haftmann@35722
  2086
    qed
haftmann@35722
  2087
  qed
haftmann@35722
  2088
qed
haftmann@35722
  2089
haftmann@35722
  2090
text{*The form of a finite set of given cardinality*}
haftmann@35722
  2091
haftmann@35722
  2092
lemma card_eq_SucD:
haftmann@35722
  2093
assumes "card A = Suc k"
haftmann@35722
  2094
shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
haftmann@35722
  2095
proof -
haftmann@35722
  2096
  have fin: "finite A" using assms by (auto intro: ccontr)
haftmann@35722
  2097
  moreover have "card A \<noteq> 0" using assms by auto
haftmann@35722
  2098
  ultimately obtain b where b: "b \<in> A" by auto
haftmann@35722
  2099
  show ?thesis
haftmann@35722
  2100
  proof (intro exI conjI)
haftmann@35722
  2101
    show "A = insert b (A-{b})" using b by blast
haftmann@35722
  2102
    show "b \<notin> A - {b}" by blast
haftmann@35722
  2103
    show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
haftmann@35722
  2104
      using assms b fin by(fastsimp dest:mk_disjoint_insert)+
haftmann@35722
  2105
  qed
haftmann@35722
  2106
qed
haftmann@35722
  2107
haftmann@35722
  2108
lemma card_Suc_eq:
haftmann@35722
  2109
  "(card A = Suc k) =
haftmann@35722
  2110
   (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
haftmann@35722
  2111
apply(rule iffI)
haftmann@35722
  2112
 apply(erule card_eq_SucD)
haftmann@35722
  2113
apply(auto)
haftmann@35722
  2114
apply(subst card_insert)
haftmann@35722
  2115
 apply(auto intro:ccontr)
haftmann@35722
  2116
done
haftmann@35722
  2117
haftmann@35722
  2118
lemma finite_fun_UNIVD2:
haftmann@35722
  2119
  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
haftmann@35722
  2120
  shows "finite (UNIV :: 'b set)"
haftmann@35722
  2121
proof -
haftmann@35722
  2122
  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
haftmann@35722
  2123
    by(rule finite_imageI)
haftmann@35722
  2124
  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
haftmann@35722
  2125
    by(rule UNIV_eq_I) auto
haftmann@35722
  2126
  ultimately show "finite (UNIV :: 'b set)" by simp
haftmann@35722
  2127
qed
haftmann@35722
  2128
haftmann@35722
  2129
lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
haftmann@35722
  2130
  unfolding UNIV_unit by simp
haftmann@35722
  2131
haftmann@35722
  2132
haftmann@35722
  2133
subsubsection {* Cardinality of image *}
haftmann@35722
  2134
haftmann@35722
  2135
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
haftmann@35722
  2136
apply (induct set: finite)
haftmann@35722
  2137
 apply simp
haftmann@35722
  2138
apply (simp add: le_SucI card_insert_if)
haftmann@35722
  2139
done
haftmann@35722
  2140
haftmann@35722
  2141
lemma card_image:
haftmann@35722
  2142
  assumes "inj_on f A"
haftmann@35722
  2143
  shows "card (f ` A) = card A"
haftmann@35722
  2144
proof (cases "finite A")
haftmann@35722
  2145
  case True then show ?thesis using assms by (induct A) simp_all
haftmann@35722
  2146
next
haftmann@35722
  2147
  case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
haftmann@35722
  2148
  with False show ?thesis by simp
haftmann@35722
  2149
qed
haftmann@35722
  2150
haftmann@35722
  2151
lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
haftmann@35722
  2152
by(auto simp: card_image bij_betw_def)
haftmann@35722
  2153
haftmann@35722
  2154
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
haftmann@35722
  2155
by (simp add: card_seteq card_image)
haftmann@35722
  2156
haftmann@35722
  2157
lemma eq_card_imp_inj_on:
haftmann@35722
  2158
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
haftmann@35722
  2159
apply (induct rule:finite_induct)
haftmann@35722
  2160
apply simp
haftmann@35722
  2161
apply(frule card_image_le[where f = f])
haftmann@35722
  2162
apply(simp add:card_insert_if split:if_splits)
haftmann@35722
  2163
done
haftmann@35722
  2164
haftmann@35722
  2165
lemma inj_on_iff_eq_card:
haftmann@35722
  2166
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
haftmann@35722
  2167
by(blast intro: card_image eq_card_imp_inj_on)
haftmann@35722
  2168
haftmann@35722
  2169
haftmann@35722
  2170
lemma card_inj_on_le:
haftmann@35722
  2171
  "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
haftmann@35722
  2172
apply (subgoal_tac "finite A") 
haftmann@35722
  2173
 apply (force intro: card_mono simp add: card_image [symmetric])
haftmann@35722
  2174
apply (blast intro: finite_imageD dest: finite_subset) 
haftmann@35722
  2175
done
haftmann@35722
  2176
haftmann@35722
  2177
lemma card_bij_eq:
haftmann@35722
  2178
  "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
haftmann@35722
  2179
     finite A; finite B |] ==> card A = card B"
haftmann@35722
  2180
by (auto intro: le_antisym card_inj_on_le)
haftmann@35722
  2181
hoelzl@40703
  2182
lemma bij_betw_finite:
hoelzl@40703
  2183
  assumes "bij_betw f A B"
hoelzl@40703
  2184
  shows "finite A \<longleftrightarrow> finite B"
hoelzl@40703
  2185
using assms unfolding bij_betw_def
hoelzl@40703
  2186
using finite_imageD[of f A] by auto
haftmann@35722
  2187
nipkow@37466
  2188
subsubsection {* Pigeonhole Principles *}
nipkow@37466
  2189
nipkow@40311
  2190
lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
nipkow@37466
  2191
by (auto dest: card_image less_irrefl_nat)
nipkow@37466
  2192
nipkow@37466
  2193
lemma pigeonhole_infinite:
nipkow@37466
  2194
assumes  "~ finite A" and "finite(f`A)"
nipkow@37466
  2195
shows "EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  2196
proof -
nipkow@37466
  2197
  have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
nipkow@37466
  2198
  proof(induct "f`A" arbitrary: A rule: finite_induct)
nipkow@37466
  2199
    case empty thus ?case by simp
nipkow@37466
  2200
  next
nipkow@37466
  2201
    case (insert b F)
nipkow@37466
  2202
    show ?case
nipkow@37466
  2203
    proof cases
nipkow@37466
  2204
      assume "finite{a:A. f a = b}"
nipkow@37466
  2205
      hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp
nipkow@37466
  2206
      also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
nipkow@37466
  2207
      finally have "~ finite({a:A. f a \<noteq> b})" .
nipkow@37466
  2208
      from insert(3)[OF _ this]
nipkow@37466
  2209
      show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
nipkow@37466
  2210
    next
nipkow@37466
  2211
      assume 1: "~finite{a:A. f a = b}"
nipkow@37466
  2212
      hence "{a \<in> A. f a = b} \<noteq> {}" by force
nipkow@37466
  2213
      thus ?thesis using 1 by blast
nipkow@37466
  2214
    qed
nipkow@37466
  2215
  qed
nipkow@37466
  2216
  from this[OF assms(2,1)] show ?thesis .
nipkow@37466
  2217
qed
nipkow@37466
  2218
nipkow@37466
  2219
lemma pigeonhole_infinite_rel:
nipkow@37466
  2220
assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
nipkow@37466
  2221
shows "EX b:B. ~finite{a:A. R a b}"
nipkow@37466
  2222
proof -
nipkow@37466
  2223
   let ?F = "%a. {b:B. R a b}"
nipkow@37466
  2224
   from finite_Pow_iff[THEN iffD2, OF `finite B`]
nipkow@37466
  2225
   have "finite(?F ` A)" by(blast intro: rev_finite_subset)
nipkow@37466
  2226
   from pigeonhole_infinite[where f = ?F, OF assms(1) this]
nipkow@37466
  2227
   obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
nipkow@37466
  2228
   obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast
nipkow@37466
  2229
   { assume "finite{a:A. R a b0}"
nipkow@37466
  2230
     then have "finite {a\<in>A. ?F a = ?F a0}"
nipkow@37466
  2231
       using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)
nipkow@37466
  2232
   }
nipkow@37466
  2233
   with 1 `b0 : B` show ?thesis by blast
nipkow@37466
  2234
qed
nipkow@37466
  2235
nipkow@37466
  2236
haftmann@35722
  2237
subsubsection {* Cardinality of sums *}
haftmann@35722
  2238
haftmann@35722
  2239
lemma card_Plus:
haftmann@35722
  2240
  assumes "finite A" and "finite B"
haftmann@35722
  2241
  shows "card (A <+> B) = card A + card B"
haftmann@35722
  2242
proof -
haftmann@35722
  2243
  have "Inl`A \<inter> Inr`B = {}" by fast
haftmann@35722
  2244
  with assms show ?thesis
haftmann@35722
  2245
    unfolding Plus_def
haftmann@35722
  2246
    by (simp add: card_Un_disjoint card_image)
haftmann@35722
  2247
qed
haftmann@35722
  2248
haftmann@35722
  2249
lemma card_Plus_conv_if:
haftmann@35722
  2250
  "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
haftmann@35722
  2251
  by (auto simp add: card_Plus)
haftmann@35722
  2252
haftmann@35722
  2253
haftmann@35722
  2254
subsubsection {* Cardinality of the Powerset *}
haftmann@35722
  2255
haftmann@35722
  2256
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
haftmann@35722
  2257
apply (induct set: finite)
haftmann@35722
  2258
 apply (simp_all add: Pow_insert)
haftmann@35722
  2259
apply (subst card_Un_disjoint, blast)
haftmann@35722
  2260
  apply (blast intro: finite_imageI, blast)
haftmann@35722
  2261
apply (subgoal_tac "inj_on (insert x) (Pow F)")
haftmann@35722
  2262
 apply (simp add: card_image Pow_insert)
haftmann@35722
  2263
apply (unfold inj_on_def)
haftmann@35722
  2264
apply (blast elim!: equalityE)
haftmann@35722
  2265
done
haftmann@35722
  2266
nipkow@37466
  2267
text {* Relates to equivalence classes.  Based on a theorem of F. Kammüller.  *}
haftmann@35722
  2268
haftmann@35722
  2269
lemma dvd_partition:
haftmann@35722
  2270
  "finite (Union C) ==>
haftmann@35722
  2271
    ALL c : C. k dvd card c ==>
haftmann@35722
  2272
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
haftmann@35722
  2273
  k dvd card (Union C)"
haftmann@35722
  2274
apply(frule finite_UnionD)
haftmann@35722
  2275
apply(rotate_tac -1)
haftmann@35722
  2276
apply (induct set: finite, simp_all, clarify)
haftmann@35722
  2277
apply (subst card_Un_disjoint)
haftmann@35722
  2278
   apply (auto simp add: disjoint_eq_subset_Compl)
haftmann@35722
  2279
done
haftmann@35722
  2280
haftmann@35722
  2281
haftmann@35722
  2282
subsubsection {* Relating injectivity and surjectivity *}
haftmann@35722
  2283
haftmann@35722
  2284
lemma finite_surj_inj: "finite(A) \<Longrightarrow> A <= f`A \<Longrightarrow> inj_on f A"
haftmann@35722
  2285
apply(rule eq_card_imp_inj_on, assumption)
haftmann@35722
  2286
apply(frule finite_imageI)
haftmann@35722
  2287
apply(drule (1) card_seteq)
haftmann@35722
  2288
 apply(erule card_image_le)
haftmann@35722
  2289
apply simp
haftmann@35722
  2290
done
haftmann@35722
  2291
haftmann@35722
  2292
lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2293
shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
hoelzl@40702
  2294
by (blast intro: finite_surj_inj subset_UNIV)
haftmann@35722
  2295
haftmann@35722
  2296
lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
haftmann@35722
  2297
shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
haftmann@35722
  2298
by(fastsimp simp:surj_def dest!: endo_inj_surj)
haftmann@35722
  2299
haftmann@35722
  2300
corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
haftmann@35722
  2301
proof
haftmann@35722
  2302
  assume "finite(UNIV::nat set)"
haftmann@35722
  2303
  with finite_UNIV_inj_surj[of Suc]
haftmann@35722
  2304
  show False by simp (blast dest: Suc_neq_Zero surjD)
haftmann@35722
  2305
qed
haftmann@35722
  2306
blanchet@35828
  2307
(* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
blanchet@35828
  2308
lemma infinite_UNIV_char_0[no_atp]:
haftmann@35722
  2309
  "\<not> finite (UNIV::'a::semiring_char_0 set)"
haftmann@35722
  2310
proof
haftmann@35722
  2311
  assume "finite (UNIV::'a set)"
haftmann@35722
  2312
  with subset_UNIV have "finite (range of_nat::'a set)"
haftmann@35722
  2313
    by (rule finite_subset)
haftmann@35722
  2314
  moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
haftmann@35722
  2315
    by (simp add: inj_on_def)
haftmann@35722
  2316
  ultimately have "finite (UNIV::nat set)"
haftmann@35722
  2317
    by (rule finite_imageD)
haftmann@35722
  2318
  then show "False"
haftmann@35722
  2319
    by simp
haftmann@35722
  2320
qed
haftmann@35722
  2321
haftmann@35722
  2322
end