src/HOL/Finite_Set.thy
author nipkow
Wed Feb 02 09:15:40 2005 +0100 (2005-02-02)
changeset 15484 2636ec211ec8
parent 15483 704b3ce6d0f7
child 15487 55497029b255
permissions -rw-r--r--
fold and fol1 changes
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(*  Title:      HOL/Finite_Set.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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                Additions by Jeremy Avigad in Feb 2004
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*)
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header {* Finite sets *}
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theory Finite_Set
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imports Divides Power Inductive
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begin
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subsection {* Definition and basic properties *}
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consts Finites :: "'a set set"
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syntax
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  finite :: "'a set => bool"
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translations
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  "finite A" == "A : Finites"
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inductive Finites
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  intros
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    emptyI [simp, intro!]: "{} : Finites"
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    insertI [simp, intro!]: "A : Finites ==> insert a A : Finites"
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axclass finite \<subseteq> type
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  finite: "finite UNIV"
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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 prems 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: Finites]:
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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 {}" .
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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 = {}" thus ?thesis using insert(4) by simp
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  next
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    assume "F \<noteq> {}" 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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  "finite F ==> F \<subseteq> A ==>
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    P {} ==> (!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==>
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    P F"
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proof -
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  assume "P {}" and insert:
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    "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
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  assume "finite F"
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  thus "F \<subseteq> A ==> P F"
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  proof induct
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    show "P {}" .
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    fix x F assume "finite F" and "x \<notin> F"
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      and 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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    qed
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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:
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assumes fin: "finite A" shows "\<exists> (n::nat) f. A = f ` {i::nat. 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}" by(simp add:image_def) qed
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next
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  case (insert a A)
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  from insert.hyps obtain n f where "A = f ` {i::nat. i < n}" by blast
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  hence "insert a A = (%i. if i<n then f i else a) ` {i. i < n+1}"
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    by (auto simp add:image_def Ball_def)
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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: finite_imp_nat_seg_image nat_seg_image_imp_finite)
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subsubsection{* Finiteness and set theoretic constructions *}
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lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
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  -- {* The union of two finite sets is finite. *}
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  by (induct set: Finites) 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})" .
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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" 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" .
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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 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_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_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_empty_induct:
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  "finite A ==>
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  P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
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proof -
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  assume "finite A"
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    and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
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  have "P (A - A)"
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  proof -
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    fix c b :: "'a set"
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    presume 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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    from c show "c \<subseteq> b ==> P (b - 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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  next
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    show "A \<subseteq> A" ..
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  qed
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  thus "P {}" by simp
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qed
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lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
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  by (rule Diff_subset [THEN finite_subset])
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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: Finites) simp_all
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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)
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  done
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lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
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proof -
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  have aux: "!!A. finite (A - {}) = finite A" by simp
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  fix B :: "'a set"
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  assume "finite B"
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  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
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    apply induct
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     apply simp
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    apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
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     apply clarify
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     apply (simp (no_asm_use) add: inj_on_def)
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     apply (blast dest!: aux [THEN iffD1], atomize)
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    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
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    apply (frule subsetD [OF equalityD2 insertI1], clarify)
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    apply (rule_tac x = xa in bexI)
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     apply (simp_all add: inj_on_image_set_diff)
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    done
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qed (rule refl)
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lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
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  -- {* The inverse image of a singleton under an injective function
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         is included in a singleton. *}
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  apply (auto simp add: inj_on_def)
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  apply (blast intro: the_equality [symmetric])
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  done
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lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
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  -- {* The inverse image of a finite set under an injective function
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         is finite. *}
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  apply (induct set: Finites, simp_all)
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  apply (subst vimage_insert)
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  apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
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  done
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text {* The finite UNION of finite sets *}
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lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
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  by (induct set: Finites) simp_all
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text {*
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  Strengthen RHS to
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  @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
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  We'd need to prove
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  @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
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  by induction. *}
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lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
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  by (blast intro: finite_UN_I finite_subset)
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text {* Sigma of finite sets *}
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lemma finite_SigmaI [simp]:
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    "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
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  by (unfold Sigma_def) (blast intro!: finite_UN_I)
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lemma finite_cartesian_product: "[| finite A; finite B |] ==>
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    finite (A <*> B)"
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  by (rule finite_SigmaI)
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lemma finite_Prod_UNIV:
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    "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
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  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
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   apply (erule ssubst)
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   apply (erule finite_SigmaI, auto)
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  done
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lemma finite_cartesian_productD1:
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     "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
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apply (auto simp add: finite_conv_nat_seg_image) 
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apply (drule_tac x=n in spec) 
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apply (drule_tac x="fst o f" in spec) 
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apply (auto simp add: o_def) 
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 prefer 2 apply (force dest!: equalityD2) 
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apply (drule equalityD1) 
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apply (rename_tac y x)
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apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
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 prefer 2 apply force
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apply clarify
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apply (rule_tac x=k in image_eqI, auto)
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done
paulson@15409
   334
paulson@15409
   335
lemma finite_cartesian_productD2:
paulson@15409
   336
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
paulson@15409
   337
apply (auto simp add: finite_conv_nat_seg_image) 
paulson@15409
   338
apply (drule_tac x=n in spec) 
paulson@15409
   339
apply (drule_tac x="snd o f" in spec) 
paulson@15409
   340
apply (auto simp add: o_def) 
paulson@15409
   341
 prefer 2 apply (force dest!: equalityD2) 
paulson@15409
   342
apply (drule equalityD1)
paulson@15409
   343
apply (rename_tac x y)
paulson@15409
   344
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
paulson@15409
   345
 prefer 2 apply force
paulson@15409
   346
apply clarify
paulson@15409
   347
apply (rule_tac x=k in image_eqI, auto)
paulson@15409
   348
done
paulson@15409
   349
paulson@15409
   350
wenzelm@12396
   351
instance unit :: finite
wenzelm@12396
   352
proof
wenzelm@12396
   353
  have "finite {()}" by simp
wenzelm@12396
   354
  also have "{()} = UNIV" by auto
wenzelm@12396
   355
  finally show "finite (UNIV :: unit set)" .
wenzelm@12396
   356
qed
wenzelm@12396
   357
wenzelm@12396
   358
instance * :: (finite, finite) finite
wenzelm@12396
   359
proof
wenzelm@12396
   360
  show "finite (UNIV :: ('a \<times> 'b) set)"
wenzelm@12396
   361
  proof (rule finite_Prod_UNIV)
wenzelm@12396
   362
    show "finite (UNIV :: 'a set)" by (rule finite)
wenzelm@12396
   363
    show "finite (UNIV :: 'b set)" by (rule finite)
wenzelm@12396
   364
  qed
wenzelm@12396
   365
qed
wenzelm@12396
   366
wenzelm@12396
   367
nipkow@15392
   368
text {* The powerset of a finite set *}
wenzelm@12396
   369
wenzelm@12396
   370
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
wenzelm@12396
   371
proof
wenzelm@12396
   372
  assume "finite (Pow A)"
wenzelm@12396
   373
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
wenzelm@12396
   374
  thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
wenzelm@12396
   375
next
wenzelm@12396
   376
  assume "finite A"
wenzelm@12396
   377
  thus "finite (Pow A)"
wenzelm@12396
   378
    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
wenzelm@12396
   379
qed
wenzelm@12396
   380
nipkow@15392
   381
nipkow@15392
   382
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
nipkow@15392
   383
by(blast intro: finite_subset[OF subset_Pow_Union])
nipkow@15392
   384
nipkow@15392
   385
wenzelm@12396
   386
lemma finite_converse [iff]: "finite (r^-1) = finite r"
wenzelm@12396
   387
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
wenzelm@12396
   388
   apply simp
wenzelm@12396
   389
   apply (rule iffI)
wenzelm@12396
   390
    apply (erule finite_imageD [unfolded inj_on_def])
wenzelm@12396
   391
    apply (simp split add: split_split)
wenzelm@12396
   392
   apply (erule finite_imageI)
paulson@14208
   393
  apply (simp add: converse_def image_def, auto)
wenzelm@12396
   394
  apply (rule bexI)
wenzelm@12396
   395
   prefer 2 apply assumption
wenzelm@12396
   396
  apply simp
wenzelm@12396
   397
  done
wenzelm@12396
   398
paulson@14430
   399
nipkow@15392
   400
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
nipkow@15392
   401
Ehmety) *}
wenzelm@12396
   402
wenzelm@12396
   403
lemma finite_Field: "finite r ==> finite (Field r)"
wenzelm@12396
   404
  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
wenzelm@12396
   405
  apply (induct set: Finites)
wenzelm@12396
   406
   apply (auto simp add: Field_def Domain_insert Range_insert)
wenzelm@12396
   407
  done
wenzelm@12396
   408
wenzelm@12396
   409
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
wenzelm@12396
   410
  apply clarify
wenzelm@12396
   411
  apply (erule trancl_induct)
wenzelm@12396
   412
   apply (auto simp add: Field_def)
wenzelm@12396
   413
  done
wenzelm@12396
   414
wenzelm@12396
   415
lemma finite_trancl: "finite (r^+) = finite r"
wenzelm@12396
   416
  apply auto
wenzelm@12396
   417
   prefer 2
wenzelm@12396
   418
   apply (rule trancl_subset_Field2 [THEN finite_subset])
wenzelm@12396
   419
   apply (rule finite_SigmaI)
wenzelm@12396
   420
    prefer 3
berghofe@13704
   421
    apply (blast intro: r_into_trancl' finite_subset)
wenzelm@12396
   422
   apply (auto simp add: finite_Field)
wenzelm@12396
   423
  done
wenzelm@12396
   424
wenzelm@12396
   425
nipkow@15392
   426
subsection {* A fold functional for finite sets *}
nipkow@15392
   427
nipkow@15392
   428
text {* The intended behaviour is
nipkow@15480
   429
@{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<dots>)"}
nipkow@15392
   430
if @{text f} is associative-commutative. For an application of @{text fold}
nipkow@15392
   431
se the definitions of sums and products over finite sets.
nipkow@15392
   432
*}
nipkow@15392
   433
nipkow@15392
   434
consts
nipkow@15392
   435
  foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
nipkow@15392
   436
nipkow@15480
   437
inductive "foldSet f g z"
nipkow@15392
   438
intros
nipkow@15480
   439
emptyI [intro]: "({}, z) : foldSet f g z"
nipkow@15480
   440
insertI [intro]: "\<lbrakk> x \<notin> A; (A, y) : foldSet f g z \<rbrakk>
nipkow@15480
   441
 \<Longrightarrow> (insert x A, f (g x) y) : foldSet f g z"
nipkow@15392
   442
nipkow@15480
   443
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g z"
nipkow@15392
   444
nipkow@15392
   445
constdefs
nipkow@15392
   446
  fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
nipkow@15480
   447
  "fold f g z A == THE x. (A, x) : foldSet f g z"
nipkow@15392
   448
nipkow@15392
   449
lemma Diff1_foldSet:
nipkow@15480
   450
  "(A - {x}, y) : foldSet f g z ==> x: A ==> (A, f (g x) y) : foldSet f g z"
nipkow@15392
   451
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
nipkow@15392
   452
nipkow@15480
   453
lemma foldSet_imp_finite: "(A, x) : foldSet f g z ==> finite A"
nipkow@15392
   454
  by (induct set: foldSet) auto
nipkow@15392
   455
nipkow@15480
   456
lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f g z"
nipkow@15392
   457
  by (induct set: Finites) auto
nipkow@15392
   458
nipkow@15392
   459
nipkow@15392
   460
subsubsection {* Commutative monoids *}
nipkow@15480
   461
nipkow@15392
   462
locale ACf =
nipkow@15392
   463
  fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
nipkow@15392
   464
  assumes commute: "x \<cdot> y = y \<cdot> x"
nipkow@15392
   465
    and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
nipkow@15392
   466
nipkow@15392
   467
locale ACe = ACf +
nipkow@15392
   468
  fixes e :: 'a
nipkow@15392
   469
  assumes ident [simp]: "x \<cdot> e = x"
nipkow@15392
   470
nipkow@15480
   471
locale ACIf = ACf +
nipkow@15480
   472
  assumes idem: "x \<cdot> x = x"
nipkow@15480
   473
nipkow@15392
   474
lemma (in ACf) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
nipkow@15392
   475
proof -
nipkow@15392
   476
  have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
nipkow@15392
   477
  also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
nipkow@15392
   478
  also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
nipkow@15392
   479
  finally show ?thesis .
nipkow@15392
   480
qed
nipkow@15392
   481
nipkow@15392
   482
lemmas (in ACf) AC = assoc commute left_commute
nipkow@15392
   483
nipkow@15392
   484
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
nipkow@15392
   485
proof -
nipkow@15392
   486
  have "x \<cdot> e = x" by (rule ident)
nipkow@15392
   487
  thus ?thesis by (subst commute)
nipkow@15392
   488
qed
nipkow@15392
   489
nipkow@15402
   490
text{* Instantiation of locales: *}
nipkow@15402
   491
nipkow@15402
   492
lemma ACf_add: "ACf (op + :: 'a::comm_monoid_add \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15402
   493
by(fastsimp intro: ACf.intro add_assoc add_commute)
nipkow@15402
   494
nipkow@15402
   495
lemma ACe_add: "ACe (op +) (0::'a::comm_monoid_add)"
nipkow@15402
   496
by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_add)
nipkow@15402
   497
nipkow@15402
   498
nipkow@15402
   499
lemma ACf_mult: "ACf (op * :: 'a::comm_monoid_mult \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15402
   500
by(fast intro: ACf.intro mult_assoc ab_semigroup_mult.mult_commute)
nipkow@15402
   501
nipkow@15402
   502
lemma ACe_mult: "ACe (op *) (1::'a::comm_monoid_mult)"
nipkow@15402
   503
by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_mult)
nipkow@15402
   504
nipkow@15402
   505
nipkow@15392
   506
subsubsection{*From @{term foldSet} to @{term fold}*}
nipkow@15392
   507
nipkow@15479
   508
(* only used in the next lemma, but in there twice *)
nipkow@15479
   509
lemma card_lemma: assumes A1: "A = insert b B" and notinB: "b \<notin> B" and
nipkow@15479
   510
  card: "A = h`{i. i<Suc n}" and new: "\<not>(EX k<n. h n = h k)"
nipkow@15479
   511
shows "EX h. B = h`{i. i<n}" (is "EX h. ?P h")
nipkow@15479
   512
proof
nipkow@15479
   513
  let ?h = "%i. if h i = b then h n else h i"
nipkow@15479
   514
  show "B = ?h`{i. i<n}" (is "_ = ?r")
nipkow@15479
   515
  proof
nipkow@15479
   516
    show "B \<subseteq> ?r"
nipkow@15479
   517
    proof
nipkow@15479
   518
      fix u assume "u \<in> B"
nipkow@15479
   519
      hence uinA: "u \<in> A" and unotb: "u \<noteq> b" using A1 notinB by blast+
nipkow@15479
   520
      then obtain i\<^isub>u where below: "i\<^isub>u < Suc n" and [simp]: "u = h i\<^isub>u"
nipkow@15479
   521
	using card by(auto simp:image_def)
nipkow@15479
   522
      show "u \<in> ?r"
nipkow@15479
   523
      proof cases
nipkow@15479
   524
	assume "i\<^isub>u < n"
nipkow@15479
   525
	thus ?thesis using unotb by(fastsimp)
nipkow@15479
   526
      next
nipkow@15479
   527
	assume "\<not> i\<^isub>u < n"
nipkow@15479
   528
	with below have [simp]: "i\<^isub>u = n" by arith
nipkow@15479
   529
	obtain i\<^isub>k where i\<^isub>k: "i\<^isub>k < Suc n" and [simp]: "b = h i\<^isub>k"
nipkow@15479
   530
	  using A1 card by blast
nipkow@15479
   531
	have "i\<^isub>k < n"
nipkow@15479
   532
	proof (rule ccontr)
nipkow@15479
   533
	  assume "\<not> i\<^isub>k < n"
nipkow@15479
   534
	  hence "i\<^isub>k = n" using i\<^isub>k by arith
nipkow@15479
   535
	  thus False using unotb by simp
nipkow@15479
   536
	qed
nipkow@15479
   537
	thus ?thesis by(auto simp add:image_def)
nipkow@15479
   538
      qed
nipkow@15479
   539
    qed
nipkow@15479
   540
  next
nipkow@15479
   541
    show "?r \<subseteq> B"
nipkow@15479
   542
    proof
nipkow@15479
   543
      fix u assume "u \<in> ?r"
nipkow@15479
   544
      then obtain i\<^isub>u where below: "i\<^isub>u < n" and
nipkow@15479
   545
        or: "b = h i\<^isub>u \<and> u = h n \<or> h i\<^isub>u \<noteq> b \<and> h i\<^isub>u = u"
nipkow@15479
   546
	by(auto simp:image_def)
nipkow@15479
   547
      from or show "u \<in> B"
nipkow@15479
   548
      proof
nipkow@15479
   549
	assume [simp]: "b = h i\<^isub>u \<and> u = h n"
nipkow@15479
   550
	have "u \<in> A" using card by auto
nipkow@15479
   551
        moreover have "u \<noteq> b" using new below by auto
nipkow@15479
   552
	ultimately show "u \<in> B" using A1 by blast
nipkow@15479
   553
      next
nipkow@15479
   554
	assume "h i\<^isub>u \<noteq> b \<and> h i\<^isub>u = u"
nipkow@15479
   555
	moreover hence "u \<in> A" using card below by auto
nipkow@15479
   556
	ultimately show "u \<in> B" using A1 by blast
nipkow@15479
   557
      qed
nipkow@15479
   558
    qed
nipkow@15479
   559
  qed
nipkow@15479
   560
qed
nipkow@15479
   561
nipkow@15392
   562
lemma (in ACf) foldSet_determ_aux:
nipkow@15480
   563
  "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; (A,x) : foldSet f g z; (A,x') : foldSet f g z \<rbrakk>
nipkow@15392
   564
   \<Longrightarrow> x' = x"
nipkow@15392
   565
proof (induct n)
nipkow@15392
   566
  case 0 thus ?case by auto
nipkow@15392
   567
next
nipkow@15392
   568
  case (Suc n)
nipkow@15480
   569
  have IH: "!!A x x' h. \<lbrakk>A = h`{i::nat. i<n}; (A,x) \<in> foldSet f g z; (A,x') \<in> foldSet f g z\<rbrakk>
nipkow@15392
   570
           \<Longrightarrow> x' = x" and card: "A = h`{i. i<Suc n}"
nipkow@15480
   571
  and Afoldx: "(A, x) \<in> foldSet f g z" and Afoldy: "(A,x') \<in> foldSet f g z" .
nipkow@15392
   572
  show ?case
nipkow@15392
   573
  proof cases
nipkow@15392
   574
    assume "EX k<n. h n = h k"
nipkow@15392
   575
    hence card': "A = h ` {i. i < n}"
nipkow@15392
   576
      using card by (auto simp:image_def less_Suc_eq)
nipkow@15392
   577
    show ?thesis by(rule IH[OF card' Afoldx Afoldy])
nipkow@15392
   578
  next
nipkow@15392
   579
    assume new: "\<not>(EX k<n. h n = h k)"
nipkow@15392
   580
    show ?thesis
nipkow@15392
   581
    proof (rule foldSet.cases[OF Afoldx])
nipkow@15480
   582
      assume "(A, x) = ({}, z)"
nipkow@15392
   583
      thus "x' = x" using Afoldy by (auto)
nipkow@15392
   584
    next
nipkow@15392
   585
      fix B b y
nipkow@15392
   586
      assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
nipkow@15480
   587
	and y: "(B,y) \<in> foldSet f g z" and notinB: "b \<notin> B"
nipkow@15392
   588
      hence A1: "A = insert b B" and x: "x = g b \<cdot> y" by auto
nipkow@15392
   589
      show ?thesis
nipkow@15392
   590
      proof (rule foldSet.cases[OF Afoldy])
nipkow@15480
   591
	assume "(A,x') = ({}, z)"
nipkow@15392
   592
	thus ?thesis using A1 by auto
nipkow@15392
   593
      next
nipkow@15480
   594
	fix C c r
nipkow@15480
   595
	assume eq2: "(A,x') = (insert c C, g c \<cdot> r)"
nipkow@15480
   596
	  and r: "(C,r) \<in> foldSet f g z" and notinC: "c \<notin> C"
nipkow@15480
   597
	hence A2: "A = insert c C" and x': "x' = g c \<cdot> r" by auto
nipkow@15479
   598
	obtain hB where lessB: "B = hB ` {i. i<n}"
nipkow@15479
   599
	  using card_lemma[OF A1 notinB card new] by auto
nipkow@15479
   600
	obtain hC where lessC: "C = hC ` {i. i<n}"
nipkow@15479
   601
	  using card_lemma[OF A2 notinC card new] by auto
nipkow@15392
   602
	show ?thesis
nipkow@15392
   603
	proof cases
nipkow@15392
   604
	  assume "b = c"
nipkow@15392
   605
	  then moreover have "B = C" using A1 A2 notinB notinC by auto
nipkow@15480
   606
	  ultimately show ?thesis using IH[OF lessB] y r x x' by auto
nipkow@15392
   607
	next
nipkow@15392
   608
	  assume diff: "b \<noteq> c"
nipkow@15392
   609
	  let ?D = "B - {c}"
nipkow@15392
   610
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
nipkow@15392
   611
	    using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
nipkow@15402
   612
	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
nipkow@15402
   613
	  with A1 have "finite ?D" by simp
nipkow@15480
   614
	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g z"
nipkow@15392
   615
	    using finite_imp_foldSet by rules
nipkow@15392
   616
	  moreover have cinB: "c \<in> B" using B by(auto)
nipkow@15480
   617
	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g z"
nipkow@15392
   618
	    by(rule Diff1_foldSet)
nipkow@15479
   619
	  hence "g c \<cdot> d = y" by(rule IH[OF lessB y])
nipkow@15480
   620
          moreover have "g b \<cdot> d = r"
nipkow@15480
   621
	  proof (rule IH[OF lessC r])
nipkow@15480
   622
	    show "(C,g b \<cdot> d) \<in> foldSet f g z" using C notinB Dfoldd
nipkow@15392
   623
	      by fastsimp
nipkow@15392
   624
	  qed
nipkow@15392
   625
	  ultimately show ?thesis using x x' by(auto simp:AC)
nipkow@15392
   626
	qed
nipkow@15392
   627
      qed
nipkow@15392
   628
    qed
nipkow@15392
   629
  qed
nipkow@15392
   630
qed
nipkow@15392
   631
nipkow@15392
   632
(* The same proof, but using card 
nipkow@15392
   633
lemma (in ACf) foldSet_determ_aux:
nipkow@15392
   634
  "!!A x x'. \<lbrakk> card A < n; (A,x) : foldSet f g e; (A,x') : foldSet f g e \<rbrakk>
nipkow@15392
   635
   \<Longrightarrow> x' = x"
nipkow@15392
   636
proof (induct n)
nipkow@15392
   637
  case 0 thus ?case by simp
nipkow@15392
   638
next
nipkow@15392
   639
  case (Suc n)
nipkow@15392
   640
  have IH: "!!A x x'. \<lbrakk>card A < n; (A,x) \<in> foldSet f g e; (A,x') \<in> foldSet f g e\<rbrakk>
nipkow@15392
   641
           \<Longrightarrow> x' = x" and card: "card A < Suc n"
nipkow@15392
   642
  and Afoldx: "(A, x) \<in> foldSet f g e" and Afoldy: "(A,x') \<in> foldSet f g e" .
nipkow@15392
   643
  from card have "card A < n \<or> card A = n" by arith
nipkow@15392
   644
  thus ?case
nipkow@15392
   645
  proof
nipkow@15392
   646
    assume less: "card A < n"
nipkow@15392
   647
    show ?thesis by(rule IH[OF less Afoldx Afoldy])
nipkow@15392
   648
  next
nipkow@15392
   649
    assume cardA: "card A = n"
nipkow@15392
   650
    show ?thesis
nipkow@15392
   651
    proof (rule foldSet.cases[OF Afoldx])
nipkow@15392
   652
      assume "(A, x) = ({}, e)"
nipkow@15392
   653
      thus "x' = x" using Afoldy by (auto)
nipkow@15392
   654
    next
nipkow@15392
   655
      fix B b y
nipkow@15392
   656
      assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
nipkow@15392
   657
	and y: "(B,y) \<in> foldSet f g e" and notinB: "b \<notin> B"
nipkow@15392
   658
      hence A1: "A = insert b B" and x: "x = g b \<cdot> y" by auto
nipkow@15392
   659
      show ?thesis
nipkow@15392
   660
      proof (rule foldSet.cases[OF Afoldy])
nipkow@15392
   661
	assume "(A,x') = ({}, e)"
nipkow@15392
   662
	thus ?thesis using A1 by auto
nipkow@15392
   663
      next
nipkow@15392
   664
	fix C c z
nipkow@15392
   665
	assume eq2: "(A,x') = (insert c C, g c \<cdot> z)"
nipkow@15392
   666
	  and z: "(C,z) \<in> foldSet f g e" and notinC: "c \<notin> C"
nipkow@15392
   667
	hence A2: "A = insert c C" and x': "x' = g c \<cdot> z" by auto
nipkow@15392
   668
	have finA: "finite A" by(rule foldSet_imp_finite[OF Afoldx])
nipkow@15392
   669
	with cardA A1 notinB have less: "card B < n" by simp
nipkow@15392
   670
	show ?thesis
nipkow@15392
   671
	proof cases
nipkow@15392
   672
	  assume "b = c"
nipkow@15392
   673
	  then moreover have "B = C" using A1 A2 notinB notinC by auto
nipkow@15392
   674
	  ultimately show ?thesis using IH[OF less] y z x x' by auto
nipkow@15392
   675
	next
nipkow@15392
   676
	  assume diff: "b \<noteq> c"
nipkow@15392
   677
	  let ?D = "B - {c}"
nipkow@15392
   678
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
nipkow@15392
   679
	    using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
nipkow@15392
   680
	  have "finite ?D" using finA A1 by simp
nipkow@15392
   681
	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g e"
nipkow@15392
   682
	    using finite_imp_foldSet by rules
nipkow@15392
   683
	  moreover have cinB: "c \<in> B" using B by(auto)
nipkow@15392
   684
	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g e"
nipkow@15392
   685
	    by(rule Diff1_foldSet)
nipkow@15392
   686
	  hence "g c \<cdot> d = y" by(rule IH[OF less y])
nipkow@15392
   687
          moreover have "g b \<cdot> d = z"
nipkow@15392
   688
	  proof (rule IH[OF _ z])
nipkow@15392
   689
	    show "card C < n" using C cardA A1 notinB finA cinB
nipkow@15392
   690
	      by(auto simp:card_Diff1_less)
nipkow@15392
   691
	  next
nipkow@15392
   692
	    show "(C,g b \<cdot> d) \<in> foldSet f g e" using C notinB Dfoldd
nipkow@15392
   693
	      by fastsimp
nipkow@15392
   694
	  qed
nipkow@15392
   695
	  ultimately show ?thesis using x x' by(auto simp:AC)
nipkow@15392
   696
	qed
nipkow@15392
   697
      qed
nipkow@15392
   698
    qed
nipkow@15392
   699
  qed
nipkow@15392
   700
qed
nipkow@15392
   701
*)
nipkow@15392
   702
nipkow@15392
   703
lemma (in ACf) foldSet_determ:
nipkow@15480
   704
  "(A, x) : foldSet f g z ==> (A, y) : foldSet f g z ==> y = x"
nipkow@15392
   705
apply(frule foldSet_imp_finite)
nipkow@15392
   706
apply(simp add:finite_conv_nat_seg_image)
nipkow@15392
   707
apply(blast intro: foldSet_determ_aux [rule_format])
nipkow@15392
   708
done
nipkow@15392
   709
nipkow@15480
   710
lemma (in ACf) fold_equality: "(A, y) : foldSet f g z ==> fold f g z A = y"
nipkow@15392
   711
  by (unfold fold_def) (blast intro: foldSet_determ)
nipkow@15392
   712
nipkow@15392
   713
text{* The base case for @{text fold}: *}
nipkow@15392
   714
nipkow@15480
   715
lemma fold_empty [simp]: "fold f g z {} = z"
nipkow@15392
   716
  by (unfold fold_def) blast
nipkow@15392
   717
nipkow@15392
   718
lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
nipkow@15480
   719
    ((insert x A, v) : foldSet f g z) =
nipkow@15480
   720
    (EX y. (A, y) : foldSet f g z & v = f (g x) y)"
nipkow@15392
   721
  apply auto
nipkow@15392
   722
  apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
nipkow@15392
   723
   apply (fastsimp dest: foldSet_imp_finite)
nipkow@15392
   724
  apply (blast intro: foldSet_determ)
nipkow@15392
   725
  done
nipkow@15392
   726
nipkow@15392
   727
text{* The recursion equation for @{text fold}: *}
nipkow@15392
   728
nipkow@15392
   729
lemma (in ACf) fold_insert[simp]:
nipkow@15480
   730
    "finite A ==> x \<notin> A ==> fold f g z (insert x A) = f (g x) (fold f g z A)"
nipkow@15392
   731
  apply (unfold fold_def)
nipkow@15392
   732
  apply (simp add: fold_insert_aux)
nipkow@15392
   733
  apply (rule the_equality)
nipkow@15392
   734
  apply (auto intro: finite_imp_foldSet
nipkow@15392
   735
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
nipkow@15392
   736
  done
nipkow@15392
   737
nipkow@15392
   738
declare
nipkow@15392
   739
  empty_foldSetE [rule del]  foldSet.intros [rule del]
nipkow@15392
   740
  -- {* Delete rules to do with @{text foldSet} relation. *}
nipkow@15392
   741
nipkow@15480
   742
text{* A simplified version for idempotent functions: *}
nipkow@15480
   743
nipkow@15480
   744
lemma (in ACIf) fold_insert2:
nipkow@15480
   745
assumes finA: "finite A"
nipkow@15480
   746
shows "fold (op \<cdot>) g z (insert a A) = g a \<cdot> fold f g z A"
nipkow@15480
   747
proof cases
nipkow@15480
   748
  assume "a \<in> A"
nipkow@15480
   749
  then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
nipkow@15480
   750
    by(blast dest: mk_disjoint_insert)
nipkow@15480
   751
  show ?thesis
nipkow@15480
   752
  proof -
nipkow@15480
   753
    from finA A have finB: "finite B" by(blast intro: finite_subset)
nipkow@15480
   754
    have "fold f g z (insert a A) = fold f g z (insert a B)" using A by simp
nipkow@15480
   755
    also have "\<dots> = (g a) \<cdot> (fold f g z B)"
nipkow@15480
   756
      using finB disj by(simp)
nipkow@15480
   757
    also have "\<dots> = g a \<cdot> fold f g z A"
nipkow@15480
   758
      using A finB disj by(simp add:idem assoc[symmetric])
nipkow@15480
   759
    finally show ?thesis .
nipkow@15480
   760
  qed
nipkow@15480
   761
next
nipkow@15480
   762
  assume "a \<notin> A"
nipkow@15480
   763
  with finA show ?thesis by simp
nipkow@15480
   764
qed
nipkow@15480
   765
nipkow@15484
   766
lemma (in ACIf) foldI_conv_id:
nipkow@15484
   767
  "finite A \<Longrightarrow> fold f g z A = fold f id z (g ` A)"
nipkow@15484
   768
by(erule finite_induct)(simp_all add: fold_insert2 del: fold_insert)
nipkow@15484
   769
nipkow@15392
   770
subsubsection{*Lemmas about @{text fold}*}
nipkow@15392
   771
nipkow@15392
   772
lemma (in ACf) fold_commute:
nipkow@15480
   773
  "finite A ==> (!!z. f (g x) (fold f g z A) = fold f g (f (g x) z) A)"
nipkow@15392
   774
  apply (induct set: Finites, simp)
nipkow@15392
   775
  apply (simp add: left_commute)
nipkow@15392
   776
  done
nipkow@15392
   777
nipkow@15392
   778
lemma (in ACf) fold_nest_Un_Int:
nipkow@15392
   779
  "finite A ==> finite B
nipkow@15480
   780
    ==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)"
nipkow@15392
   781
  apply (induct set: Finites, simp)
nipkow@15392
   782
  apply (simp add: fold_commute Int_insert_left insert_absorb)
nipkow@15392
   783
  done
nipkow@15392
   784
nipkow@15392
   785
lemma (in ACf) fold_nest_Un_disjoint:
nipkow@15392
   786
  "finite A ==> finite B ==> A Int B = {}
nipkow@15480
   787
    ==> fold f g z (A Un B) = fold f g (fold f g z B) A"
nipkow@15392
   788
  by (simp add: fold_nest_Un_Int)
nipkow@15392
   789
nipkow@15392
   790
lemma (in ACf) fold_reindex:
nipkow@15392
   791
assumes fin: "finite B"
nipkow@15480
   792
shows "inj_on h B \<Longrightarrow> fold f g z (h ` B) = fold f (g \<circ> h) z B"
nipkow@15392
   793
using fin apply (induct)
nipkow@15392
   794
 apply simp
nipkow@15392
   795
apply simp
nipkow@15392
   796
done
nipkow@15392
   797
nipkow@15392
   798
lemma (in ACe) fold_Un_Int:
nipkow@15392
   799
  "finite A ==> finite B ==>
nipkow@15392
   800
    fold f g e A \<cdot> fold f g e B =
nipkow@15392
   801
    fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
nipkow@15392
   802
  apply (induct set: Finites, simp)
nipkow@15392
   803
  apply (simp add: AC insert_absorb Int_insert_left)
nipkow@15392
   804
  done
nipkow@15392
   805
nipkow@15392
   806
corollary (in ACe) fold_Un_disjoint:
nipkow@15392
   807
  "finite A ==> finite B ==> A Int B = {} ==>
nipkow@15392
   808
    fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B"
nipkow@15392
   809
  by (simp add: fold_Un_Int)
nipkow@15392
   810
nipkow@15392
   811
lemma (in ACe) fold_UN_disjoint:
nipkow@15392
   812
  "\<lbrakk> finite I; ALL i:I. finite (A i);
nipkow@15392
   813
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
nipkow@15392
   814
   \<Longrightarrow> fold f g e (UNION I A) =
nipkow@15392
   815
       fold f (%i. fold f g e (A i)) e I"
nipkow@15392
   816
  apply (induct set: Finites, simp, atomize)
nipkow@15392
   817
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
nipkow@15392
   818
   prefer 2 apply blast
nipkow@15392
   819
  apply (subgoal_tac "A x Int UNION F A = {}")
nipkow@15392
   820
   prefer 2 apply blast
nipkow@15392
   821
  apply (simp add: fold_Un_disjoint)
nipkow@15392
   822
  done
nipkow@15392
   823
nipkow@15392
   824
lemma (in ACf) fold_cong:
nipkow@15480
   825
  "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A"
nipkow@15480
   826
  apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g z C = fold f h z C")
nipkow@15392
   827
   apply simp
nipkow@15392
   828
  apply (erule finite_induct, simp)
nipkow@15392
   829
  apply (simp add: subset_insert_iff, clarify)
nipkow@15392
   830
  apply (subgoal_tac "finite C")
nipkow@15392
   831
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
nipkow@15392
   832
  apply (subgoal_tac "C = insert x (C - {x})")
nipkow@15392
   833
   prefer 2 apply blast
nipkow@15392
   834
  apply (erule ssubst)
nipkow@15392
   835
  apply (drule spec)
nipkow@15392
   836
  apply (erule (1) notE impE)
nipkow@15392
   837
  apply (simp add: Ball_def del: insert_Diff_single)
nipkow@15392
   838
  done
nipkow@15392
   839
nipkow@15392
   840
lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@15392
   841
  fold f (%x. fold f (g x) e (B x)) e A =
nipkow@15392
   842
  fold f (split g) e (SIGMA x:A. B x)"
nipkow@15392
   843
apply (subst Sigma_def)
nipkow@15392
   844
apply (subst fold_UN_disjoint)
nipkow@15392
   845
   apply assumption
nipkow@15392
   846
  apply simp
nipkow@15392
   847
 apply blast
nipkow@15392
   848
apply (erule fold_cong)
nipkow@15392
   849
apply (subst fold_UN_disjoint)
nipkow@15392
   850
   apply simp
nipkow@15392
   851
  apply simp
nipkow@15392
   852
 apply blast
nipkow@15392
   853
apply (simp)
nipkow@15392
   854
done
nipkow@15392
   855
nipkow@15392
   856
lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
nipkow@15392
   857
   fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
nipkow@15392
   858
apply (erule finite_induct)
nipkow@15392
   859
 apply simp
nipkow@15392
   860
apply (simp add:AC)
nipkow@15392
   861
done
nipkow@15392
   862
nipkow@15392
   863
nipkow@15402
   864
subsection {* Generalized summation over a set *}
nipkow@15402
   865
nipkow@15402
   866
constdefs
nipkow@15402
   867
  setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
nipkow@15402
   868
  "setsum f A == if finite A then fold (op +) f 0 A else 0"
nipkow@15402
   869
nipkow@15402
   870
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
nipkow@15402
   871
written @{text"\<Sum>x\<in>A. e"}. *}
nipkow@15402
   872
nipkow@15402
   873
syntax
nipkow@15402
   874
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
nipkow@15402
   875
syntax (xsymbols)
nipkow@15402
   876
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
   877
syntax (HTML output)
nipkow@15402
   878
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
   879
nipkow@15402
   880
translations -- {* Beware of argument permutation! *}
nipkow@15402
   881
  "SUM i:A. b" == "setsum (%i. b) A"
nipkow@15402
   882
  "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
nipkow@15402
   883
nipkow@15402
   884
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
nipkow@15402
   885
 @{text"\<Sum>x|P. e"}. *}
nipkow@15402
   886
nipkow@15402
   887
syntax
nipkow@15402
   888
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
nipkow@15402
   889
syntax (xsymbols)
nipkow@15402
   890
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402
   891
syntax (HTML output)
nipkow@15402
   892
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402
   893
nipkow@15402
   894
translations
nipkow@15402
   895
  "SUM x|P. t" => "setsum (%x. t) {x. P}"
nipkow@15402
   896
  "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
nipkow@15402
   897
nipkow@15402
   898
text{* Finally we abbreviate @{term"\<Sum>x\<in>A. x"} by @{text"\<Sum>A"}. *}
nipkow@15402
   899
nipkow@15402
   900
syntax
nipkow@15402
   901
  "_Setsum" :: "'a set => 'a::comm_monoid_mult"  ("\<Sum>_" [1000] 999)
nipkow@15402
   902
nipkow@15402
   903
parse_translation {*
nipkow@15402
   904
  let
nipkow@15402
   905
    fun Setsum_tr [A] = Syntax.const "setsum" $ Abs ("", dummyT, Bound 0) $ A
nipkow@15402
   906
  in [("_Setsum", Setsum_tr)] end;
nipkow@15402
   907
*}
nipkow@15402
   908
nipkow@15402
   909
print_translation {*
nipkow@15402
   910
let
nipkow@15402
   911
  fun setsum_tr' [Abs(_,_,Bound 0), A] = Syntax.const "_Setsum" $ A
nipkow@15402
   912
    | setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
nipkow@15402
   913
       if x<>y then raise Match
nipkow@15402
   914
       else let val x' = Syntax.mark_bound x
nipkow@15402
   915
                val t' = subst_bound(x',t)
nipkow@15402
   916
                val P' = subst_bound(x',P)
nipkow@15402
   917
            in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
nipkow@15402
   918
in
nipkow@15402
   919
[("setsum", setsum_tr')]
nipkow@15402
   920
end
nipkow@15402
   921
*}
nipkow@15402
   922
nipkow@15402
   923
lemma setsum_empty [simp]: "setsum f {} = 0"
nipkow@15402
   924
  by (simp add: setsum_def)
nipkow@15402
   925
nipkow@15402
   926
lemma setsum_insert [simp]:
nipkow@15402
   927
    "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
nipkow@15402
   928
  by (simp add: setsum_def ACf.fold_insert [OF ACf_add])
nipkow@15402
   929
paulson@15409
   930
lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
paulson@15409
   931
  by (simp add: setsum_def)
paulson@15409
   932
nipkow@15402
   933
lemma setsum_reindex:
nipkow@15402
   934
     "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
nipkow@15402
   935
by(auto simp add: setsum_def ACf.fold_reindex[OF ACf_add] dest!:finite_imageD)
nipkow@15402
   936
nipkow@15402
   937
lemma setsum_reindex_id:
nipkow@15402
   938
     "inj_on f B ==> setsum f B = setsum id (f ` B)"
nipkow@15402
   939
by (auto simp add: setsum_reindex)
nipkow@15402
   940
nipkow@15402
   941
lemma setsum_cong:
nipkow@15402
   942
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
nipkow@15402
   943
by(fastsimp simp: setsum_def intro: ACf.fold_cong[OF ACf_add])
nipkow@15402
   944
nipkow@15402
   945
lemma setsum_reindex_cong:
nipkow@15402
   946
     "[|inj_on f A; B = f ` A; !!a. g a = h (f a)|] 
nipkow@15402
   947
      ==> setsum h B = setsum g A"
nipkow@15402
   948
  by (simp add: setsum_reindex cong: setsum_cong)
nipkow@15402
   949
nipkow@15402
   950
lemma setsum_0: "setsum (%i. 0) A = 0"
nipkow@15402
   951
apply (clarsimp simp: setsum_def)
nipkow@15402
   952
apply (erule finite_induct, auto simp:ACf.fold_insert [OF ACf_add])
nipkow@15402
   953
done
nipkow@15402
   954
nipkow@15402
   955
lemma setsum_0': "ALL a:F. f a = 0 ==> setsum f F = 0"
nipkow@15402
   956
  apply (subgoal_tac "setsum f F = setsum (%x. 0) F")
nipkow@15402
   957
  apply (erule ssubst, rule setsum_0)
nipkow@15402
   958
  apply (rule setsum_cong, auto)
nipkow@15402
   959
  done
nipkow@15402
   960
nipkow@15402
   961
lemma setsum_Un_Int: "finite A ==> finite B ==>
nipkow@15402
   962
  setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
nipkow@15402
   963
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
nipkow@15402
   964
by(simp add: setsum_def ACe.fold_Un_Int[OF ACe_add,symmetric])
nipkow@15402
   965
nipkow@15402
   966
lemma setsum_Un_disjoint: "finite A ==> finite B
nipkow@15402
   967
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
nipkow@15402
   968
by (subst setsum_Un_Int [symmetric], auto)
nipkow@15402
   969
paulson@15409
   970
(*But we can't get rid of finite I. If infinite, although the rhs is 0, 
paulson@15409
   971
  the lhs need not be, since UNION I A could still be finite.*)
nipkow@15402
   972
lemma setsum_UN_disjoint:
nipkow@15402
   973
    "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402
   974
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402
   975
      setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
nipkow@15402
   976
by(simp add: setsum_def ACe.fold_UN_disjoint[OF ACe_add] cong: setsum_cong)
nipkow@15402
   977
paulson@15409
   978
text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
paulson@15409
   979
directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
nipkow@15402
   980
lemma setsum_Union_disjoint:
paulson@15409
   981
  "[| (ALL A:C. finite A);
paulson@15409
   982
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
paulson@15409
   983
   ==> setsum f (Union C) = setsum (setsum f) C"
paulson@15409
   984
apply (cases "finite C") 
paulson@15409
   985
 prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
nipkow@15402
   986
  apply (frule setsum_UN_disjoint [of C id f])
paulson@15409
   987
 apply (unfold Union_def id_def, assumption+)
paulson@15409
   988
done
nipkow@15402
   989
paulson@15409
   990
(*But we can't get rid of finite A. If infinite, although the lhs is 0, 
paulson@15409
   991
  the rhs need not be, since SIGMA A B could still be finite.*)
nipkow@15402
   992
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@15402
   993
    (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) =
nipkow@15402
   994
    (\<Sum>z\<in>(SIGMA x:A. B x). f (fst z) (snd z))"
nipkow@15402
   995
by(simp add:setsum_def ACe.fold_Sigma[OF ACe_add] split_def cong:setsum_cong)
nipkow@15402
   996
paulson@15409
   997
text{*Here we can eliminate the finiteness assumptions, by cases.*}
paulson@15409
   998
lemma setsum_cartesian_product: 
paulson@15409
   999
   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>z\<in>A <*> B. f (fst z) (snd z))"
paulson@15409
  1000
apply (cases "finite A") 
paulson@15409
  1001
 apply (cases "finite B") 
paulson@15409
  1002
  apply (simp add: setsum_Sigma)
paulson@15409
  1003
 apply (cases "A={}", simp)
paulson@15409
  1004
 apply (simp add: setsum_0) 
paulson@15409
  1005
apply (auto simp add: setsum_def
paulson@15409
  1006
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
paulson@15409
  1007
done
nipkow@15402
  1008
nipkow@15402
  1009
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
nipkow@15402
  1010
by(simp add:setsum_def ACe.fold_distrib[OF ACe_add])
nipkow@15402
  1011
nipkow@15402
  1012
nipkow@15402
  1013
subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402
  1014
nipkow@15402
  1015
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
nipkow@15402
  1016
  apply (case_tac "finite A")
nipkow@15402
  1017
   prefer 2 apply (simp add: setsum_def)
nipkow@15402
  1018
  apply (erule rev_mp)
nipkow@15402
  1019
  apply (erule finite_induct, auto)
nipkow@15402
  1020
  done
nipkow@15402
  1021
nipkow@15402
  1022
lemma setsum_eq_0_iff [simp]:
nipkow@15402
  1023
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
nipkow@15402
  1024
  by (induct set: Finites) auto
nipkow@15402
  1025
nipkow@15402
  1026
lemma setsum_Un_nat: "finite A ==> finite B ==>
nipkow@15402
  1027
    (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
nipkow@15402
  1028
  -- {* For the natural numbers, we have subtraction. *}
nipkow@15402
  1029
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
nipkow@15402
  1030
nipkow@15402
  1031
lemma setsum_Un: "finite A ==> finite B ==>
nipkow@15402
  1032
    (setsum f (A Un B) :: 'a :: ab_group_add) =
nipkow@15402
  1033
      setsum f A + setsum f B - setsum f (A Int B)"
nipkow@15402
  1034
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
nipkow@15402
  1035
nipkow@15402
  1036
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
nipkow@15402
  1037
    (if a:A then setsum f A - f a else setsum f A)"
nipkow@15402
  1038
  apply (case_tac "finite A")
nipkow@15402
  1039
   prefer 2 apply (simp add: setsum_def)
nipkow@15402
  1040
  apply (erule finite_induct)
nipkow@15402
  1041
   apply (auto simp add: insert_Diff_if)
nipkow@15402
  1042
  apply (drule_tac a = a in mk_disjoint_insert, auto)
nipkow@15402
  1043
  done
nipkow@15402
  1044
nipkow@15402
  1045
lemma setsum_diff1: "finite A \<Longrightarrow>
nipkow@15402
  1046
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
nipkow@15402
  1047
  (if a:A then setsum f A - f a else setsum f A)"
nipkow@15402
  1048
  by (erule finite_induct) (auto simp add: insert_Diff_if)
nipkow@15402
  1049
nipkow@15402
  1050
(* By Jeremy Siek: *)
nipkow@15402
  1051
nipkow@15402
  1052
lemma setsum_diff_nat: 
nipkow@15402
  1053
  assumes finB: "finite B"
nipkow@15402
  1054
  shows "B \<subseteq> A \<Longrightarrow> (setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
nipkow@15402
  1055
using finB
nipkow@15402
  1056
proof (induct)
nipkow@15402
  1057
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
nipkow@15402
  1058
next
nipkow@15402
  1059
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
nipkow@15402
  1060
    and xFinA: "insert x F \<subseteq> A"
nipkow@15402
  1061
    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
nipkow@15402
  1062
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
nipkow@15402
  1063
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
nipkow@15402
  1064
    by (simp add: setsum_diff1_nat)
nipkow@15402
  1065
  from xFinA have "F \<subseteq> A" by simp
nipkow@15402
  1066
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
nipkow@15402
  1067
  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
nipkow@15402
  1068
    by simp
nipkow@15402
  1069
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
nipkow@15402
  1070
  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
nipkow@15402
  1071
    by simp
nipkow@15402
  1072
  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
nipkow@15402
  1073
  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
nipkow@15402
  1074
    by simp
nipkow@15402
  1075
  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
nipkow@15402
  1076
qed
nipkow@15402
  1077
nipkow@15402
  1078
lemma setsum_diff:
nipkow@15402
  1079
  assumes le: "finite A" "B \<subseteq> A"
nipkow@15402
  1080
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
nipkow@15402
  1081
proof -
nipkow@15402
  1082
  from le have finiteB: "finite B" using finite_subset by auto
nipkow@15402
  1083
  show ?thesis using finiteB le
nipkow@15402
  1084
    proof (induct)
nipkow@15402
  1085
      case empty
nipkow@15402
  1086
      thus ?case by auto
nipkow@15402
  1087
    next
nipkow@15402
  1088
      case (insert x F)
nipkow@15402
  1089
      thus ?case using le finiteB 
nipkow@15402
  1090
	by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
nipkow@15402
  1091
    qed
nipkow@15402
  1092
  qed
nipkow@15402
  1093
nipkow@15402
  1094
lemma setsum_mono:
nipkow@15402
  1095
  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
nipkow@15402
  1096
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
nipkow@15402
  1097
proof (cases "finite K")
nipkow@15402
  1098
  case True
nipkow@15402
  1099
  thus ?thesis using le
nipkow@15402
  1100
  proof (induct)
nipkow@15402
  1101
    case empty
nipkow@15402
  1102
    thus ?case by simp
nipkow@15402
  1103
  next
nipkow@15402
  1104
    case insert
nipkow@15402
  1105
    thus ?case using add_mono 
nipkow@15402
  1106
      by force
nipkow@15402
  1107
  qed
nipkow@15402
  1108
next
nipkow@15402
  1109
  case False
nipkow@15402
  1110
  thus ?thesis
nipkow@15402
  1111
    by (simp add: setsum_def)
nipkow@15402
  1112
qed
nipkow@15402
  1113
nipkow@15402
  1114
lemma setsum_mono2_nat:
nipkow@15402
  1115
  assumes fin: "finite B" and sub: "A \<subseteq> B"
nipkow@15402
  1116
shows "setsum f A \<le> (setsum f B :: nat)"
nipkow@15402
  1117
proof -
nipkow@15402
  1118
  have "setsum f A \<le> setsum f A + setsum f (B-A)" by arith
nipkow@15402
  1119
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
nipkow@15402
  1120
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
nipkow@15402
  1121
  also have "A \<union> (B-A) = B" using sub by blast
nipkow@15402
  1122
  finally show ?thesis .
nipkow@15402
  1123
qed
nipkow@15402
  1124
nipkow@15402
  1125
lemma setsum_negf: "finite A ==> setsum (%x. - (f x)::'a::ab_group_add) A =
nipkow@15402
  1126
  - setsum f A"
nipkow@15402
  1127
  by (induct set: Finites, auto)
nipkow@15402
  1128
nipkow@15402
  1129
lemma setsum_subtractf: "finite A ==> setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
nipkow@15402
  1130
  setsum f A - setsum g A"
nipkow@15402
  1131
  by (simp add: diff_minus setsum_addf setsum_negf)
nipkow@15402
  1132
nipkow@15402
  1133
lemma setsum_nonneg: "[| finite A;
nipkow@15402
  1134
    \<forall>x \<in> A. (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) \<le> f x |] ==>
nipkow@15402
  1135
    0 \<le> setsum f A";
nipkow@15402
  1136
  apply (induct set: Finites, auto)
nipkow@15402
  1137
  apply (subgoal_tac "0 + 0 \<le> f x + setsum f F", simp)
nipkow@15402
  1138
  apply (blast intro: add_mono)
nipkow@15402
  1139
  done
nipkow@15402
  1140
nipkow@15402
  1141
lemma setsum_nonpos: "[| finite A;
nipkow@15402
  1142
    \<forall>x \<in> A. f x \<le> (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) |] ==>
nipkow@15402
  1143
    setsum f A \<le> 0";
nipkow@15402
  1144
  apply (induct set: Finites, auto)
nipkow@15402
  1145
  apply (subgoal_tac "f x + setsum f F \<le> 0 + 0", simp)
nipkow@15402
  1146
  apply (blast intro: add_mono)
nipkow@15402
  1147
  done
nipkow@15402
  1148
nipkow@15402
  1149
lemma setsum_mult: 
nipkow@15402
  1150
  fixes f :: "'a => ('b::semiring_0_cancel)"
nipkow@15402
  1151
  shows "r * setsum f A = setsum (%n. r * f n) A"
nipkow@15402
  1152
proof (cases "finite A")
nipkow@15402
  1153
  case True
nipkow@15402
  1154
  thus ?thesis
nipkow@15402
  1155
  proof (induct)
nipkow@15402
  1156
    case empty thus ?case by simp
nipkow@15402
  1157
  next
nipkow@15402
  1158
    case (insert x A) thus ?case by (simp add: right_distrib)
nipkow@15402
  1159
  qed
nipkow@15402
  1160
next
nipkow@15402
  1161
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15402
  1162
qed
nipkow@15402
  1163
nipkow@15402
  1164
lemma setsum_abs: 
nipkow@15402
  1165
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
nipkow@15402
  1166
  assumes fin: "finite A" 
nipkow@15402
  1167
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
nipkow@15402
  1168
using fin 
nipkow@15402
  1169
proof (induct) 
nipkow@15402
  1170
  case empty thus ?case by simp
nipkow@15402
  1171
next
nipkow@15402
  1172
  case (insert x A)
nipkow@15402
  1173
  thus ?case by (auto intro: abs_triangle_ineq order_trans)
nipkow@15402
  1174
qed
nipkow@15402
  1175
nipkow@15402
  1176
lemma setsum_abs_ge_zero: 
nipkow@15402
  1177
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
nipkow@15402
  1178
  assumes fin: "finite A" 
nipkow@15402
  1179
  shows "0 \<le> setsum (%i. abs(f i)) A"
nipkow@15402
  1180
using fin 
nipkow@15402
  1181
proof (induct) 
nipkow@15402
  1182
  case empty thus ?case by simp
nipkow@15402
  1183
next
nipkow@15402
  1184
  case (insert x A) thus ?case by (auto intro: order_trans)
nipkow@15402
  1185
qed
nipkow@15402
  1186
nipkow@15402
  1187
nipkow@15402
  1188
subsection {* Generalized product over a set *}
nipkow@15402
  1189
nipkow@15402
  1190
constdefs
nipkow@15402
  1191
  setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
nipkow@15402
  1192
  "setprod f A == if finite A then fold (op *) f 1 A else 1"
nipkow@15402
  1193
nipkow@15402
  1194
syntax
nipkow@15402
  1195
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_:_. _)" [0, 51, 10] 10)
nipkow@15402
  1196
nipkow@15402
  1197
syntax (xsymbols)
nipkow@15402
  1198
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
  1199
syntax (HTML output)
nipkow@15402
  1200
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
  1201
translations
nipkow@15402
  1202
  "\<Prod>i:A. b" == "setprod (%i. b) A"  -- {* Beware of argument permutation! *}
nipkow@15402
  1203
nipkow@15402
  1204
syntax
nipkow@15402
  1205
  "_Setprod" :: "'a set => 'a::comm_monoid_mult"  ("\<Prod>_" [1000] 999)
nipkow@15402
  1206
nipkow@15402
  1207
parse_translation {*
nipkow@15402
  1208
  let
nipkow@15402
  1209
    fun Setprod_tr [A] = Syntax.const "setprod" $ Abs ("", dummyT, Bound 0) $ A
nipkow@15402
  1210
  in [("_Setprod", Setprod_tr)] end;
nipkow@15402
  1211
*}
nipkow@15402
  1212
print_translation {*
nipkow@15402
  1213
let fun setprod_tr' [Abs(x,Tx,t), A] =
nipkow@15402
  1214
    if t = Bound 0 then Syntax.const "_Setprod" $ A else raise Match
nipkow@15402
  1215
in
nipkow@15402
  1216
[("setprod", setprod_tr')]
nipkow@15402
  1217
end
nipkow@15402
  1218
*}
nipkow@15402
  1219
nipkow@15402
  1220
nipkow@15402
  1221
lemma setprod_empty [simp]: "setprod f {} = 1"
nipkow@15402
  1222
  by (auto simp add: setprod_def)
nipkow@15402
  1223
nipkow@15402
  1224
lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
nipkow@15402
  1225
    setprod f (insert a A) = f a * setprod f A"
nipkow@15402
  1226
by (simp add: setprod_def ACf.fold_insert [OF ACf_mult])
nipkow@15402
  1227
paulson@15409
  1228
lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
paulson@15409
  1229
  by (simp add: setprod_def)
paulson@15409
  1230
nipkow@15402
  1231
lemma setprod_reindex:
nipkow@15402
  1232
     "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
nipkow@15402
  1233
by(auto simp: setprod_def ACf.fold_reindex[OF ACf_mult] dest!:finite_imageD)
nipkow@15402
  1234
nipkow@15402
  1235
lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
nipkow@15402
  1236
by (auto simp add: setprod_reindex)
nipkow@15402
  1237
nipkow@15402
  1238
lemma setprod_cong:
nipkow@15402
  1239
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
nipkow@15402
  1240
by(fastsimp simp: setprod_def intro: ACf.fold_cong[OF ACf_mult])
nipkow@15402
  1241
nipkow@15402
  1242
lemma setprod_reindex_cong: "inj_on f A ==>
nipkow@15402
  1243
    B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
nipkow@15402
  1244
  by (frule setprod_reindex, simp)
nipkow@15402
  1245
nipkow@15402
  1246
nipkow@15402
  1247
lemma setprod_1: "setprod (%i. 1) A = 1"
nipkow@15402
  1248
  apply (case_tac "finite A")
nipkow@15402
  1249
  apply (erule finite_induct, auto simp add: mult_ac)
nipkow@15402
  1250
  done
nipkow@15402
  1251
nipkow@15402
  1252
lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
nipkow@15402
  1253
  apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
nipkow@15402
  1254
  apply (erule ssubst, rule setprod_1)
nipkow@15402
  1255
  apply (rule setprod_cong, auto)
nipkow@15402
  1256
  done
nipkow@15402
  1257
nipkow@15402
  1258
lemma setprod_Un_Int: "finite A ==> finite B
nipkow@15402
  1259
    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
nipkow@15402
  1260
by(simp add: setprod_def ACe.fold_Un_Int[OF ACe_mult,symmetric])
nipkow@15402
  1261
nipkow@15402
  1262
lemma setprod_Un_disjoint: "finite A ==> finite B
nipkow@15402
  1263
  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
nipkow@15402
  1264
by (subst setprod_Un_Int [symmetric], auto)
nipkow@15402
  1265
nipkow@15402
  1266
lemma setprod_UN_disjoint:
nipkow@15402
  1267
    "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402
  1268
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402
  1269
      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
nipkow@15402
  1270
by(simp add: setprod_def ACe.fold_UN_disjoint[OF ACe_mult] cong: setprod_cong)
nipkow@15402
  1271
nipkow@15402
  1272
lemma setprod_Union_disjoint:
paulson@15409
  1273
  "[| (ALL A:C. finite A);
paulson@15409
  1274
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |] 
paulson@15409
  1275
   ==> setprod f (Union C) = setprod (setprod f) C"
paulson@15409
  1276
apply (cases "finite C") 
paulson@15409
  1277
 prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
nipkow@15402
  1278
  apply (frule setprod_UN_disjoint [of C id f])
paulson@15409
  1279
 apply (unfold Union_def id_def, assumption+)
paulson@15409
  1280
done
nipkow@15402
  1281
nipkow@15402
  1282
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@15402
  1283
    (\<Prod>x:A. (\<Prod>y: B x. f x y)) =
nipkow@15402
  1284
    (\<Prod>z:(SIGMA x:A. B x). f (fst z) (snd z))"
nipkow@15402
  1285
by(simp add:setprod_def ACe.fold_Sigma[OF ACe_mult] split_def cong:setprod_cong)
nipkow@15402
  1286
paulson@15409
  1287
text{*Here we can eliminate the finiteness assumptions, by cases.*}
paulson@15409
  1288
lemma setprod_cartesian_product: 
paulson@15409
  1289
     "(\<Prod>x:A. (\<Prod>y: B. f x y)) = (\<Prod>z:(A <*> B). f (fst z) (snd z))"
paulson@15409
  1290
apply (cases "finite A") 
paulson@15409
  1291
 apply (cases "finite B") 
paulson@15409
  1292
  apply (simp add: setprod_Sigma)
paulson@15409
  1293
 apply (cases "A={}", simp)
paulson@15409
  1294
 apply (simp add: setprod_1) 
paulson@15409
  1295
apply (auto simp add: setprod_def
paulson@15409
  1296
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
paulson@15409
  1297
done
nipkow@15402
  1298
nipkow@15402
  1299
lemma setprod_timesf:
paulson@15409
  1300
     "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
nipkow@15402
  1301
by(simp add:setprod_def ACe.fold_distrib[OF ACe_mult])
nipkow@15402
  1302
nipkow@15402
  1303
nipkow@15402
  1304
subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402
  1305
nipkow@15402
  1306
lemma setprod_eq_1_iff [simp]:
nipkow@15402
  1307
    "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
nipkow@15402
  1308
  by (induct set: Finites) auto
nipkow@15402
  1309
nipkow@15402
  1310
lemma setprod_zero:
nipkow@15402
  1311
     "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
nipkow@15402
  1312
  apply (induct set: Finites, force, clarsimp)
nipkow@15402
  1313
  apply (erule disjE, auto)
nipkow@15402
  1314
  done
nipkow@15402
  1315
nipkow@15402
  1316
lemma setprod_nonneg [rule_format]:
nipkow@15402
  1317
     "(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
nipkow@15402
  1318
  apply (case_tac "finite A")
nipkow@15402
  1319
  apply (induct set: Finites, force, clarsimp)
nipkow@15402
  1320
  apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
nipkow@15402
  1321
  apply (rule mult_mono, assumption+)
nipkow@15402
  1322
  apply (auto simp add: setprod_def)
nipkow@15402
  1323
  done
nipkow@15402
  1324
nipkow@15402
  1325
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
nipkow@15402
  1326
     --> 0 < setprod f A"
nipkow@15402
  1327
  apply (case_tac "finite A")
nipkow@15402
  1328
  apply (induct set: Finites, force, clarsimp)
nipkow@15402
  1329
  apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
nipkow@15402
  1330
  apply (rule mult_strict_mono, assumption+)
nipkow@15402
  1331
  apply (auto simp add: setprod_def)
nipkow@15402
  1332
  done
nipkow@15402
  1333
nipkow@15402
  1334
lemma setprod_nonzero [rule_format]:
nipkow@15402
  1335
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
nipkow@15402
  1336
      finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
nipkow@15402
  1337
  apply (erule finite_induct, auto)
nipkow@15402
  1338
  done
nipkow@15402
  1339
nipkow@15402
  1340
lemma setprod_zero_eq:
nipkow@15402
  1341
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
nipkow@15402
  1342
     finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
nipkow@15402
  1343
  apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
nipkow@15402
  1344
  done
nipkow@15402
  1345
nipkow@15402
  1346
lemma setprod_nonzero_field:
nipkow@15402
  1347
    "finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0"
nipkow@15402
  1348
  apply (rule setprod_nonzero, auto)
nipkow@15402
  1349
  done
nipkow@15402
  1350
nipkow@15402
  1351
lemma setprod_zero_eq_field:
nipkow@15402
  1352
    "finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)"
nipkow@15402
  1353
  apply (rule setprod_zero_eq, auto)
nipkow@15402
  1354
  done
nipkow@15402
  1355
nipkow@15402
  1356
lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
nipkow@15402
  1357
    (setprod f (A Un B) :: 'a ::{field})
nipkow@15402
  1358
      = setprod f A * setprod f B / setprod f (A Int B)"
nipkow@15402
  1359
  apply (subst setprod_Un_Int [symmetric], auto)
nipkow@15402
  1360
  apply (subgoal_tac "finite (A Int B)")
nipkow@15402
  1361
  apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
nipkow@15402
  1362
  apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self)
nipkow@15402
  1363
  done
nipkow@15402
  1364
nipkow@15402
  1365
lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
nipkow@15402
  1366
    (setprod f (A - {a}) :: 'a :: {field}) =
nipkow@15402
  1367
      (if a:A then setprod f A / f a else setprod f A)"
nipkow@15402
  1368
  apply (erule finite_induct)
nipkow@15402
  1369
   apply (auto simp add: insert_Diff_if)
nipkow@15402
  1370
  apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a")
nipkow@15402
  1371
  apply (erule ssubst)
nipkow@15402
  1372
  apply (subst times_divide_eq_right [THEN sym])
nipkow@15402
  1373
  apply (auto simp add: mult_ac times_divide_eq_right divide_self)
nipkow@15402
  1374
  done
nipkow@15402
  1375
nipkow@15402
  1376
lemma setprod_inversef: "finite A ==>
nipkow@15402
  1377
    ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
nipkow@15402
  1378
      setprod (inverse \<circ> f) A = inverse (setprod f A)"
nipkow@15402
  1379
  apply (erule finite_induct)
nipkow@15402
  1380
  apply (simp, simp)
nipkow@15402
  1381
  done
nipkow@15402
  1382
nipkow@15402
  1383
lemma setprod_dividef:
nipkow@15402
  1384
     "[|finite A;
nipkow@15402
  1385
        \<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
nipkow@15402
  1386
      ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
nipkow@15402
  1387
  apply (subgoal_tac
nipkow@15402
  1388
         "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
nipkow@15402
  1389
  apply (erule ssubst)
nipkow@15402
  1390
  apply (subst divide_inverse)
nipkow@15402
  1391
  apply (subst setprod_timesf)
nipkow@15402
  1392
  apply (subst setprod_inversef, assumption+, rule refl)
nipkow@15402
  1393
  apply (rule setprod_cong, rule refl)
nipkow@15402
  1394
  apply (subst divide_inverse, auto)
nipkow@15402
  1395
  done
nipkow@15402
  1396
wenzelm@12396
  1397
subsection {* Finite cardinality *}
wenzelm@12396
  1398
nipkow@15402
  1399
text {* This definition, although traditional, is ugly to work with:
nipkow@15402
  1400
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
nipkow@15402
  1401
But now that we have @{text setsum} things are easy:
wenzelm@12396
  1402
*}
wenzelm@12396
  1403
wenzelm@12396
  1404
constdefs
wenzelm@12396
  1405
  card :: "'a set => nat"
nipkow@15402
  1406
  "card A == setsum (%x. 1::nat) A"
wenzelm@12396
  1407
wenzelm@12396
  1408
lemma card_empty [simp]: "card {} = 0"
nipkow@15402
  1409
  by (simp add: card_def)
nipkow@15402
  1410
paulson@15409
  1411
lemma card_infinite [simp]: "~ finite A ==> card A = 0"
paulson@15409
  1412
  by (simp add: card_def)
paulson@15409
  1413
nipkow@15402
  1414
lemma card_eq_setsum: "card A = setsum (%x. 1) A"
nipkow@15402
  1415
by (simp add: card_def)
wenzelm@12396
  1416
wenzelm@12396
  1417
lemma card_insert_disjoint [simp]:
wenzelm@12396
  1418
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
nipkow@15402
  1419
by(simp add: card_def ACf.fold_insert[OF ACf_add])
nipkow@15402
  1420
nipkow@15402
  1421
lemma card_insert_if:
nipkow@15402
  1422
    "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
nipkow@15402
  1423
  by (simp add: insert_absorb)
wenzelm@12396
  1424
wenzelm@12396
  1425
lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
wenzelm@12396
  1426
  apply auto
paulson@14208
  1427
  apply (drule_tac a = x in mk_disjoint_insert, clarify)
nipkow@15402
  1428
  apply (auto)
wenzelm@12396
  1429
  done
wenzelm@12396
  1430
paulson@15409
  1431
lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
paulson@15409
  1432
by auto
paulson@15409
  1433
wenzelm@12396
  1434
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
nipkow@14302
  1435
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
nipkow@14302
  1436
apply(simp del:insert_Diff_single)
nipkow@14302
  1437
done
wenzelm@12396
  1438
wenzelm@12396
  1439
lemma card_Diff_singleton:
wenzelm@12396
  1440
    "finite A ==> x: A ==> card (A - {x}) = card A - 1"
wenzelm@12396
  1441
  by (simp add: card_Suc_Diff1 [symmetric])
wenzelm@12396
  1442
wenzelm@12396
  1443
lemma card_Diff_singleton_if:
wenzelm@12396
  1444
    "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
wenzelm@12396
  1445
  by (simp add: card_Diff_singleton)
wenzelm@12396
  1446
wenzelm@12396
  1447
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
wenzelm@12396
  1448
  by (simp add: card_insert_if card_Suc_Diff1)
wenzelm@12396
  1449
wenzelm@12396
  1450
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
wenzelm@12396
  1451
  by (simp add: card_insert_if)
wenzelm@12396
  1452
nipkow@15402
  1453
lemma card_mono: "\<lbrakk> finite B; A \<subseteq> B \<rbrakk> \<Longrightarrow> card A \<le> card B"
nipkow@15402
  1454
by (simp add: card_def setsum_mono2_nat)
nipkow@15402
  1455
wenzelm@12396
  1456
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
paulson@14208
  1457
  apply (induct set: Finites, simp, clarify)
wenzelm@12396
  1458
  apply (subgoal_tac "finite A & A - {x} <= F")
paulson@14208
  1459
   prefer 2 apply (blast intro: finite_subset, atomize)
wenzelm@12396
  1460
  apply (drule_tac x = "A - {x}" in spec)
wenzelm@12396
  1461
  apply (simp add: card_Diff_singleton_if split add: split_if_asm)
paulson@14208
  1462
  apply (case_tac "card A", auto)
wenzelm@12396
  1463
  done
wenzelm@12396
  1464
wenzelm@12396
  1465
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
wenzelm@12396
  1466
  apply (simp add: psubset_def linorder_not_le [symmetric])
wenzelm@12396
  1467
  apply (blast dest: card_seteq)
wenzelm@12396
  1468
  done
wenzelm@12396
  1469
wenzelm@12396
  1470
lemma card_Un_Int: "finite A ==> finite B
wenzelm@12396
  1471
    ==> card A + card B = card (A Un B) + card (A Int B)"
nipkow@15402
  1472
by(simp add:card_def setsum_Un_Int)
wenzelm@12396
  1473
wenzelm@12396
  1474
lemma card_Un_disjoint: "finite A ==> finite B
wenzelm@12396
  1475
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
wenzelm@12396
  1476
  by (simp add: card_Un_Int)
wenzelm@12396
  1477
wenzelm@12396
  1478
lemma card_Diff_subset:
nipkow@15402
  1479
  "finite B ==> B <= A ==> card (A - B) = card A - card B"
nipkow@15402
  1480
by(simp add:card_def setsum_diff_nat)
wenzelm@12396
  1481
wenzelm@12396
  1482
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
wenzelm@12396
  1483
  apply (rule Suc_less_SucD)
wenzelm@12396
  1484
  apply (simp add: card_Suc_Diff1)
wenzelm@12396
  1485
  done
wenzelm@12396
  1486
wenzelm@12396
  1487
lemma card_Diff2_less:
wenzelm@12396
  1488
    "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
wenzelm@12396
  1489
  apply (case_tac "x = y")
wenzelm@12396
  1490
   apply (simp add: card_Diff1_less)
wenzelm@12396
  1491
  apply (rule less_trans)
wenzelm@12396
  1492
   prefer 2 apply (auto intro!: card_Diff1_less)
wenzelm@12396
  1493
  done
wenzelm@12396
  1494
wenzelm@12396
  1495
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
wenzelm@12396
  1496
  apply (case_tac "x : A")
wenzelm@12396
  1497
   apply (simp_all add: card_Diff1_less less_imp_le)
wenzelm@12396
  1498
  done
wenzelm@12396
  1499
wenzelm@12396
  1500
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
paulson@14208
  1501
by (erule psubsetI, blast)
wenzelm@12396
  1502
paulson@14889
  1503
lemma insert_partition:
nipkow@15402
  1504
  "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
nipkow@15402
  1505
  \<Longrightarrow> x \<inter> \<Union> F = {}"
paulson@14889
  1506
by auto
paulson@14889
  1507
paulson@14889
  1508
(* main cardinality theorem *)
paulson@14889
  1509
lemma card_partition [rule_format]:
paulson@14889
  1510
     "finite C ==>  
paulson@14889
  1511
        finite (\<Union> C) -->  
paulson@14889
  1512
        (\<forall>c\<in>C. card c = k) -->   
paulson@14889
  1513
        (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->  
paulson@14889
  1514
        k * card(C) = card (\<Union> C)"
paulson@14889
  1515
apply (erule finite_induct, simp)
paulson@14889
  1516
apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition 
paulson@14889
  1517
       finite_subset [of _ "\<Union> (insert x F)"])
paulson@14889
  1518
done
paulson@14889
  1519
wenzelm@12396
  1520
paulson@15409
  1521
lemma setsum_constant_nat: "(\<Sum>x\<in>A. y) = (card A) * y"
nipkow@15402
  1522
  -- {* Generalized to any @{text comm_semiring_1_cancel} in
nipkow@15402
  1523
        @{text IntDef} as @{text setsum_constant}. *}
paulson@15409
  1524
apply (cases "finite A") 
paulson@15409
  1525
apply (erule finite_induct, auto)
paulson@15409
  1526
done
nipkow@15402
  1527
nipkow@15402
  1528
lemma setprod_constant: "finite A ==> (\<Prod>x: A. (y::'a::recpower)) = y^(card A)"
nipkow@15402
  1529
  apply (erule finite_induct)
nipkow@15402
  1530
  apply (auto simp add: power_Suc)
nipkow@15402
  1531
  done
nipkow@15402
  1532
nipkow@15402
  1533
nipkow@15402
  1534
subsubsection {* Cardinality of unions *}
nipkow@15402
  1535
nipkow@15402
  1536
lemma card_UN_disjoint:
nipkow@15402
  1537
    "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402
  1538
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402
  1539
      card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
nipkow@15402
  1540
  apply (simp add: card_def)
nipkow@15402
  1541
  apply (subgoal_tac
nipkow@15402
  1542
           "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
nipkow@15402
  1543
  apply (simp add: setsum_UN_disjoint)
nipkow@15402
  1544
  apply (simp add: setsum_constant_nat cong: setsum_cong)
nipkow@15402
  1545
  done
nipkow@15402
  1546
nipkow@15402
  1547
lemma card_Union_disjoint:
nipkow@15402
  1548
  "finite C ==> (ALL A:C. finite A) ==>
nipkow@15402
  1549
        (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
nipkow@15402
  1550
      card (Union C) = setsum card C"
nipkow@15402
  1551
  apply (frule card_UN_disjoint [of C id])
nipkow@15402
  1552
  apply (unfold Union_def id_def, assumption+)
nipkow@15402
  1553
  done
nipkow@15402
  1554
wenzelm@12396
  1555
subsubsection {* Cardinality of image *}
wenzelm@12396
  1556
paulson@15447
  1557
text{*The image of a finite set can be expressed using @{term fold}.*}
paulson@15447
  1558
lemma image_eq_fold: "finite A ==> f ` A = fold (op Un) (%x. {f x}) {} A"
paulson@15447
  1559
  apply (erule finite_induct, simp)
paulson@15447
  1560
  apply (subst ACf.fold_insert) 
paulson@15447
  1561
  apply (auto simp add: ACf_def) 
paulson@15447
  1562
  done
paulson@15447
  1563
wenzelm@12396
  1564
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
paulson@14208
  1565
  apply (induct set: Finites, simp)
wenzelm@12396
  1566
  apply (simp add: le_SucI finite_imageI card_insert_if)
wenzelm@12396
  1567
  done
wenzelm@12396
  1568
nipkow@15402
  1569
lemma card_image: "inj_on f A ==> card (f ` A) = card A"
nipkow@15402
  1570
by(simp add:card_def setsum_reindex o_def)
wenzelm@12396
  1571
wenzelm@12396
  1572
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
wenzelm@12396
  1573
  by (simp add: card_seteq card_image)
wenzelm@12396
  1574
nipkow@15111
  1575
lemma eq_card_imp_inj_on:
nipkow@15111
  1576
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
nipkow@15111
  1577
apply(induct rule:finite_induct)
nipkow@15111
  1578
 apply simp
nipkow@15111
  1579
apply(frule card_image_le[where f = f])
nipkow@15111
  1580
apply(simp add:card_insert_if split:if_splits)
nipkow@15111
  1581
done
nipkow@15111
  1582
nipkow@15111
  1583
lemma inj_on_iff_eq_card:
nipkow@15111
  1584
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
nipkow@15111
  1585
by(blast intro: card_image eq_card_imp_inj_on)
nipkow@15111
  1586
wenzelm@12396
  1587
nipkow@15402
  1588
lemma card_inj_on_le:
nipkow@15402
  1589
    "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
nipkow@15402
  1590
apply (subgoal_tac "finite A") 
nipkow@15402
  1591
 apply (force intro: card_mono simp add: card_image [symmetric])
nipkow@15402
  1592
apply (blast intro: finite_imageD dest: finite_subset) 
nipkow@15402
  1593
done
nipkow@15402
  1594
nipkow@15402
  1595
lemma card_bij_eq:
nipkow@15402
  1596
    "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
nipkow@15402
  1597
       finite A; finite B |] ==> card A = card B"
nipkow@15402
  1598
  by (auto intro: le_anti_sym card_inj_on_le)
nipkow@15402
  1599
nipkow@15402
  1600
nipkow@15402
  1601
subsubsection {* Cardinality of products *}
nipkow@15402
  1602
nipkow@15402
  1603
(*
nipkow@15402
  1604
lemma SigmaI_insert: "y \<notin> A ==>
nipkow@15402
  1605
  (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
nipkow@15402
  1606
  by auto
nipkow@15402
  1607
*)
nipkow@15402
  1608
nipkow@15402
  1609
lemma card_SigmaI [simp]:
nipkow@15402
  1610
  "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
nipkow@15402
  1611
  \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
nipkow@15402
  1612
by(simp add:card_def setsum_Sigma)
nipkow@15402
  1613
paulson@15409
  1614
lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
paulson@15409
  1615
apply (cases "finite A") 
paulson@15409
  1616
apply (cases "finite B") 
paulson@15409
  1617
  apply (simp add: setsum_constant_nat) 
paulson@15409
  1618
apply (auto simp add: card_eq_0_iff
paulson@15409
  1619
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
paulson@15409
  1620
done
nipkow@15402
  1621
nipkow@15402
  1622
lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
paulson@15409
  1623
by (simp add: card_cartesian_product) 
paulson@15409
  1624
nipkow@15402
  1625
nipkow@15402
  1626
wenzelm@12396
  1627
subsubsection {* Cardinality of the Powerset *}
wenzelm@12396
  1628
wenzelm@12396
  1629
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
wenzelm@12396
  1630
  apply (induct set: Finites)
wenzelm@12396
  1631
   apply (simp_all add: Pow_insert)
paulson@14208
  1632
  apply (subst card_Un_disjoint, blast)
paulson@14208
  1633
    apply (blast intro: finite_imageI, blast)
wenzelm@12396
  1634
  apply (subgoal_tac "inj_on (insert x) (Pow F)")
wenzelm@12396
  1635
   apply (simp add: card_image Pow_insert)
wenzelm@12396
  1636
  apply (unfold inj_on_def)
wenzelm@12396
  1637
  apply (blast elim!: equalityE)
wenzelm@12396
  1638
  done
wenzelm@12396
  1639
nipkow@15392
  1640
text {* Relates to equivalence classes.  Based on a theorem of
nipkow@15392
  1641
F. Kammüller's.  *}
wenzelm@12396
  1642
wenzelm@12396
  1643
lemma dvd_partition:
nipkow@15392
  1644
  "finite (Union C) ==>
wenzelm@12396
  1645
    ALL c : C. k dvd card c ==>
paulson@14430
  1646
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
wenzelm@12396
  1647
  k dvd card (Union C)"
nipkow@15392
  1648
apply(frule finite_UnionD)
nipkow@15392
  1649
apply(rotate_tac -1)
paulson@14208
  1650
  apply (induct set: Finites, simp_all, clarify)
wenzelm@12396
  1651
  apply (subst card_Un_disjoint)
wenzelm@12396
  1652
  apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
wenzelm@12396
  1653
  done
wenzelm@12396
  1654
wenzelm@12396
  1655
nipkow@15392
  1656
subsubsection {* Theorems about @{text "choose"} *}
wenzelm@12396
  1657
wenzelm@12396
  1658
text {*
nipkow@15392
  1659
  \medskip Basic theorem about @{text "choose"}.  By Florian
nipkow@15392
  1660
  Kamm\"uller, tidied by LCP.
wenzelm@12396
  1661
*}
wenzelm@12396
  1662
nipkow@15392
  1663
lemma card_s_0_eq_empty:
nipkow@15392
  1664
    "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
nipkow@15392
  1665
  apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
nipkow@15392
  1666
  apply (simp cong add: rev_conj_cong)
nipkow@15392
  1667
  done
wenzelm@12396
  1668
nipkow@15392
  1669
lemma choose_deconstruct: "finite M ==> x \<notin> M
nipkow@15392
  1670
  ==> {s. s <= insert x M & card(s) = Suc k}
nipkow@15392
  1671
       = {s. s <= M & card(s) = Suc k} Un
nipkow@15392
  1672
         {s. EX t. t <= M & card(t) = k & s = insert x t}"
nipkow@15392
  1673
  apply safe
nipkow@15392
  1674
   apply (auto intro: finite_subset [THEN card_insert_disjoint])
nipkow@15392
  1675
  apply (drule_tac x = "xa - {x}" in spec)
nipkow@15392
  1676
  apply (subgoal_tac "x \<notin> xa", auto)
nipkow@15392
  1677
  apply (erule rev_mp, subst card_Diff_singleton)
nipkow@15392
  1678
  apply (auto intro: finite_subset)
wenzelm@12396
  1679
  done
wenzelm@12396
  1680
nipkow@15392
  1681
text{*There are as many subsets of @{term A} having cardinality @{term k}
nipkow@15392
  1682
 as there are sets obtained from the former by inserting a fixed element
nipkow@15392
  1683
 @{term x} into each.*}
nipkow@15392
  1684
lemma constr_bij:
nipkow@15392
  1685
   "[|finite A; x \<notin> A|] ==>
nipkow@15392
  1686
    card {B. EX C. C <= A & card(C) = k & B = insert x C} =
nipkow@15392
  1687
    card {B. B <= A & card(B) = k}"
nipkow@15392
  1688
  apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
nipkow@15392
  1689
       apply (auto elim!: equalityE simp add: inj_on_def)
nipkow@15392
  1690
    apply (subst Diff_insert0, auto)
nipkow@15392
  1691
   txt {* finiteness of the two sets *}
nipkow@15392
  1692
   apply (rule_tac [2] B = "Pow (A)" in finite_subset)
nipkow@15392
  1693
   apply (rule_tac B = "Pow (insert x A)" in finite_subset)
nipkow@15392
  1694
   apply fast+
wenzelm@12396
  1695
  done
wenzelm@12396
  1696
nipkow@15392
  1697
text {*
nipkow@15392
  1698
  Main theorem: combinatorial statement about number of subsets of a set.
nipkow@15392
  1699
*}
wenzelm@12396
  1700
nipkow@15392
  1701
lemma n_sub_lemma:
nipkow@15392
  1702
  "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
nipkow@15392
  1703
  apply (induct k)
nipkow@15392
  1704
   apply (simp add: card_s_0_eq_empty, atomize)
nipkow@15392
  1705
  apply (rotate_tac -1, erule finite_induct)
nipkow@15392
  1706
   apply (simp_all (no_asm_simp) cong add: conj_cong
nipkow@15392
  1707
     add: card_s_0_eq_empty choose_deconstruct)
nipkow@15392
  1708
  apply (subst card_Un_disjoint)
nipkow@15392
  1709
     prefer 4 apply (force simp add: constr_bij)
nipkow@15392
  1710
    prefer 3 apply force
nipkow@15392
  1711
   prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
nipkow@15392
  1712
     finite_subset [of _ "Pow (insert x F)", standard])
nipkow@15392
  1713
  apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
wenzelm@12396
  1714
  done
wenzelm@12396
  1715
nipkow@15392
  1716
theorem n_subsets:
nipkow@15392
  1717
    "finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
nipkow@15392
  1718
  by (simp add: n_sub_lemma)
nipkow@15392
  1719
nipkow@15392
  1720
nipkow@15392
  1721
subsection{* A fold functional for non-empty sets *}
nipkow@15392
  1722
nipkow@15392
  1723
text{* Does not require start value. *}
wenzelm@12396
  1724
nipkow@15392
  1725
consts
nipkow@15392
  1726
  foldSet1 :: "('a => 'a => 'a) => ('a set \<times> 'a) set"
nipkow@15392
  1727
nipkow@15392
  1728
inductive "foldSet1 f"
nipkow@15392
  1729
intros
nipkow@15392
  1730
foldSet1_singletonI [intro]: "({a}, a) : foldSet1 f"
nipkow@15392
  1731
foldSet1_insertI [intro]:
nipkow@15392
  1732
 "\<lbrakk> (A, x) : foldSet1 f; a \<notin> A; A \<noteq> {} \<rbrakk>
nipkow@15392
  1733
  \<Longrightarrow> (insert a A, f a x) : foldSet1 f"
wenzelm@12396
  1734
nipkow@15392
  1735
constdefs
nipkow@15392
  1736
  fold1 :: "('a => 'a => 'a) => 'a set => 'a"
nipkow@15392
  1737
  "fold1 f A == THE x. (A, x) : foldSet1 f"
nipkow@15392
  1738
nipkow@15392
  1739
lemma foldSet1_nonempty:
nipkow@15392
  1740
 "(A, x) : foldSet1 f \<Longrightarrow> A \<noteq> {}"
nipkow@15392
  1741
by(erule foldSet1.cases, simp_all) 
nipkow@15392
  1742
wenzelm@12396
  1743
nipkow@15392
  1744
inductive_cases empty_foldSet1E [elim!]: "({}, x) : foldSet1 f"
nipkow@15392
  1745
nipkow@15392
  1746
lemma foldSet1_sing[iff]: "(({a},b) : foldSet1 f) = (a = b)"
nipkow@15392
  1747
apply(rule iffI)
nipkow@15392
  1748
 prefer 2 apply fast
nipkow@15392
  1749
apply (erule foldSet1.cases)
nipkow@15392
  1750
 apply blast
nipkow@15392
  1751
apply (erule foldSet1.cases)
nipkow@15392
  1752
 apply blast
nipkow@15392
  1753
apply blast
nipkow@15376
  1754
done
wenzelm@12396
  1755
nipkow@15392
  1756
lemma Diff1_foldSet1:
nipkow@15392
  1757
  "(A - {x}, y) : foldSet1 f ==> x: A ==> (A, f x y) : foldSet1 f"
nipkow@15392
  1758
by (erule insert_Diff [THEN subst], rule foldSet1.intros,
nipkow@15392
  1759
    auto dest!:foldSet1_nonempty)
wenzelm@12396
  1760
nipkow@15392
  1761
lemma foldSet1_imp_finite: "(A, x) : foldSet1 f ==> finite A"
nipkow@15392
  1762
  by (induct set: foldSet1) auto
wenzelm@12396
  1763
nipkow@15392
  1764
lemma finite_nonempty_imp_foldSet1:
nipkow@15392
  1765
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. (A, x) : foldSet1 f"
nipkow@15392
  1766
  by (induct set: Finites) auto
nipkow@15376
  1767
nipkow@15392
  1768
lemma (in ACf) foldSet1_determ_aux:
nipkow@15392
  1769
  "!!A x y. \<lbrakk> card A < n; (A, x) : foldSet1 f; (A, y) : foldSet1 f \<rbrakk> \<Longrightarrow> y = x"
nipkow@15392
  1770
proof (induct n)
nipkow@15392
  1771
  case 0 thus ?case by simp
nipkow@15392
  1772
next
nipkow@15392
  1773
  case (Suc n)
nipkow@15392
  1774
  have IH: "!!A x y. \<lbrakk>card A < n; (A, x) \<in> foldSet1 f; (A, y) \<in> foldSet1 f\<rbrakk>
nipkow@15392
  1775
           \<Longrightarrow> y = x" and card: "card A < Suc n"
nipkow@15392
  1776
  and Afoldx: "(A, x) \<in> foldSet1 f" and Afoldy: "(A, y) \<in> foldSet1 f" .
nipkow@15392
  1777
  from card have "card A < n \<or> card A = n" by arith
nipkow@15392
  1778
  thus ?case
nipkow@15392
  1779
  proof
nipkow@15392
  1780
    assume less: "card A < n"
nipkow@15392
  1781
    show ?thesis by(rule IH[OF less Afoldx Afoldy])
nipkow@15392
  1782
  next
nipkow@15392
  1783
    assume cardA: "card A = n"
nipkow@15392
  1784
    show ?thesis
nipkow@15392
  1785
    proof (rule foldSet1.cases[OF Afoldx])
nipkow@15392
  1786
      fix a assume "(A, x) = ({a}, a)"
nipkow@15392
  1787
      thus "y = x" using Afoldy by (simp add:foldSet1_sing)
nipkow@15392
  1788
    next
nipkow@15392
  1789
      fix Ax ax x'
nipkow@15392
  1790
      assume eq1: "(A, x) = (insert ax Ax, ax \<cdot> x')"
nipkow@15392
  1791
	and x': "(Ax, x') \<in> foldSet1 f" and notinx: "ax \<notin> Ax"
nipkow@15392
  1792
	and Axnon: "Ax \<noteq> {}"
nipkow@15392
  1793
      hence A1: "A = insert ax Ax" and x: "x = ax \<cdot> x'" by auto
nipkow@15392
  1794
      show ?thesis
nipkow@15392
  1795
      proof (rule foldSet1.cases[OF Afoldy])
nipkow@15392
  1796
	fix ay assume "(A, y) = ({ay}, ay)"
nipkow@15392
  1797
	thus ?thesis using eq1 x' Axnon notinx
nipkow@15392
  1798
	  by (fastsimp simp:foldSet1_sing)
nipkow@15392
  1799
      next
nipkow@15392
  1800
	fix Ay ay y'
nipkow@15392
  1801
	assume eq2: "(A, y) = (insert ay Ay, ay \<cdot> y')"
nipkow@15392
  1802
	  and y': "(Ay, y') \<in> foldSet1 f" and notiny: "ay \<notin> Ay"
nipkow@15392
  1803
	  and Aynon: "Ay \<noteq> {}"
nipkow@15392
  1804
	hence A2: "A = insert ay Ay" and y: "y = ay \<cdot> y'" by auto
nipkow@15392
  1805
	have finA: "finite A" by(rule foldSet1_imp_finite[OF Afoldx])
nipkow@15392
  1806
	with cardA A1 notinx have less: "card Ax < n" by simp
nipkow@15392
  1807
	show ?thesis
nipkow@15392
  1808
	proof cases
nipkow@15392
  1809
	  assume "ax = ay"
nipkow@15392
  1810
	  then moreover have "Ax = Ay" using A1 A2 notinx notiny by auto
nipkow@15392
  1811
	  ultimately show ?thesis using IH[OF less x'] y' eq1 eq2 by auto
nipkow@15392
  1812
	next
nipkow@15392
  1813
	  assume diff: "ax \<noteq> ay"
nipkow@15392
  1814
	  let ?B = "Ax - {ay}"
nipkow@15392
  1815
	  have Ax: "Ax = insert ay ?B" and Ay: "Ay = insert ax ?B"
nipkow@15392
  1816
	    using A1 A2 notinx notiny diff by(blast elim!:equalityE)+
nipkow@15392
  1817
	  show ?thesis
nipkow@15392
  1818
	  proof cases
nipkow@15392
  1819
	    assume "?B = {}"
nipkow@15392
  1820
	    with Ax Ay show ?thesis using x' y' x y by(simp add:commute)
nipkow@15392
  1821
	  next
nipkow@15392
  1822
	    assume Bnon: "?B \<noteq> {}"
nipkow@15392
  1823
	    moreover have "finite ?B" using finA A1 by simp
nipkow@15392
  1824
	    ultimately obtain b where Bfoldb: "(?B,b) \<in> foldSet1 f"
nipkow@15392
  1825
	      using finite_nonempty_imp_foldSet1 by(blast)
nipkow@15392
  1826
	    moreover have ayinAx: "ay \<in> Ax" using Ax by(auto)
nipkow@15392
  1827
	    ultimately have "(Ax,ay\<cdot>b) \<in> foldSet1 f" by(rule Diff1_foldSet1)
nipkow@15392
  1828
	    hence "ay\<cdot>b = x'" by(rule IH[OF less x'])
nipkow@15392
  1829
            moreover have "ax\<cdot>b = y'"
nipkow@15392
  1830
	    proof (rule IH[OF _ y'])
nipkow@15392
  1831
	      show "card Ay < n" using Ay cardA A1 notinx finA ayinAx
nipkow@15392
  1832
		by(auto simp:card_Diff1_less)
nipkow@15392
  1833
	    next
nipkow@15392
  1834
	      show "(Ay,ax\<cdot>b) \<in> foldSet1 f" using Ay notinx Bfoldb Bnon
nipkow@15392
  1835
		by fastsimp
nipkow@15392
  1836
	    qed
nipkow@15392
  1837
	    ultimately show ?thesis using x y by(auto simp:AC)
nipkow@15392
  1838
	  qed
nipkow@15392
  1839
	qed
nipkow@15392
  1840
      qed
nipkow@15392
  1841
    qed
nipkow@15392
  1842
  qed
wenzelm@12396
  1843
qed
wenzelm@12396
  1844
nipkow@15392
  1845
nipkow@15392
  1846
lemma (in ACf) foldSet1_determ:
nipkow@15392
  1847
  "(A, x) : foldSet1 f ==> (A, y) : foldSet1 f ==> y = x"
nipkow@15392
  1848
by (blast intro: foldSet1_determ_aux [rule_format])
nipkow@15392
  1849
nipkow@15392
  1850
lemma (in ACf) foldSet1_equality: "(A, y) : foldSet1 f ==> fold1 f A = y"
nipkow@15392
  1851
  by (unfold fold1_def) (blast intro: foldSet1_determ)
nipkow@15392
  1852
nipkow@15483
  1853
lemma fold1_singleton[simp]: "fold1 f {a} = a"
nipkow@15392
  1854
  by (unfold fold1_def) blast
wenzelm@12396
  1855
nipkow@15392
  1856
lemma (in ACf) foldSet1_insert_aux: "x \<notin> A ==> A \<noteq> {} \<Longrightarrow> 
nipkow@15392
  1857
    ((insert x A, v) : foldSet1 f) =
nipkow@15392
  1858
    (EX y. (A, y) : foldSet1 f & v = f x y)"
nipkow@15392
  1859
apply auto
nipkow@15392
  1860
apply (rule_tac A1 = A and f1 = f in finite_nonempty_imp_foldSet1 [THEN exE])
nipkow@15392
  1861
  apply (fastsimp dest: foldSet1_imp_finite)
nipkow@15392
  1862
 apply blast
nipkow@15392
  1863
apply (blast intro: foldSet1_determ)
nipkow@15392
  1864
done
nipkow@15376
  1865
nipkow@15392
  1866
lemma (in ACf) fold1_insert:
nipkow@15392
  1867
  "finite A ==> x \<notin> A ==> A \<noteq> {} \<Longrightarrow> fold1 f (insert x A) = f x (fold1 f A)"
nipkow@15392
  1868
apply (unfold fold1_def)
nipkow@15392
  1869
apply (simp add: foldSet1_insert_aux)
nipkow@15392
  1870
apply (rule the_equality)
nipkow@15392
  1871
apply (auto intro: finite_nonempty_imp_foldSet1
nipkow@15392
  1872
    cong add: conj_cong simp add: fold1_def [symmetric] foldSet1_equality)
nipkow@15392
  1873
done
nipkow@15376
  1874
nipkow@15484
  1875
lemma (in ACIf) fold1_insert2[simp]:
nipkow@15392
  1876
assumes finA: "finite A" and nonA: "A \<noteq> {}"
nipkow@15392
  1877
shows "fold1 f (insert a A) = f a (fold1 f A)"
nipkow@15392
  1878
proof cases
nipkow@15392
  1879
  assume "a \<in> A"
nipkow@15392
  1880
  then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
nipkow@15392
  1881
    by(blast dest: mk_disjoint_insert)
nipkow@15392
  1882
  show ?thesis
nipkow@15392
  1883
  proof cases
nipkow@15392
  1884
    assume "B = {}"
nipkow@15392
  1885
    thus ?thesis using A by(simp add:idem fold1_singleton)
nipkow@15392
  1886
  next
nipkow@15392
  1887
    assume nonB: "B \<noteq> {}"
nipkow@15392
  1888
    from finA A have finB: "finite B" by(blast intro: finite_subset)
nipkow@15392
  1889
    have "fold1 f (insert a A) = fold1 f (insert a B)" using A by simp
nipkow@15392
  1890
    also have "\<dots> = f a (fold1 f B)"
nipkow@15392
  1891
      using finB nonB disj by(simp add: fold1_insert)
nipkow@15392
  1892
    also have "\<dots> = f a (fold1 f A)"
nipkow@15392
  1893
      using A finB nonB disj by(simp add:idem fold1_insert assoc[symmetric])
nipkow@15392
  1894
    finally show ?thesis .
nipkow@15392
  1895
  qed
nipkow@15392
  1896
next
nipkow@15392
  1897
  assume "a \<notin> A"
nipkow@15392
  1898
  with finA nonA show ?thesis by(simp add:fold1_insert)
nipkow@15392
  1899
qed
nipkow@15392
  1900
nipkow@15392
  1901
text{* Now the recursion rules for definitions: *}
nipkow@15392
  1902
nipkow@15392
  1903
lemma fold1_singleton_def: "g \<equiv> fold1 f \<Longrightarrow> g {a} = a"
nipkow@15392
  1904
by(simp add:fold1_singleton)
nipkow@15392
  1905
nipkow@15392
  1906
lemma (in ACf) fold1_insert_def:
nipkow@15392
  1907
  "\<lbrakk> g \<equiv> fold1 f; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
nipkow@15392
  1908
by(simp add:fold1_insert)
nipkow@15392
  1909
nipkow@15392
  1910
lemma (in ACIf) fold1_insert2_def:
nipkow@15392
  1911
  "\<lbrakk> g \<equiv> fold1 f; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
nipkow@15392
  1912
by(simp add:fold1_insert2)
nipkow@15392
  1913
nipkow@15376
  1914
nipkow@15484
  1915
subsubsection{* Lemmas about {@text fold1} *}
nipkow@15484
  1916
nipkow@15484
  1917
lemma (in ACf) fold1_Un:
nipkow@15484
  1918
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  1919
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
nipkow@15484
  1920
       fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
nipkow@15484
  1921
using A
nipkow@15484
  1922
proof(induct rule:finite_ne_induct)
nipkow@15484
  1923
  case singleton thus ?case by(simp add:fold1_insert)
nipkow@15484
  1924
next
nipkow@15484
  1925
  case insert thus ?case by (simp add:fold1_insert assoc)
nipkow@15484
  1926
qed
nipkow@15484
  1927
nipkow@15484
  1928
lemma (in ACIf) fold1_Un2:
nipkow@15484
  1929
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  1930
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
nipkow@15484
  1931
       fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
nipkow@15484
  1932
using A
nipkow@15484
  1933
proof(induct rule:finite_ne_induct)
nipkow@15484
  1934
  case singleton thus ?case by(simp add:fold1_insert2)
nipkow@15484
  1935
next
nipkow@15484
  1936
  case insert thus ?case by (simp add:fold1_insert2 assoc)
nipkow@15484
  1937
qed
nipkow@15484
  1938
nipkow@15484
  1939
lemma (in ACf) fold1_in:
nipkow@15484
  1940
  assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x\<cdot>y \<in> {x,y}"
nipkow@15484
  1941
  shows "fold1 f A \<in> A"
nipkow@15484
  1942
using A
nipkow@15484
  1943
proof (induct rule:finite_ne_induct)
nipkow@15484
  1944
  case singleton thus ?case by(simp)
nipkow@15484
  1945
next
nipkow@15484
  1946
  case insert thus ?case using elem by (force simp add:fold1_insert)
nipkow@15484
  1947
qed
nipkow@15484
  1948
nipkow@15484
  1949
lemma fold1_le:
nipkow@15484
  1950
  assumes ACf: "ACf(f::'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15484
  1951
  and A: "finite (A)" "A \<noteq> {}" and le: "\<And>x y. f x y \<le> x"
nipkow@15484
  1952
  shows "a \<in> A \<Longrightarrow> fold1 f A \<le> a"
nipkow@15484
  1953
using A
nipkow@15484
  1954
proof (induct rule:finite_ne_induct)
nipkow@15484
  1955
  case singleton thus ?case by(simp)
nipkow@15484
  1956
next
nipkow@15484
  1957
  from le have le2: "\<And>x y. f x y \<le> y" by (fastsimp simp:ACf.commute[OF ACf])
nipkow@15484
  1958
  case (insert x F) thus ?case using le le2
nipkow@15484
  1959
    by (simp add:ACf.fold1_insert[OF ACf]) (blast intro: le order_trans)
nipkow@15484
  1960
qed
nipkow@15484
  1961
nipkow@15484
  1962
lemma fold1_ge:
nipkow@15484
  1963
  assumes ACf: "ACf(f::'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15484
  1964
  and A: "finite (A)" "A \<noteq> {}" and ge: "\<And>x y. x \<le> f x y"
nipkow@15484
  1965
  shows "a \<in> A \<Longrightarrow> a \<le> fold1 f A"
nipkow@15484
  1966
using A
nipkow@15484
  1967
proof (induct rule:finite_ne_induct)
nipkow@15484
  1968
  case singleton thus ?case by(simp)
nipkow@15484
  1969
next
nipkow@15484
  1970
  from ge have ge2: "\<And>x y. y \<le> f x y" by (fastsimp simp:ACf.commute[OF ACf])
nipkow@15484
  1971
  case (insert x F) thus ?case using ge ge2
nipkow@15484
  1972
    by (simp add:ACf.fold1_insert[OF ACf]) (blast intro: ge order_trans)
nipkow@15484
  1973
qed
nipkow@15484
  1974
nipkow@15484
  1975
nipkow@15392
  1976
subsection{*Min and Max*}
nipkow@15392
  1977
nipkow@15392
  1978
text{* As an application of @{text fold1} we define the minimal and
nipkow@15392
  1979
maximal element of a (non-empty) set over a linear order. First we
nipkow@15392
  1980
show that @{text min} and @{text max} are ACI: *}
nipkow@15392
  1981
nipkow@15392
  1982
lemma ACf_min: "ACf(min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15392
  1983
apply(rule ACf.intro)
nipkow@15392
  1984
apply(auto simp:min_def)
nipkow@15392
  1985
done
nipkow@15392
  1986
nipkow@15392
  1987
lemma ACIf_min: "ACIf(min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15392
  1988
apply(rule ACIf.intro[OF ACf_min])
nipkow@15392
  1989
apply(rule ACIf_axioms.intro)
nipkow@15392
  1990
apply(auto simp:min_def)
nipkow@15376
  1991
done
nipkow@15376
  1992
nipkow@15392
  1993
lemma ACf_max: "ACf(max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15392
  1994
apply(rule ACf.intro)
nipkow@15392
  1995
apply(auto simp:max_def)
nipkow@15392
  1996
done
nipkow@15392
  1997
nipkow@15392
  1998
lemma ACIf_max: "ACIf(max:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15392
  1999
apply(rule ACIf.intro[OF ACf_max])
nipkow@15392
  2000
apply(rule ACIf_axioms.intro)
nipkow@15392
  2001
apply(auto simp:max_def)
nipkow@15376
  2002
done
wenzelm@12396
  2003
nipkow@15392
  2004
text{* Now we make the definitions: *}
nipkow@15392
  2005
nipkow@15392
  2006
constdefs
nipkow@15392
  2007
  Min :: "('a::linorder)set => 'a"
nipkow@15392
  2008
  "Min  ==  fold1 min"
nipkow@15392
  2009
nipkow@15392
  2010
  Max :: "('a::linorder)set => 'a"
nipkow@15392
  2011
  "Max  ==  fold1 max"
nipkow@15392
  2012
nipkow@15402
  2013
text{* Now we instantiate the recursion equations and declare them
nipkow@15392
  2014
simplification rules: *}
nipkow@15392
  2015
nipkow@15392
  2016
declare
nipkow@15392
  2017
  fold1_singleton_def[OF Min_def, simp]
nipkow@15392
  2018
  ACIf.fold1_insert2_def[OF ACIf_min Min_def, simp]
nipkow@15392
  2019
  fold1_singleton_def[OF Max_def, simp]
nipkow@15392
  2020
  ACIf.fold1_insert2_def[OF ACIf_max Max_def, simp]
nipkow@15392
  2021
nipkow@15484
  2022
text{* Now we instantiate some @{text fold1} properties: *}
nipkow@15392
  2023
nipkow@15392
  2024
lemma Min_in [simp]:
nipkow@15484
  2025
  shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Min A \<in> A"
nipkow@15484
  2026
using ACf.fold1_in[OF ACf_min]
nipkow@15484
  2027
by(fastsimp simp: Min_def min_def)
nipkow@15392
  2028
nipkow@15392
  2029
lemma Max_in [simp]:
nipkow@15484
  2030
  shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Max A \<in> A"
nipkow@15484
  2031
using ACf.fold1_in[OF ACf_max]
nipkow@15484
  2032
by(fastsimp simp: Max_def max_def)
nipkow@15392
  2033
nipkow@15484
  2034
lemma Min_le [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> Min A \<le> x"
nipkow@15484
  2035
by(simp add: Min_def fold1_le[OF ACf_min] min_le_iff_disj)
nipkow@15392
  2036
nipkow@15484
  2037
lemma Max_ge [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> x \<le> Max A"
nipkow@15484
  2038
by(simp add: Max_def fold1_ge[OF ACf_max] le_max_iff_disj)
wenzelm@12396
  2039
nipkow@15042
  2040
end