src/HOL/Finite_Set.thy
author ballarin
Fri Aug 26 10:01:06 2005 +0200 (2005-08-26)
changeset 17149 e2b19c92ef51
parent 17085 5b57f995a179
child 17189 b15f8e094874
permissions -rw-r--r--
Lemmas on dvd, power and finite summation added or strengthened.
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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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                with contributions by Jeremy Avigad
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*)
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header {* Finite sets *}
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theory Finite_Set
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imports Power Inductive Lattice_Locales
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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_inj_on:
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  assumes fin: "finite A" 
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  shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}"
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using fin
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proof induct
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  case empty
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  show ?case  
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  proof show "\<exists>f. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp 
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  qed
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next
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  case (insert a A)
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  have notinA: "a \<notin> A" .
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  from insert.hyps obtain n f
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    where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
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  hence "insert a A = f(n:=a) ` {i. i < Suc n}"
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        "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
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    by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
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  thus ?case by blast
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qed
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lemma nat_seg_image_imp_finite:
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  "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
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proof (induct n)
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  case 0 thus ?case by simp
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next
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  case (Suc n)
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  let ?B = "f ` {i. i < n}"
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  have finB: "finite ?B" by(rule Suc.hyps[OF refl])
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  show ?case
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  proof cases
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    assume "\<exists>k<n. f n = f k"
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    hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  next
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    assume "\<not>(\<exists> k<n. f n = f k)"
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    hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  qed
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qed
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lemma finite_conv_nat_seg_image:
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  "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
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by(blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
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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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lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"
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by (simp add: Plus_def)
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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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   331
apply (drule_tac x=n in spec) 
paulson@15409
   332
apply (drule_tac x="fst o f" in spec) 
paulson@15409
   333
apply (auto simp add: o_def) 
paulson@15409
   334
 prefer 2 apply (force dest!: equalityD2) 
paulson@15409
   335
apply (drule equalityD1) 
paulson@15409
   336
apply (rename_tac y x)
paulson@15409
   337
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
paulson@15409
   338
 prefer 2 apply force
paulson@15409
   339
apply clarify
paulson@15409
   340
apply (rule_tac x=k in image_eqI, auto)
paulson@15409
   341
done
paulson@15409
   342
paulson@15409
   343
lemma finite_cartesian_productD2:
paulson@15409
   344
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
paulson@15409
   345
apply (auto simp add: finite_conv_nat_seg_image) 
paulson@15409
   346
apply (drule_tac x=n in spec) 
paulson@15409
   347
apply (drule_tac x="snd o f" in spec) 
paulson@15409
   348
apply (auto simp add: o_def) 
paulson@15409
   349
 prefer 2 apply (force dest!: equalityD2) 
paulson@15409
   350
apply (drule equalityD1)
paulson@15409
   351
apply (rename_tac x y)
paulson@15409
   352
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
paulson@15409
   353
 prefer 2 apply force
paulson@15409
   354
apply clarify
paulson@15409
   355
apply (rule_tac x=k in image_eqI, auto)
paulson@15409
   356
done
paulson@15409
   357
paulson@15409
   358
nipkow@15392
   359
text {* The powerset of a finite set *}
wenzelm@12396
   360
wenzelm@12396
   361
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
wenzelm@12396
   362
proof
wenzelm@12396
   363
  assume "finite (Pow A)"
wenzelm@12396
   364
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
wenzelm@12396
   365
  thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
wenzelm@12396
   366
next
wenzelm@12396
   367
  assume "finite A"
wenzelm@12396
   368
  thus "finite (Pow A)"
wenzelm@12396
   369
    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
wenzelm@12396
   370
qed
wenzelm@12396
   371
nipkow@15392
   372
nipkow@15392
   373
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
nipkow@15392
   374
by(blast intro: finite_subset[OF subset_Pow_Union])
nipkow@15392
   375
nipkow@15392
   376
wenzelm@12396
   377
lemma finite_converse [iff]: "finite (r^-1) = finite r"
wenzelm@12396
   378
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
wenzelm@12396
   379
   apply simp
wenzelm@12396
   380
   apply (rule iffI)
wenzelm@12396
   381
    apply (erule finite_imageD [unfolded inj_on_def])
wenzelm@12396
   382
    apply (simp split add: split_split)
wenzelm@12396
   383
   apply (erule finite_imageI)
paulson@14208
   384
  apply (simp add: converse_def image_def, auto)
wenzelm@12396
   385
  apply (rule bexI)
wenzelm@12396
   386
   prefer 2 apply assumption
wenzelm@12396
   387
  apply simp
wenzelm@12396
   388
  done
wenzelm@12396
   389
paulson@14430
   390
nipkow@15392
   391
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
nipkow@15392
   392
Ehmety) *}
wenzelm@12396
   393
wenzelm@12396
   394
lemma finite_Field: "finite r ==> finite (Field r)"
wenzelm@12396
   395
  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
wenzelm@12396
   396
  apply (induct set: Finites)
wenzelm@12396
   397
   apply (auto simp add: Field_def Domain_insert Range_insert)
wenzelm@12396
   398
  done
wenzelm@12396
   399
wenzelm@12396
   400
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
wenzelm@12396
   401
  apply clarify
wenzelm@12396
   402
  apply (erule trancl_induct)
wenzelm@12396
   403
   apply (auto simp add: Field_def)
wenzelm@12396
   404
  done
wenzelm@12396
   405
wenzelm@12396
   406
lemma finite_trancl: "finite (r^+) = finite r"
wenzelm@12396
   407
  apply auto
wenzelm@12396
   408
   prefer 2
wenzelm@12396
   409
   apply (rule trancl_subset_Field2 [THEN finite_subset])
wenzelm@12396
   410
   apply (rule finite_SigmaI)
wenzelm@12396
   411
    prefer 3
berghofe@13704
   412
    apply (blast intro: r_into_trancl' finite_subset)
wenzelm@12396
   413
   apply (auto simp add: finite_Field)
wenzelm@12396
   414
  done
wenzelm@12396
   415
wenzelm@12396
   416
nipkow@15392
   417
subsection {* A fold functional for finite sets *}
nipkow@15392
   418
nipkow@15392
   419
text {* The intended behaviour is
nipkow@15480
   420
@{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
   421
if @{text f} is associative-commutative. For an application of @{text fold}
nipkow@15392
   422
se the definitions of sums and products over finite sets.
nipkow@15392
   423
*}
nipkow@15392
   424
nipkow@15392
   425
consts
nipkow@15392
   426
  foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
nipkow@15392
   427
nipkow@15480
   428
inductive "foldSet f g z"
nipkow@15392
   429
intros
nipkow@15480
   430
emptyI [intro]: "({}, z) : foldSet f g z"
paulson@15506
   431
insertI [intro]:
paulson@15506
   432
     "\<lbrakk> x \<notin> A; (A, y) : foldSet f g z \<rbrakk>
paulson@15506
   433
      \<Longrightarrow> (insert x A, f (g x) y) : foldSet f g z"
nipkow@15392
   434
nipkow@15480
   435
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g z"
nipkow@15392
   436
nipkow@15392
   437
constdefs
nipkow@15392
   438
  fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
nipkow@15480
   439
  "fold f g z A == THE x. (A, x) : foldSet f g z"
nipkow@15392
   440
paulson@15498
   441
text{*A tempting alternative for the definiens is
paulson@15498
   442
@{term "if finite A then THE x. (A, x) : foldSet f g e else e"}.
paulson@15498
   443
It allows the removal of finiteness assumptions from the theorems
paulson@15498
   444
@{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}.
paulson@15498
   445
The proofs become ugly, with @{text rule_format}. It is not worth the effort.*}
paulson@15498
   446
paulson@15498
   447
nipkow@15392
   448
lemma Diff1_foldSet:
nipkow@15480
   449
  "(A - {x}, y) : foldSet f g z ==> x: A ==> (A, f (g x) y) : foldSet f g z"
nipkow@15392
   450
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
nipkow@15392
   451
nipkow@15480
   452
lemma foldSet_imp_finite: "(A, x) : foldSet f g z ==> finite A"
nipkow@15392
   453
  by (induct set: foldSet) auto
nipkow@15392
   454
nipkow@15480
   455
lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f g z"
nipkow@15392
   456
  by (induct set: Finites) auto
nipkow@15392
   457
nipkow@15392
   458
nipkow@15392
   459
subsubsection {* Commutative monoids *}
nipkow@15480
   460
nipkow@15392
   461
locale ACf =
nipkow@15392
   462
  fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
nipkow@15392
   463
  assumes commute: "x \<cdot> y = y \<cdot> x"
nipkow@15392
   464
    and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
nipkow@15392
   465
nipkow@15392
   466
locale ACe = ACf +
nipkow@15392
   467
  fixes e :: 'a
nipkow@15392
   468
  assumes ident [simp]: "x \<cdot> e = x"
nipkow@15392
   469
nipkow@15480
   470
locale ACIf = ACf +
nipkow@15480
   471
  assumes idem: "x \<cdot> x = x"
nipkow@15480
   472
nipkow@15392
   473
lemma (in ACf) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
nipkow@15392
   474
proof -
nipkow@15392
   475
  have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
nipkow@15392
   476
  also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
nipkow@15392
   477
  also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
nipkow@15392
   478
  finally show ?thesis .
nipkow@15392
   479
qed
nipkow@15392
   480
nipkow@15392
   481
lemmas (in ACf) AC = assoc commute left_commute
nipkow@15392
   482
nipkow@15392
   483
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
nipkow@15392
   484
proof -
nipkow@15392
   485
  have "x \<cdot> e = x" by (rule ident)
nipkow@15392
   486
  thus ?thesis by (subst commute)
nipkow@15392
   487
qed
nipkow@15392
   488
nipkow@15497
   489
lemma (in ACIf) idem2: "x \<cdot> (x \<cdot> y) = x \<cdot> y"
nipkow@15497
   490
proof -
nipkow@15497
   491
  have "x \<cdot> (x \<cdot> y) = (x \<cdot> x) \<cdot> y" by(simp add:assoc)
nipkow@15497
   492
  also have "\<dots> = x \<cdot> y" by(simp add:idem)
nipkow@15497
   493
  finally show ?thesis .
nipkow@15497
   494
qed
nipkow@15497
   495
nipkow@15497
   496
lemmas (in ACIf) ACI = AC idem idem2
nipkow@15497
   497
ballarin@15765
   498
text{* Interpretation of locales: *}
ballarin@15765
   499
ballarin@15765
   500
interpretation AC_add: ACe ["op +" "0::'a::comm_monoid_add"]
ballarin@15765
   501
by(auto intro: ACf.intro ACe_axioms.intro add_assoc add_commute)
nipkow@15402
   502
ballarin@15765
   503
interpretation AC_mult: ACe ["op *" "1::'a::comm_monoid_mult"]
ballarin@15765
   504
  apply -
nipkow@15780
   505
   apply (fast intro: ACf.intro mult_assoc mult_commute)
nipkow@15780
   506
  apply (fastsimp intro: ACe_axioms.intro mult_assoc mult_commute)
ballarin@15765
   507
  done
ballarin@15765
   508
nipkow@15402
   509
nipkow@15392
   510
subsubsection{*From @{term foldSet} to @{term fold}*}
nipkow@15392
   511
paulson@15510
   512
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
paulson@15510
   513
by (auto simp add: less_Suc_eq) 
paulson@15510
   514
paulson@15510
   515
lemma insert_image_inj_on_eq:
paulson@15510
   516
     "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; 
paulson@15510
   517
        inj_on h {i. i < Suc m}|] 
paulson@15510
   518
      ==> A = h ` {i. i < m}"
paulson@15510
   519
apply (auto simp add: image_less_Suc inj_on_def)
paulson@15510
   520
apply (blast intro: less_trans) 
paulson@15510
   521
done
paulson@15510
   522
paulson@15510
   523
lemma insert_inj_onE:
paulson@15510
   524
  assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" 
paulson@15510
   525
      and inj_on: "inj_on h {i::nat. i<n}"
paulson@15510
   526
  shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"
paulson@15510
   527
proof (cases n)
paulson@15510
   528
  case 0 thus ?thesis using aA by auto
paulson@15510
   529
next
paulson@15510
   530
  case (Suc m)
paulson@15510
   531
  have nSuc: "n = Suc m" . 
paulson@15510
   532
  have mlessn: "m<n" by (simp add: nSuc)
paulson@15532
   533
  from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
paulson@15520
   534
  let ?hm = "swap k m h"
paulson@15520
   535
  have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn 
paulson@15520
   536
    by (simp add: inj_on_swap_iff inj_on)
paulson@15510
   537
  show ?thesis
paulson@15520
   538
  proof (intro exI conjI)
paulson@15520
   539
    show "inj_on ?hm {i. i < m}" using inj_hm
paulson@15510
   540
      by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
paulson@15520
   541
    show "m<n" by (rule mlessn)
paulson@15520
   542
    show "A = ?hm ` {i. i < m}" 
paulson@15520
   543
    proof (rule insert_image_inj_on_eq)
paulson@15520
   544
      show "inj_on (swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
paulson@15520
   545
      show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) 
paulson@15520
   546
      show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
paulson@15520
   547
	using aA hkeq nSuc klessn
paulson@15520
   548
	by (auto simp add: swap_def image_less_Suc fun_upd_image 
paulson@15520
   549
			   less_Suc_eq inj_on_image_set_diff [OF inj_on])
nipkow@15479
   550
    qed
nipkow@15479
   551
  qed
nipkow@15479
   552
qed
nipkow@15479
   553
nipkow@15392
   554
lemma (in ACf) foldSet_determ_aux:
paulson@15510
   555
  "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n}; 
paulson@15510
   556
                (A,x) : foldSet f g z; (A,x') : foldSet f g z \<rbrakk>
nipkow@15392
   557
   \<Longrightarrow> x' = x"
paulson@15510
   558
proof (induct n rule: less_induct)
paulson@15510
   559
  case (less n)
paulson@15510
   560
    have IH: "!!m h A x x'. 
paulson@15510
   561
               \<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m}; 
paulson@15510
   562
                (A,x) \<in> foldSet f g z; (A, x') \<in> foldSet f g z\<rbrakk> \<Longrightarrow> x' = x" .
paulson@15510
   563
    have Afoldx: "(A,x) \<in> foldSet f g z" and Afoldx': "(A,x') \<in> foldSet f g z"
paulson@15510
   564
     and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" .
paulson@15510
   565
    show ?case
paulson@15510
   566
    proof (rule foldSet.cases [OF Afoldx])
paulson@15510
   567
      assume "(A, x) = ({}, z)"
paulson@15510
   568
      with Afoldx' show "x' = x" by blast
nipkow@15392
   569
    next
paulson@15510
   570
      fix B b u
paulson@15510
   571
      assume "(A,x) = (insert b B, g b \<cdot> u)" and notinB: "b \<notin> B"
paulson@15510
   572
         and Bu: "(B,u) \<in> foldSet f g z"
paulson@15510
   573
      hence AbB: "A = insert b B" and x: "x = g b \<cdot> u" by auto
paulson@15510
   574
      show "x'=x" 
paulson@15510
   575
      proof (rule foldSet.cases [OF Afoldx'])
paulson@15510
   576
        assume "(A, x') = ({}, z)"
paulson@15510
   577
        with AbB show "x' = x" by blast
nipkow@15392
   578
      next
paulson@15510
   579
	fix C c v
paulson@15510
   580
	assume "(A,x') = (insert c C, g c \<cdot> v)" and notinC: "c \<notin> C"
paulson@15510
   581
	   and Cv: "(C,v) \<in> foldSet f g z"
paulson@15510
   582
	hence AcC: "A = insert c C" and x': "x' = g c \<cdot> v" by auto
paulson@15510
   583
	from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
paulson@15510
   584
        from insert_inj_onE [OF Beq notinB injh]
paulson@15510
   585
        obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" 
paulson@15510
   586
                     and Beq: "B = hB ` {i. i < mB}"
paulson@15510
   587
                     and lessB: "mB < n" by auto 
paulson@15510
   588
	from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
paulson@15510
   589
        from insert_inj_onE [OF Ceq notinC injh]
paulson@15510
   590
        obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
paulson@15510
   591
                       and Ceq: "C = hC ` {i. i < mC}"
paulson@15510
   592
                       and lessC: "mC < n" by auto 
paulson@15510
   593
	show "x'=x"
nipkow@15392
   594
	proof cases
paulson@15510
   595
          assume "b=c"
paulson@15510
   596
	  then moreover have "B = C" using AbB AcC notinB notinC by auto
paulson@15510
   597
	  ultimately show ?thesis  using Bu Cv x x' IH[OF lessC Ceq inj_onC]
paulson@15510
   598
            by auto
nipkow@15392
   599
	next
nipkow@15392
   600
	  assume diff: "b \<noteq> c"
nipkow@15392
   601
	  let ?D = "B - {c}"
nipkow@15392
   602
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
paulson@15510
   603
	    using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
nipkow@15402
   604
	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
paulson@15510
   605
	  with AbB have "finite ?D" by simp
nipkow@15480
   606
	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g z"
nipkow@15392
   607
	    using finite_imp_foldSet by rules
paulson@15506
   608
	  moreover have cinB: "c \<in> B" using B by auto
nipkow@15480
   609
	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g z"
nipkow@15392
   610
	    by(rule Diff1_foldSet)
paulson@15510
   611
	  hence "g c \<cdot> d = u" by (rule IH [OF lessB Beq inj_onB Bu]) 
paulson@15510
   612
          moreover have "g b \<cdot> d = v"
paulson@15510
   613
	  proof (rule IH[OF lessC Ceq inj_onC Cv])
paulson@15510
   614
	    show "(C, g b \<cdot> d) \<in> foldSet f g z" using C notinB Dfoldd
nipkow@15392
   615
	      by fastsimp
nipkow@15392
   616
	  qed
paulson@15510
   617
	  ultimately show ?thesis using x x' by (auto simp: AC)
nipkow@15392
   618
	qed
nipkow@15392
   619
      qed
nipkow@15392
   620
    qed
nipkow@15392
   621
  qed
nipkow@15392
   622
nipkow@15392
   623
nipkow@15392
   624
lemma (in ACf) foldSet_determ:
paulson@15510
   625
  "(A,x) : foldSet f g z ==> (A,y) : foldSet f g z ==> y = x"
paulson@15510
   626
apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) 
paulson@15510
   627
apply (blast intro: foldSet_determ_aux [rule_format])
nipkow@15392
   628
done
nipkow@15392
   629
nipkow@15480
   630
lemma (in ACf) fold_equality: "(A, y) : foldSet f g z ==> fold f g z A = y"
nipkow@15392
   631
  by (unfold fold_def) (blast intro: foldSet_determ)
nipkow@15392
   632
nipkow@15392
   633
text{* The base case for @{text fold}: *}
nipkow@15392
   634
nipkow@15480
   635
lemma fold_empty [simp]: "fold f g z {} = z"
nipkow@15392
   636
  by (unfold fold_def) blast
nipkow@15392
   637
nipkow@15392
   638
lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
nipkow@15480
   639
    ((insert x A, v) : foldSet f g z) =
nipkow@15480
   640
    (EX y. (A, y) : foldSet f g z & v = f (g x) y)"
nipkow@15392
   641
  apply auto
nipkow@15392
   642
  apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
nipkow@15392
   643
   apply (fastsimp dest: foldSet_imp_finite)
nipkow@15392
   644
  apply (blast intro: foldSet_determ)
nipkow@15392
   645
  done
nipkow@15392
   646
nipkow@15392
   647
text{* The recursion equation for @{text fold}: *}
nipkow@15392
   648
nipkow@15392
   649
lemma (in ACf) fold_insert[simp]:
nipkow@15480
   650
    "finite A ==> x \<notin> A ==> fold f g z (insert x A) = f (g x) (fold f g z A)"
nipkow@15392
   651
  apply (unfold fold_def)
nipkow@15392
   652
  apply (simp add: fold_insert_aux)
nipkow@15392
   653
  apply (rule the_equality)
nipkow@15392
   654
  apply (auto intro: finite_imp_foldSet
nipkow@15392
   655
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
nipkow@15392
   656
  done
nipkow@15392
   657
nipkow@15535
   658
lemma (in ACf) fold_rec:
nipkow@15535
   659
assumes fin: "finite A" and a: "a:A"
nipkow@15535
   660
shows "fold f g z A = f (g a) (fold f g z (A - {a}))"
nipkow@15535
   661
proof-
nipkow@15535
   662
  have A: "A = insert a (A - {a})" using a by blast
nipkow@15535
   663
  hence "fold f g z A = fold f g z (insert a (A - {a}))" by simp
nipkow@15535
   664
  also have "\<dots> = f (g a) (fold f g z (A - {a}))"
nipkow@15535
   665
    by(rule fold_insert) (simp add:fin)+
nipkow@15535
   666
  finally show ?thesis .
nipkow@15535
   667
qed
nipkow@15535
   668
nipkow@15392
   669
nipkow@15480
   670
text{* A simplified version for idempotent functions: *}
nipkow@15480
   671
paulson@15509
   672
lemma (in ACIf) fold_insert_idem:
nipkow@15480
   673
assumes finA: "finite A"
paulson@15508
   674
shows "fold f g z (insert a A) = g a \<cdot> fold f g z A"
nipkow@15480
   675
proof cases
nipkow@15480
   676
  assume "a \<in> A"
nipkow@15480
   677
  then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
nipkow@15480
   678
    by(blast dest: mk_disjoint_insert)
nipkow@15480
   679
  show ?thesis
nipkow@15480
   680
  proof -
nipkow@15480
   681
    from finA A have finB: "finite B" by(blast intro: finite_subset)
nipkow@15480
   682
    have "fold f g z (insert a A) = fold f g z (insert a B)" using A by simp
nipkow@15480
   683
    also have "\<dots> = (g a) \<cdot> (fold f g z B)"
paulson@15506
   684
      using finB disj by simp
nipkow@15480
   685
    also have "\<dots> = g a \<cdot> fold f g z A"
nipkow@15480
   686
      using A finB disj by(simp add:idem assoc[symmetric])
nipkow@15480
   687
    finally show ?thesis .
nipkow@15480
   688
  qed
nipkow@15480
   689
next
nipkow@15480
   690
  assume "a \<notin> A"
nipkow@15480
   691
  with finA show ?thesis by simp
nipkow@15480
   692
qed
nipkow@15480
   693
nipkow@15484
   694
lemma (in ACIf) foldI_conv_id:
nipkow@15484
   695
  "finite A \<Longrightarrow> fold f g z A = fold f id z (g ` A)"
paulson@15509
   696
by(erule finite_induct)(simp_all add: fold_insert_idem del: fold_insert)
nipkow@15484
   697
nipkow@15392
   698
subsubsection{*Lemmas about @{text fold}*}
nipkow@15392
   699
nipkow@15392
   700
lemma (in ACf) fold_commute:
paulson@15487
   701
  "finite A ==> (!!z. f x (fold f g z A) = fold f g (f x z) A)"
nipkow@15392
   702
  apply (induct set: Finites, simp)
paulson@15487
   703
  apply (simp add: left_commute [of x])
nipkow@15392
   704
  done
nipkow@15392
   705
nipkow@15392
   706
lemma (in ACf) fold_nest_Un_Int:
nipkow@15392
   707
  "finite A ==> finite B
nipkow@15480
   708
    ==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)"
nipkow@15392
   709
  apply (induct set: Finites, simp)
nipkow@15392
   710
  apply (simp add: fold_commute Int_insert_left insert_absorb)
nipkow@15392
   711
  done
nipkow@15392
   712
nipkow@15392
   713
lemma (in ACf) fold_nest_Un_disjoint:
nipkow@15392
   714
  "finite A ==> finite B ==> A Int B = {}
nipkow@15480
   715
    ==> fold f g z (A Un B) = fold f g (fold f g z B) A"
nipkow@15392
   716
  by (simp add: fold_nest_Un_Int)
nipkow@15392
   717
nipkow@15392
   718
lemma (in ACf) fold_reindex:
paulson@15487
   719
assumes fin: "finite A"
paulson@15487
   720
shows "inj_on h A \<Longrightarrow> fold f g z (h ` A) = fold f (g \<circ> h) z A"
paulson@15506
   721
using fin apply induct
nipkow@15392
   722
 apply simp
nipkow@15392
   723
apply simp
nipkow@15392
   724
done
nipkow@15392
   725
nipkow@15392
   726
lemma (in ACe) fold_Un_Int:
nipkow@15392
   727
  "finite A ==> finite B ==>
nipkow@15392
   728
    fold f g e A \<cdot> fold f g e B =
nipkow@15392
   729
    fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
nipkow@15392
   730
  apply (induct set: Finites, simp)
nipkow@15392
   731
  apply (simp add: AC insert_absorb Int_insert_left)
nipkow@15392
   732
  done
nipkow@15392
   733
nipkow@15392
   734
corollary (in ACe) fold_Un_disjoint:
nipkow@15392
   735
  "finite A ==> finite B ==> A Int B = {} ==>
nipkow@15392
   736
    fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B"
nipkow@15392
   737
  by (simp add: fold_Un_Int)
nipkow@15392
   738
nipkow@15392
   739
lemma (in ACe) fold_UN_disjoint:
nipkow@15392
   740
  "\<lbrakk> finite I; ALL i:I. finite (A i);
nipkow@15392
   741
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
nipkow@15392
   742
   \<Longrightarrow> fold f g e (UNION I A) =
nipkow@15392
   743
       fold f (%i. fold f g e (A i)) e I"
nipkow@15392
   744
  apply (induct set: Finites, simp, atomize)
nipkow@15392
   745
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
nipkow@15392
   746
   prefer 2 apply blast
nipkow@15392
   747
  apply (subgoal_tac "A x Int UNION F A = {}")
nipkow@15392
   748
   prefer 2 apply blast
nipkow@15392
   749
  apply (simp add: fold_Un_disjoint)
nipkow@15392
   750
  done
nipkow@15392
   751
paulson@15506
   752
text{*Fusion theorem, as described in
paulson@15506
   753
Graham Hutton's paper,
paulson@15506
   754
A Tutorial on the Universality and Expressiveness of Fold,
paulson@15506
   755
JFP 9:4 (355-372), 1999.*}
paulson@15506
   756
lemma (in ACf) fold_fusion:
paulson@15506
   757
      includes ACf g
paulson@15506
   758
      shows
paulson@15506
   759
	"finite A ==> 
paulson@15506
   760
	 (!!x y. h (g x y) = f x (h y)) ==>
paulson@15506
   761
         h (fold g j w A) = fold f j (h w) A"
paulson@15506
   762
  by (induct set: Finites, simp_all)
paulson@15506
   763
nipkow@15392
   764
lemma (in ACf) fold_cong:
nipkow@15480
   765
  "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A"
nipkow@15480
   766
  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
   767
   apply simp
nipkow@15392
   768
  apply (erule finite_induct, simp)
nipkow@15392
   769
  apply (simp add: subset_insert_iff, clarify)
nipkow@15392
   770
  apply (subgoal_tac "finite C")
nipkow@15392
   771
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
nipkow@15392
   772
  apply (subgoal_tac "C = insert x (C - {x})")
nipkow@15392
   773
   prefer 2 apply blast
nipkow@15392
   774
  apply (erule ssubst)
nipkow@15392
   775
  apply (drule spec)
nipkow@15392
   776
  apply (erule (1) notE impE)
nipkow@15392
   777
  apply (simp add: Ball_def del: insert_Diff_single)
nipkow@15392
   778
  done
nipkow@15392
   779
nipkow@15392
   780
lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@15392
   781
  fold f (%x. fold f (g x) e (B x)) e A =
nipkow@15392
   782
  fold f (split g) e (SIGMA x:A. B x)"
nipkow@15392
   783
apply (subst Sigma_def)
paulson@15506
   784
apply (subst fold_UN_disjoint, assumption, simp)
nipkow@15392
   785
 apply blast
nipkow@15392
   786
apply (erule fold_cong)
paulson@15506
   787
apply (subst fold_UN_disjoint, simp, simp)
nipkow@15392
   788
 apply blast
paulson@15506
   789
apply simp
nipkow@15392
   790
done
nipkow@15392
   791
nipkow@15392
   792
lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
nipkow@15392
   793
   fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
paulson@15506
   794
apply (erule finite_induct, simp)
nipkow@15392
   795
apply (simp add:AC)
nipkow@15392
   796
done
nipkow@15392
   797
nipkow@15392
   798
nipkow@15402
   799
subsection {* Generalized summation over a set *}
nipkow@15402
   800
nipkow@15402
   801
constdefs
nipkow@15402
   802
  setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
nipkow@15402
   803
  "setsum f A == if finite A then fold (op +) f 0 A else 0"
nipkow@15402
   804
nipkow@15402
   805
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
nipkow@15402
   806
written @{text"\<Sum>x\<in>A. e"}. *}
nipkow@15402
   807
nipkow@15402
   808
syntax
nipkow@15402
   809
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
nipkow@15402
   810
syntax (xsymbols)
nipkow@15402
   811
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
   812
syntax (HTML output)
nipkow@15402
   813
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
   814
nipkow@15402
   815
translations -- {* Beware of argument permutation! *}
nipkow@15402
   816
  "SUM i:A. b" == "setsum (%i. b) A"
nipkow@15402
   817
  "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
nipkow@15402
   818
nipkow@15402
   819
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
nipkow@15402
   820
 @{text"\<Sum>x|P. e"}. *}
nipkow@15402
   821
nipkow@15402
   822
syntax
nipkow@15402
   823
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
nipkow@15402
   824
syntax (xsymbols)
nipkow@15402
   825
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402
   826
syntax (HTML output)
nipkow@15402
   827
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402
   828
nipkow@15402
   829
translations
nipkow@15402
   830
  "SUM x|P. t" => "setsum (%x. t) {x. P}"
nipkow@15402
   831
  "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
nipkow@15402
   832
nipkow@15402
   833
text{* Finally we abbreviate @{term"\<Sum>x\<in>A. x"} by @{text"\<Sum>A"}. *}
nipkow@15402
   834
nipkow@15402
   835
syntax
nipkow@15402
   836
  "_Setsum" :: "'a set => 'a::comm_monoid_mult"  ("\<Sum>_" [1000] 999)
nipkow@15402
   837
nipkow@15402
   838
parse_translation {*
nipkow@15402
   839
  let
nipkow@15402
   840
    fun Setsum_tr [A] = Syntax.const "setsum" $ Abs ("", dummyT, Bound 0) $ A
nipkow@15402
   841
  in [("_Setsum", Setsum_tr)] end;
nipkow@15402
   842
*}
nipkow@15402
   843
nipkow@15402
   844
print_translation {*
nipkow@15402
   845
let
nipkow@15402
   846
  fun setsum_tr' [Abs(_,_,Bound 0), A] = Syntax.const "_Setsum" $ A
nipkow@15402
   847
    | setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
nipkow@15402
   848
       if x<>y then raise Match
nipkow@15402
   849
       else let val x' = Syntax.mark_bound x
nipkow@15402
   850
                val t' = subst_bound(x',t)
nipkow@15402
   851
                val P' = subst_bound(x',P)
nipkow@15402
   852
            in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
nipkow@15402
   853
in
nipkow@15402
   854
[("setsum", setsum_tr')]
nipkow@15402
   855
end
nipkow@15402
   856
*}
nipkow@15402
   857
nipkow@15402
   858
lemma setsum_empty [simp]: "setsum f {} = 0"
nipkow@15402
   859
  by (simp add: setsum_def)
nipkow@15402
   860
nipkow@15402
   861
lemma setsum_insert [simp]:
nipkow@15402
   862
    "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
ballarin@15765
   863
  by (simp add: setsum_def)
nipkow@15402
   864
paulson@15409
   865
lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
paulson@15409
   866
  by (simp add: setsum_def)
paulson@15409
   867
nipkow@15402
   868
lemma setsum_reindex:
nipkow@15402
   869
     "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
ballarin@15765
   870
by(auto simp add: setsum_def AC_add.fold_reindex dest!:finite_imageD)
nipkow@15402
   871
nipkow@15402
   872
lemma setsum_reindex_id:
nipkow@15402
   873
     "inj_on f B ==> setsum f B = setsum id (f ` B)"
nipkow@15402
   874
by (auto simp add: setsum_reindex)
nipkow@15402
   875
nipkow@15402
   876
lemma setsum_cong:
nipkow@15402
   877
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
ballarin@15765
   878
by(fastsimp simp: setsum_def intro: AC_add.fold_cong)
nipkow@15402
   879
nipkow@16733
   880
lemma strong_setsum_cong[cong]:
nipkow@16733
   881
  "A = B ==> (!!x. x:B =simp=> f x = g x)
nipkow@16733
   882
   ==> setsum (%x. f x) A = setsum (%x. g x) B"
berghofe@16632
   883
by(fastsimp simp: simp_implies_def setsum_def intro: AC_add.fold_cong)
berghofe@16632
   884
nipkow@15554
   885
lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A";
nipkow@15554
   886
  by (rule setsum_cong[OF refl], auto);
nipkow@15554
   887
nipkow@15402
   888
lemma setsum_reindex_cong:
nipkow@15554
   889
     "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] 
nipkow@15402
   890
      ==> setsum h B = setsum g A"
nipkow@15402
   891
  by (simp add: setsum_reindex cong: setsum_cong)
nipkow@15402
   892
nipkow@15542
   893
lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
nipkow@15402
   894
apply (clarsimp simp: setsum_def)
ballarin@15765
   895
apply (erule finite_induct, auto)
nipkow@15402
   896
done
nipkow@15402
   897
nipkow@15543
   898
lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
nipkow@15543
   899
by(simp add:setsum_cong)
nipkow@15402
   900
nipkow@15402
   901
lemma setsum_Un_Int: "finite A ==> finite B ==>
nipkow@15402
   902
  setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
nipkow@15402
   903
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
ballarin@15765
   904
by(simp add: setsum_def AC_add.fold_Un_Int [symmetric])
nipkow@15402
   905
nipkow@15402
   906
lemma setsum_Un_disjoint: "finite A ==> finite B
nipkow@15402
   907
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
nipkow@15402
   908
by (subst setsum_Un_Int [symmetric], auto)
nipkow@15402
   909
paulson@15409
   910
(*But we can't get rid of finite I. If infinite, although the rhs is 0, 
paulson@15409
   911
  the lhs need not be, since UNION I A could still be finite.*)
nipkow@15402
   912
lemma setsum_UN_disjoint:
nipkow@15402
   913
    "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402
   914
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402
   915
      setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
ballarin@15765
   916
by(simp add: setsum_def AC_add.fold_UN_disjoint cong: setsum_cong)
nipkow@15402
   917
paulson@15409
   918
text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
paulson@15409
   919
directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
nipkow@15402
   920
lemma setsum_Union_disjoint:
paulson@15409
   921
  "[| (ALL A:C. finite A);
paulson@15409
   922
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
paulson@15409
   923
   ==> setsum f (Union C) = setsum (setsum f) C"
paulson@15409
   924
apply (cases "finite C") 
paulson@15409
   925
 prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
nipkow@15402
   926
  apply (frule setsum_UN_disjoint [of C id f])
paulson@15409
   927
 apply (unfold Union_def id_def, assumption+)
paulson@15409
   928
done
nipkow@15402
   929
paulson@15409
   930
(*But we can't get rid of finite A. If infinite, although the lhs is 0, 
paulson@15409
   931
  the rhs need not be, since SIGMA A B could still be finite.*)
nipkow@15402
   932
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@15402
   933
    (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) =
nipkow@15402
   934
    (\<Sum>z\<in>(SIGMA x:A. B x). f (fst z) (snd z))"
ballarin@15765
   935
by(simp add:setsum_def AC_add.fold_Sigma split_def cong:setsum_cong)
nipkow@15402
   936
paulson@15409
   937
text{*Here we can eliminate the finiteness assumptions, by cases.*}
paulson@15409
   938
lemma setsum_cartesian_product: 
paulson@15409
   939
   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>z\<in>A <*> B. f (fst z) (snd z))"
paulson@15409
   940
apply (cases "finite A") 
paulson@15409
   941
 apply (cases "finite B") 
paulson@15409
   942
  apply (simp add: setsum_Sigma)
paulson@15409
   943
 apply (cases "A={}", simp)
nipkow@15543
   944
 apply (simp) 
paulson@15409
   945
apply (auto simp add: setsum_def
paulson@15409
   946
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
paulson@15409
   947
done
nipkow@15402
   948
nipkow@15402
   949
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
ballarin@15765
   950
by(simp add:setsum_def AC_add.fold_distrib)
nipkow@15402
   951
nipkow@15402
   952
nipkow@15402
   953
subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402
   954
nipkow@15402
   955
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
nipkow@15402
   956
  apply (case_tac "finite A")
nipkow@15402
   957
   prefer 2 apply (simp add: setsum_def)
nipkow@15402
   958
  apply (erule rev_mp)
nipkow@15402
   959
  apply (erule finite_induct, auto)
nipkow@15402
   960
  done
nipkow@15402
   961
nipkow@15402
   962
lemma setsum_eq_0_iff [simp]:
nipkow@15402
   963
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
nipkow@15402
   964
  by (induct set: Finites) auto
nipkow@15402
   965
nipkow@15402
   966
lemma setsum_Un_nat: "finite A ==> finite B ==>
nipkow@15402
   967
    (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
nipkow@15402
   968
  -- {* For the natural numbers, we have subtraction. *}
nipkow@15402
   969
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
nipkow@15402
   970
nipkow@15402
   971
lemma setsum_Un: "finite A ==> finite B ==>
nipkow@15402
   972
    (setsum f (A Un B) :: 'a :: ab_group_add) =
nipkow@15402
   973
      setsum f A + setsum f B - setsum f (A Int B)"
nipkow@15402
   974
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
nipkow@15402
   975
nipkow@15402
   976
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
nipkow@15402
   977
    (if a:A then setsum f A - f a else setsum f A)"
nipkow@15402
   978
  apply (case_tac "finite A")
nipkow@15402
   979
   prefer 2 apply (simp add: setsum_def)
nipkow@15402
   980
  apply (erule finite_induct)
nipkow@15402
   981
   apply (auto simp add: insert_Diff_if)
nipkow@15402
   982
  apply (drule_tac a = a in mk_disjoint_insert, auto)
nipkow@15402
   983
  done
nipkow@15402
   984
nipkow@15402
   985
lemma setsum_diff1: "finite A \<Longrightarrow>
nipkow@15402
   986
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
nipkow@15402
   987
  (if a:A then setsum f A - f a else setsum f A)"
nipkow@15402
   988
  by (erule finite_induct) (auto simp add: insert_Diff_if)
nipkow@15402
   989
obua@15552
   990
lemma setsum_diff1'[rule_format]: "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
obua@15552
   991
  apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
obua@15552
   992
  apply (auto simp add: insert_Diff_if add_ac)
obua@15552
   993
  done
obua@15552
   994
nipkow@15402
   995
(* By Jeremy Siek: *)
nipkow@15402
   996
nipkow@15402
   997
lemma setsum_diff_nat: 
nipkow@15402
   998
  assumes finB: "finite B"
nipkow@15402
   999
  shows "B \<subseteq> A \<Longrightarrow> (setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
nipkow@15402
  1000
using finB
nipkow@15402
  1001
proof (induct)
nipkow@15402
  1002
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
nipkow@15402
  1003
next
nipkow@15402
  1004
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
nipkow@15402
  1005
    and xFinA: "insert x F \<subseteq> A"
nipkow@15402
  1006
    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
nipkow@15402
  1007
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
nipkow@15402
  1008
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
nipkow@15402
  1009
    by (simp add: setsum_diff1_nat)
nipkow@15402
  1010
  from xFinA have "F \<subseteq> A" by simp
nipkow@15402
  1011
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
nipkow@15402
  1012
  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
nipkow@15402
  1013
    by simp
nipkow@15402
  1014
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
nipkow@15402
  1015
  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
nipkow@15402
  1016
    by simp
nipkow@15402
  1017
  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
nipkow@15402
  1018
  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
nipkow@15402
  1019
    by simp
nipkow@15402
  1020
  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
nipkow@15402
  1021
qed
nipkow@15402
  1022
nipkow@15402
  1023
lemma setsum_diff:
nipkow@15402
  1024
  assumes le: "finite A" "B \<subseteq> A"
nipkow@15402
  1025
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
nipkow@15402
  1026
proof -
nipkow@15402
  1027
  from le have finiteB: "finite B" using finite_subset by auto
nipkow@15402
  1028
  show ?thesis using finiteB le
nipkow@15402
  1029
    proof (induct)
nipkow@15402
  1030
      case empty
nipkow@15402
  1031
      thus ?case by auto
nipkow@15402
  1032
    next
nipkow@15402
  1033
      case (insert x F)
nipkow@15402
  1034
      thus ?case using le finiteB 
nipkow@15402
  1035
	by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
nipkow@15402
  1036
    qed
nipkow@15402
  1037
  qed
nipkow@15402
  1038
nipkow@15402
  1039
lemma setsum_mono:
nipkow@15402
  1040
  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
nipkow@15402
  1041
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
nipkow@15402
  1042
proof (cases "finite K")
nipkow@15402
  1043
  case True
nipkow@15402
  1044
  thus ?thesis using le
nipkow@15402
  1045
  proof (induct)
nipkow@15402
  1046
    case empty
nipkow@15402
  1047
    thus ?case by simp
nipkow@15402
  1048
  next
nipkow@15402
  1049
    case insert
nipkow@15402
  1050
    thus ?case using add_mono 
nipkow@15402
  1051
      by force
nipkow@15402
  1052
  qed
nipkow@15402
  1053
next
nipkow@15402
  1054
  case False
nipkow@15402
  1055
  thus ?thesis
nipkow@15402
  1056
    by (simp add: setsum_def)
nipkow@15402
  1057
qed
nipkow@15402
  1058
nipkow@15554
  1059
lemma setsum_strict_mono:
nipkow@15554
  1060
fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
nipkow@15554
  1061
assumes fin_ne: "finite A"  "A \<noteq> {}"
nipkow@15554
  1062
shows "(!!x. x:A \<Longrightarrow> f x < g x) \<Longrightarrow> setsum f A < setsum g A"
nipkow@15554
  1063
using fin_ne
nipkow@15554
  1064
proof (induct rule: finite_ne_induct)
nipkow@15554
  1065
  case singleton thus ?case by simp
nipkow@15554
  1066
next
nipkow@15554
  1067
  case insert thus ?case by (auto simp: add_strict_mono)
nipkow@15554
  1068
qed
nipkow@15554
  1069
nipkow@15535
  1070
lemma setsum_negf:
nipkow@15535
  1071
 "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
nipkow@15535
  1072
proof (cases "finite A")
nipkow@15535
  1073
  case True thus ?thesis by (induct set: Finites, auto)
nipkow@15535
  1074
next
nipkow@15535
  1075
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15535
  1076
qed
nipkow@15402
  1077
nipkow@15535
  1078
lemma setsum_subtractf:
nipkow@15535
  1079
 "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
nipkow@15402
  1080
  setsum f A - setsum g A"
nipkow@15535
  1081
proof (cases "finite A")
nipkow@15535
  1082
  case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
nipkow@15535
  1083
next
nipkow@15535
  1084
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15535
  1085
qed
nipkow@15402
  1086
nipkow@15535
  1087
lemma setsum_nonneg:
nipkow@15535
  1088
assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
nipkow@15535
  1089
shows "0 \<le> setsum f A"
nipkow@15535
  1090
proof (cases "finite A")
nipkow@15535
  1091
  case True thus ?thesis using nn
nipkow@15402
  1092
  apply (induct set: Finites, auto)
nipkow@15402
  1093
  apply (subgoal_tac "0 + 0 \<le> f x + setsum f F", simp)
nipkow@15402
  1094
  apply (blast intro: add_mono)
nipkow@15402
  1095
  done
nipkow@15535
  1096
next
nipkow@15535
  1097
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15535
  1098
qed
nipkow@15402
  1099
nipkow@15535
  1100
lemma setsum_nonpos:
nipkow@15535
  1101
assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
nipkow@15535
  1102
shows "setsum f A \<le> 0"
nipkow@15535
  1103
proof (cases "finite A")
nipkow@15535
  1104
  case True thus ?thesis using np
nipkow@15402
  1105
  apply (induct set: Finites, auto)
nipkow@15402
  1106
  apply (subgoal_tac "f x + setsum f F \<le> 0 + 0", simp)
nipkow@15402
  1107
  apply (blast intro: add_mono)
nipkow@15402
  1108
  done
nipkow@15535
  1109
next
nipkow@15535
  1110
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15535
  1111
qed
nipkow@15402
  1112
nipkow@15539
  1113
lemma setsum_mono2:
nipkow@15539
  1114
fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}"
nipkow@15539
  1115
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
nipkow@15539
  1116
shows "setsum f A \<le> setsum f B"
nipkow@15539
  1117
proof -
nipkow@15539
  1118
  have "setsum f A \<le> setsum f A + setsum f (B-A)"
nipkow@15539
  1119
    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
nipkow@15539
  1120
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
nipkow@15539
  1121
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
nipkow@15539
  1122
  also have "A \<union> (B-A) = B" using sub by blast
nipkow@15539
  1123
  finally show ?thesis .
nipkow@15539
  1124
qed
nipkow@15542
  1125
avigad@16775
  1126
lemma setsum_mono3: "finite B ==> A <= B ==> 
avigad@16775
  1127
    ALL x: B - A. 
avigad@16775
  1128
      0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==>
avigad@16775
  1129
        setsum f A <= setsum f B"
avigad@16775
  1130
  apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
avigad@16775
  1131
  apply (erule ssubst)
avigad@16775
  1132
  apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
avigad@16775
  1133
  apply simp
avigad@16775
  1134
  apply (rule add_left_mono)
avigad@16775
  1135
  apply (erule setsum_nonneg)
avigad@16775
  1136
  apply (subst setsum_Un_disjoint [THEN sym])
avigad@16775
  1137
  apply (erule finite_subset, assumption)
avigad@16775
  1138
  apply (rule finite_subset)
avigad@16775
  1139
  prefer 2
avigad@16775
  1140
  apply assumption
avigad@16775
  1141
  apply auto
avigad@16775
  1142
  apply (rule setsum_cong)
avigad@16775
  1143
  apply auto
avigad@16775
  1144
done
avigad@16775
  1145
ballarin@15837
  1146
(* FIXME: this is distributitivty, name as such! *)
ballarin@17149
  1147
(* suggested name: setsum_right_distrib (CB) *)
ballarin@15837
  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
ballarin@17149
  1164
lemma setsum_left_distrib:
ballarin@17149
  1165
  "setsum f A * (r::'a::semiring_0_cancel) = (\<Sum>n\<in>A. f n * r)"
ballarin@17149
  1166
proof (cases "finite A")
ballarin@17149
  1167
  case True
ballarin@17149
  1168
  then show ?thesis
ballarin@17149
  1169
  proof induct
ballarin@17149
  1170
    case empty thus ?case by simp
ballarin@17149
  1171
  next
ballarin@17149
  1172
    case (insert x A) thus ?case by (simp add: left_distrib)
ballarin@17149
  1173
  qed
ballarin@17149
  1174
next
ballarin@17149
  1175
  case False thus ?thesis by (simp add: setsum_def)
ballarin@17149
  1176
qed
ballarin@17149
  1177
ballarin@17149
  1178
lemma setsum_divide_distrib:
ballarin@17149
  1179
  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
ballarin@17149
  1180
proof (cases "finite A")
ballarin@17149
  1181
  case True
ballarin@17149
  1182
  then show ?thesis
ballarin@17149
  1183
  proof induct
ballarin@17149
  1184
    case empty thus ?case by simp
ballarin@17149
  1185
  next
ballarin@17149
  1186
    case (insert x A) thus ?case by (simp add: add_divide_distrib)
ballarin@17149
  1187
  qed
ballarin@17149
  1188
next
ballarin@17149
  1189
  case False thus ?thesis by (simp add: setsum_def)
ballarin@17149
  1190
qed
ballarin@17149
  1191
nipkow@15535
  1192
lemma setsum_abs[iff]: 
nipkow@15402
  1193
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
nipkow@15402
  1194
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
nipkow@15535
  1195
proof (cases "finite A")
nipkow@15535
  1196
  case True
nipkow@15535
  1197
  thus ?thesis
nipkow@15535
  1198
  proof (induct)
nipkow@15535
  1199
    case empty thus ?case by simp
nipkow@15535
  1200
  next
nipkow@15535
  1201
    case (insert x A)
nipkow@15535
  1202
    thus ?case by (auto intro: abs_triangle_ineq order_trans)
nipkow@15535
  1203
  qed
nipkow@15402
  1204
next
nipkow@15535
  1205
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15402
  1206
qed
nipkow@15402
  1207
nipkow@15535
  1208
lemma setsum_abs_ge_zero[iff]: 
nipkow@15402
  1209
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
nipkow@15402
  1210
  shows "0 \<le> setsum (%i. abs(f i)) A"
nipkow@15535
  1211
proof (cases "finite A")
nipkow@15535
  1212
  case True
nipkow@15535
  1213
  thus ?thesis
nipkow@15535
  1214
  proof (induct)
nipkow@15535
  1215
    case empty thus ?case by simp
nipkow@15535
  1216
  next
nipkow@15535
  1217
    case (insert x A) thus ?case by (auto intro: order_trans)
nipkow@15535
  1218
  qed
nipkow@15402
  1219
next
nipkow@15535
  1220
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15402
  1221
qed
nipkow@15402
  1222
nipkow@15539
  1223
lemma abs_setsum_abs[simp]: 
nipkow@15539
  1224
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
nipkow@15539
  1225
  shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
nipkow@15539
  1226
proof (cases "finite A")
nipkow@15539
  1227
  case True
nipkow@15539
  1228
  thus ?thesis
nipkow@15539
  1229
  proof (induct)
nipkow@15539
  1230
    case empty thus ?case by simp
nipkow@15539
  1231
  next
nipkow@15539
  1232
    case (insert a A)
nipkow@15539
  1233
    hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
nipkow@15539
  1234
    also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
avigad@16775
  1235
    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
avigad@16775
  1236
      by (simp del: abs_of_nonneg)
nipkow@15539
  1237
    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
nipkow@15539
  1238
    finally show ?case .
nipkow@15539
  1239
  qed
nipkow@15539
  1240
next
nipkow@15539
  1241
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15539
  1242
qed
nipkow@15539
  1243
nipkow@15402
  1244
ballarin@17149
  1245
text {* Commuting outer and inner summation *}
ballarin@17149
  1246
ballarin@17149
  1247
lemma swap_inj_on:
ballarin@17149
  1248
  "inj_on (%(i, j). (j, i)) (A \<times> B)"
ballarin@17149
  1249
  by (unfold inj_on_def) fast
ballarin@17149
  1250
ballarin@17149
  1251
lemma swap_product:
ballarin@17149
  1252
  "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
ballarin@17149
  1253
  by (simp add: split_def image_def) blast
ballarin@17149
  1254
ballarin@17149
  1255
lemma setsum_commute:
ballarin@17149
  1256
  "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
ballarin@17149
  1257
proof (simp add: setsum_cartesian_product)
ballarin@17149
  1258
  have "(\<Sum>z\<in>A \<times> B. f (fst z) (snd z)) =
ballarin@17149
  1259
    (\<Sum>z\<in>(%(i, j). (j, i)) ` (A \<times> B). f (snd z) (fst z))"
ballarin@17149
  1260
    (is "?s = _")
ballarin@17149
  1261
    apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
ballarin@17149
  1262
    apply (simp add: split_def)
ballarin@17149
  1263
    done
ballarin@17149
  1264
  also have "... = (\<Sum>z\<in>B \<times> A. f (snd z) (fst z))"
ballarin@17149
  1265
    (is "_ = ?t")
ballarin@17149
  1266
    apply (simp add: swap_product)
ballarin@17149
  1267
    done
ballarin@17149
  1268
  finally show "?s = ?t" .
ballarin@17149
  1269
qed
ballarin@17149
  1270
ballarin@17149
  1271
nipkow@15402
  1272
subsection {* Generalized product over a set *}
nipkow@15402
  1273
nipkow@15402
  1274
constdefs
nipkow@15402
  1275
  setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
nipkow@15402
  1276
  "setprod f A == if finite A then fold (op *) f 1 A else 1"
nipkow@15402
  1277
nipkow@15402
  1278
syntax
nipkow@16550
  1279
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
nipkow@15402
  1280
syntax (xsymbols)
nipkow@15402
  1281
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
  1282
syntax (HTML output)
nipkow@15402
  1283
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@16550
  1284
nipkow@16550
  1285
translations -- {* Beware of argument permutation! *}
nipkow@16550
  1286
  "PROD i:A. b" == "setprod (%i. b) A" 
nipkow@16550
  1287
  "\<Prod>i\<in>A. b" == "setprod (%i. b) A" 
nipkow@16550
  1288
nipkow@16550
  1289
text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
nipkow@16550
  1290
 @{text"\<Prod>x|P. e"}. *}
nipkow@16550
  1291
nipkow@16550
  1292
syntax
nipkow@16550
  1293
  "_qsetprod" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
nipkow@16550
  1294
syntax (xsymbols)
nipkow@16550
  1295
  "_qsetprod" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
nipkow@16550
  1296
syntax (HTML output)
nipkow@16550
  1297
  "_qsetprod" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
nipkow@16550
  1298
nipkow@15402
  1299
translations
nipkow@16550
  1300
  "PROD x|P. t" => "setprod (%x. t) {x. P}"
nipkow@16550
  1301
  "\<Prod>x|P. t" => "setprod (%x. t) {x. P}"
nipkow@16550
  1302
nipkow@16550
  1303
text{* Finally we abbreviate @{term"\<Prod>x\<in>A. x"} by @{text"\<Prod>A"}. *}
nipkow@15402
  1304
nipkow@15402
  1305
syntax
nipkow@15402
  1306
  "_Setprod" :: "'a set => 'a::comm_monoid_mult"  ("\<Prod>_" [1000] 999)
nipkow@15402
  1307
nipkow@15402
  1308
parse_translation {*
nipkow@15402
  1309
  let
nipkow@15402
  1310
    fun Setprod_tr [A] = Syntax.const "setprod" $ Abs ("", dummyT, Bound 0) $ A
nipkow@15402
  1311
  in [("_Setprod", Setprod_tr)] end;
nipkow@15402
  1312
*}
nipkow@15402
  1313
print_translation {*
nipkow@15402
  1314
let fun setprod_tr' [Abs(x,Tx,t), A] =
nipkow@15402
  1315
    if t = Bound 0 then Syntax.const "_Setprod" $ A else raise Match
nipkow@15402
  1316
in
nipkow@15402
  1317
[("setprod", setprod_tr')]
nipkow@15402
  1318
end
nipkow@15402
  1319
*}
nipkow@15402
  1320
nipkow@15402
  1321
nipkow@15402
  1322
lemma setprod_empty [simp]: "setprod f {} = 1"
nipkow@15402
  1323
  by (auto simp add: setprod_def)
nipkow@15402
  1324
nipkow@15402
  1325
lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
nipkow@15402
  1326
    setprod f (insert a A) = f a * setprod f A"
ballarin@15765
  1327
by (simp add: setprod_def)
nipkow@15402
  1328
paulson@15409
  1329
lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
paulson@15409
  1330
  by (simp add: setprod_def)
paulson@15409
  1331
nipkow@15402
  1332
lemma setprod_reindex:
nipkow@15402
  1333
     "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
ballarin@15765
  1334
by(auto simp: setprod_def AC_mult.fold_reindex dest!:finite_imageD)
nipkow@15402
  1335
nipkow@15402
  1336
lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
nipkow@15402
  1337
by (auto simp add: setprod_reindex)
nipkow@15402
  1338
nipkow@15402
  1339
lemma setprod_cong:
nipkow@15402
  1340
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
ballarin@15765
  1341
by(fastsimp simp: setprod_def intro: AC_mult.fold_cong)
nipkow@15402
  1342
berghofe@16632
  1343
lemma strong_setprod_cong:
berghofe@16632
  1344
  "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
berghofe@16632
  1345
by(fastsimp simp: simp_implies_def setprod_def intro: AC_mult.fold_cong)
berghofe@16632
  1346
nipkow@15402
  1347
lemma setprod_reindex_cong: "inj_on f A ==>
nipkow@15402
  1348
    B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
nipkow@15402
  1349
  by (frule setprod_reindex, simp)
nipkow@15402
  1350
nipkow@15402
  1351
nipkow@15402
  1352
lemma setprod_1: "setprod (%i. 1) A = 1"
nipkow@15402
  1353
  apply (case_tac "finite A")
nipkow@15402
  1354
  apply (erule finite_induct, auto simp add: mult_ac)
nipkow@15402
  1355
  done
nipkow@15402
  1356
nipkow@15402
  1357
lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
nipkow@15402
  1358
  apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
nipkow@15402
  1359
  apply (erule ssubst, rule setprod_1)
nipkow@15402
  1360
  apply (rule setprod_cong, auto)
nipkow@15402
  1361
  done
nipkow@15402
  1362
nipkow@15402
  1363
lemma setprod_Un_Int: "finite A ==> finite B
nipkow@15402
  1364
    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
ballarin@15765
  1365
by(simp add: setprod_def AC_mult.fold_Un_Int[symmetric])
nipkow@15402
  1366
nipkow@15402
  1367
lemma setprod_Un_disjoint: "finite A ==> finite B
nipkow@15402
  1368
  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
nipkow@15402
  1369
by (subst setprod_Un_Int [symmetric], auto)
nipkow@15402
  1370
nipkow@15402
  1371
lemma setprod_UN_disjoint:
nipkow@15402
  1372
    "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402
  1373
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402
  1374
      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
ballarin@15765
  1375
by(simp add: setprod_def AC_mult.fold_UN_disjoint cong: setprod_cong)
nipkow@15402
  1376
nipkow@15402
  1377
lemma setprod_Union_disjoint:
paulson@15409
  1378
  "[| (ALL A:C. finite A);
paulson@15409
  1379
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |] 
paulson@15409
  1380
   ==> setprod f (Union C) = setprod (setprod f) C"
paulson@15409
  1381
apply (cases "finite C") 
paulson@15409
  1382
 prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
nipkow@15402
  1383
  apply (frule setprod_UN_disjoint [of C id f])
paulson@15409
  1384
 apply (unfold Union_def id_def, assumption+)
paulson@15409
  1385
done
nipkow@15402
  1386
nipkow@15402
  1387
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@16550
  1388
    (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
nipkow@16550
  1389
    (\<Prod>z\<in>(SIGMA x:A. B x). f (fst z) (snd z))"
ballarin@15765
  1390
by(simp add:setprod_def AC_mult.fold_Sigma split_def cong:setprod_cong)
nipkow@15402
  1391
paulson@15409
  1392
text{*Here we can eliminate the finiteness assumptions, by cases.*}
paulson@15409
  1393
lemma setprod_cartesian_product: 
nipkow@16550
  1394
     "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>z\<in>(A <*> B). f (fst z) (snd z))"
paulson@15409
  1395
apply (cases "finite A") 
paulson@15409
  1396
 apply (cases "finite B") 
paulson@15409
  1397
  apply (simp add: setprod_Sigma)
paulson@15409
  1398
 apply (cases "A={}", simp)
paulson@15409
  1399
 apply (simp add: setprod_1) 
paulson@15409
  1400
apply (auto simp add: setprod_def
paulson@15409
  1401
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
paulson@15409
  1402
done
nipkow@15402
  1403
nipkow@15402
  1404
lemma setprod_timesf:
paulson@15409
  1405
     "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
ballarin@15765
  1406
by(simp add:setprod_def AC_mult.fold_distrib)
nipkow@15402
  1407
nipkow@15402
  1408
nipkow@15402
  1409
subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402
  1410
nipkow@15402
  1411
lemma setprod_eq_1_iff [simp]:
nipkow@15402
  1412
    "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
nipkow@15402
  1413
  by (induct set: Finites) auto
nipkow@15402
  1414
nipkow@15402
  1415
lemma setprod_zero:
nipkow@15402
  1416
     "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
nipkow@15402
  1417
  apply (induct set: Finites, force, clarsimp)
nipkow@15402
  1418
  apply (erule disjE, auto)
nipkow@15402
  1419
  done
nipkow@15402
  1420
nipkow@15402
  1421
lemma setprod_nonneg [rule_format]:
nipkow@15402
  1422
     "(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
nipkow@15402
  1423
  apply (case_tac "finite A")
nipkow@15402
  1424
  apply (induct set: Finites, force, clarsimp)
nipkow@15402
  1425
  apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
nipkow@15402
  1426
  apply (rule mult_mono, assumption+)
nipkow@15402
  1427
  apply (auto simp add: setprod_def)
nipkow@15402
  1428
  done
nipkow@15402
  1429
nipkow@15402
  1430
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
nipkow@15402
  1431
     --> 0 < setprod f A"
nipkow@15402
  1432
  apply (case_tac "finite A")
nipkow@15402
  1433
  apply (induct set: Finites, force, clarsimp)
nipkow@15402
  1434
  apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
nipkow@15402
  1435
  apply (rule mult_strict_mono, assumption+)
nipkow@15402
  1436
  apply (auto simp add: setprod_def)
nipkow@15402
  1437
  done
nipkow@15402
  1438
nipkow@15402
  1439
lemma setprod_nonzero [rule_format]:
nipkow@15402
  1440
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
nipkow@15402
  1441
      finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
nipkow@15402
  1442
  apply (erule finite_induct, auto)
nipkow@15402
  1443
  done
nipkow@15402
  1444
nipkow@15402
  1445
lemma setprod_zero_eq:
nipkow@15402
  1446
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
nipkow@15402
  1447
     finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
nipkow@15402
  1448
  apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
nipkow@15402
  1449
  done
nipkow@15402
  1450
nipkow@15402
  1451
lemma setprod_nonzero_field:
nipkow@15402
  1452
    "finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0"
nipkow@15402
  1453
  apply (rule setprod_nonzero, auto)
nipkow@15402
  1454
  done
nipkow@15402
  1455
nipkow@15402
  1456
lemma setprod_zero_eq_field:
nipkow@15402
  1457
    "finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)"
nipkow@15402
  1458
  apply (rule setprod_zero_eq, auto)
nipkow@15402
  1459
  done
nipkow@15402
  1460
nipkow@15402
  1461
lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
nipkow@15402
  1462
    (setprod f (A Un B) :: 'a ::{field})
nipkow@15402
  1463
      = setprod f A * setprod f B / setprod f (A Int B)"
nipkow@15402
  1464
  apply (subst setprod_Un_Int [symmetric], auto)
nipkow@15402
  1465
  apply (subgoal_tac "finite (A Int B)")
nipkow@15402
  1466
  apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
nipkow@15402
  1467
  apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self)
nipkow@15402
  1468
  done
nipkow@15402
  1469
nipkow@15402
  1470
lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
nipkow@15402
  1471
    (setprod f (A - {a}) :: 'a :: {field}) =
nipkow@15402
  1472
      (if a:A then setprod f A / f a else setprod f A)"
nipkow@15402
  1473
  apply (erule finite_induct)
nipkow@15402
  1474
   apply (auto simp add: insert_Diff_if)
nipkow@15402
  1475
  apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a")
nipkow@15402
  1476
  apply (erule ssubst)
nipkow@15402
  1477
  apply (subst times_divide_eq_right [THEN sym])
nipkow@15402
  1478
  apply (auto simp add: mult_ac times_divide_eq_right divide_self)
nipkow@15402
  1479
  done
nipkow@15402
  1480
nipkow@15402
  1481
lemma setprod_inversef: "finite A ==>
nipkow@15402
  1482
    ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
nipkow@15402
  1483
      setprod (inverse \<circ> f) A = inverse (setprod f A)"
nipkow@15402
  1484
  apply (erule finite_induct)
nipkow@15402
  1485
  apply (simp, simp)
nipkow@15402
  1486
  done
nipkow@15402
  1487
nipkow@15402
  1488
lemma setprod_dividef:
nipkow@15402
  1489
     "[|finite A;
nipkow@15402
  1490
        \<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
nipkow@15402
  1491
      ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
nipkow@15402
  1492
  apply (subgoal_tac
nipkow@15402
  1493
         "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
nipkow@15402
  1494
  apply (erule ssubst)
nipkow@15402
  1495
  apply (subst divide_inverse)
nipkow@15402
  1496
  apply (subst setprod_timesf)
nipkow@15402
  1497
  apply (subst setprod_inversef, assumption+, rule refl)
nipkow@15402
  1498
  apply (rule setprod_cong, rule refl)
nipkow@15402
  1499
  apply (subst divide_inverse, auto)
nipkow@15402
  1500
  done
nipkow@15402
  1501
wenzelm@12396
  1502
subsection {* Finite cardinality *}
wenzelm@12396
  1503
nipkow@15402
  1504
text {* This definition, although traditional, is ugly to work with:
nipkow@15402
  1505
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
nipkow@15402
  1506
But now that we have @{text setsum} things are easy:
wenzelm@12396
  1507
*}
wenzelm@12396
  1508
wenzelm@12396
  1509
constdefs
wenzelm@12396
  1510
  card :: "'a set => nat"
nipkow@15402
  1511
  "card A == setsum (%x. 1::nat) A"
wenzelm@12396
  1512
wenzelm@12396
  1513
lemma card_empty [simp]: "card {} = 0"
nipkow@15402
  1514
  by (simp add: card_def)
nipkow@15402
  1515
paulson@15409
  1516
lemma card_infinite [simp]: "~ finite A ==> card A = 0"
paulson@15409
  1517
  by (simp add: card_def)
paulson@15409
  1518
nipkow@15402
  1519
lemma card_eq_setsum: "card A = setsum (%x. 1) A"
nipkow@15402
  1520
by (simp add: card_def)
wenzelm@12396
  1521
wenzelm@12396
  1522
lemma card_insert_disjoint [simp]:
wenzelm@12396
  1523
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
ballarin@15765
  1524
by(simp add: card_def)
nipkow@15402
  1525
nipkow@15402
  1526
lemma card_insert_if:
nipkow@15402
  1527
    "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
nipkow@15402
  1528
  by (simp add: insert_absorb)
wenzelm@12396
  1529
wenzelm@12396
  1530
lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
wenzelm@12396
  1531
  apply auto
paulson@15506
  1532
  apply (drule_tac a = x in mk_disjoint_insert, clarify, auto)
wenzelm@12396
  1533
  done
wenzelm@12396
  1534
paulson@15409
  1535
lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
paulson@15409
  1536
by auto
paulson@15409
  1537
wenzelm@12396
  1538
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
nipkow@14302
  1539
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
nipkow@14302
  1540
apply(simp del:insert_Diff_single)
nipkow@14302
  1541
done
wenzelm@12396
  1542
wenzelm@12396
  1543
lemma card_Diff_singleton:
wenzelm@12396
  1544
    "finite A ==> x: A ==> card (A - {x}) = card A - 1"
wenzelm@12396
  1545
  by (simp add: card_Suc_Diff1 [symmetric])
wenzelm@12396
  1546
wenzelm@12396
  1547
lemma card_Diff_singleton_if:
wenzelm@12396
  1548
    "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
wenzelm@12396
  1549
  by (simp add: card_Diff_singleton)
wenzelm@12396
  1550
wenzelm@12396
  1551
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
wenzelm@12396
  1552
  by (simp add: card_insert_if card_Suc_Diff1)
wenzelm@12396
  1553
wenzelm@12396
  1554
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
wenzelm@12396
  1555
  by (simp add: card_insert_if)
wenzelm@12396
  1556
nipkow@15402
  1557
lemma card_mono: "\<lbrakk> finite B; A \<subseteq> B \<rbrakk> \<Longrightarrow> card A \<le> card B"
nipkow@15539
  1558
by (simp add: card_def setsum_mono2)
nipkow@15402
  1559
wenzelm@12396
  1560
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
paulson@14208
  1561
  apply (induct set: Finites, simp, clarify)
wenzelm@12396
  1562
  apply (subgoal_tac "finite A & A - {x} <= F")
paulson@14208
  1563
   prefer 2 apply (blast intro: finite_subset, atomize)
wenzelm@12396
  1564
  apply (drule_tac x = "A - {x}" in spec)
wenzelm@12396
  1565
  apply (simp add: card_Diff_singleton_if split add: split_if_asm)
paulson@14208
  1566
  apply (case_tac "card A", auto)
wenzelm@12396
  1567
  done
wenzelm@12396
  1568
wenzelm@12396
  1569
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
wenzelm@12396
  1570
  apply (simp add: psubset_def linorder_not_le [symmetric])
wenzelm@12396
  1571
  apply (blast dest: card_seteq)
wenzelm@12396
  1572
  done
wenzelm@12396
  1573
wenzelm@12396
  1574
lemma card_Un_Int: "finite A ==> finite B
wenzelm@12396
  1575
    ==> card A + card B = card (A Un B) + card (A Int B)"
nipkow@15402
  1576
by(simp add:card_def setsum_Un_Int)
wenzelm@12396
  1577
wenzelm@12396
  1578
lemma card_Un_disjoint: "finite A ==> finite B
wenzelm@12396
  1579
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
wenzelm@12396
  1580
  by (simp add: card_Un_Int)
wenzelm@12396
  1581
wenzelm@12396
  1582
lemma card_Diff_subset:
nipkow@15402
  1583
  "finite B ==> B <= A ==> card (A - B) = card A - card B"
nipkow@15402
  1584
by(simp add:card_def setsum_diff_nat)
wenzelm@12396
  1585
wenzelm@12396
  1586
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
wenzelm@12396
  1587
  apply (rule Suc_less_SucD)
wenzelm@12396
  1588
  apply (simp add: card_Suc_Diff1)
wenzelm@12396
  1589
  done
wenzelm@12396
  1590
wenzelm@12396
  1591
lemma card_Diff2_less:
wenzelm@12396
  1592
    "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
wenzelm@12396
  1593
  apply (case_tac "x = y")
wenzelm@12396
  1594
   apply (simp add: card_Diff1_less)
wenzelm@12396
  1595
  apply (rule less_trans)
wenzelm@12396
  1596
   prefer 2 apply (auto intro!: card_Diff1_less)
wenzelm@12396
  1597
  done
wenzelm@12396
  1598
wenzelm@12396
  1599
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
wenzelm@12396
  1600
  apply (case_tac "x : A")
wenzelm@12396
  1601
   apply (simp_all add: card_Diff1_less less_imp_le)
wenzelm@12396
  1602
  done
wenzelm@12396
  1603
wenzelm@12396
  1604
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
paulson@14208
  1605
by (erule psubsetI, blast)
wenzelm@12396
  1606
paulson@14889
  1607
lemma insert_partition:
nipkow@15402
  1608
  "\<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
  1609
  \<Longrightarrow> x \<inter> \<Union> F = {}"
paulson@14889
  1610
by auto
paulson@14889
  1611
paulson@14889
  1612
(* main cardinality theorem *)
paulson@14889
  1613
lemma card_partition [rule_format]:
paulson@14889
  1614
     "finite C ==>  
paulson@14889
  1615
        finite (\<Union> C) -->  
paulson@14889
  1616
        (\<forall>c\<in>C. card c = k) -->   
paulson@14889
  1617
        (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->  
paulson@14889
  1618
        k * card(C) = card (\<Union> C)"
paulson@14889
  1619
apply (erule finite_induct, simp)
paulson@14889
  1620
apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition 
paulson@14889
  1621
       finite_subset [of _ "\<Union> (insert x F)"])
paulson@14889
  1622
done
paulson@14889
  1623
wenzelm@12396
  1624
nipkow@15539
  1625
lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
nipkow@15539
  1626
apply (cases "finite A")
nipkow@15539
  1627
apply (erule finite_induct)
nipkow@15539
  1628
apply (auto simp add: ring_distrib add_ac)
paulson@15409
  1629
done
nipkow@15402
  1630
nipkow@15539
  1631
nipkow@16550
  1632
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::recpower)) = y^(card A)"
nipkow@15402
  1633
  apply (erule finite_induct)
nipkow@15402
  1634
  apply (auto simp add: power_Suc)
nipkow@15402
  1635
  done
nipkow@15402
  1636
nipkow@15542
  1637
lemma setsum_bounded:
nipkow@15542
  1638
  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{comm_semiring_1_cancel, pordered_ab_semigroup_add})"
nipkow@15542
  1639
  shows "setsum f A \<le> of_nat(card A) * K"
nipkow@15542
  1640
proof (cases "finite A")
nipkow@15542
  1641
  case True
nipkow@15542
  1642
  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
nipkow@15542
  1643
next
nipkow@15542
  1644
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15542
  1645
qed
nipkow@15542
  1646
nipkow@15402
  1647
nipkow@15402
  1648
subsubsection {* Cardinality of unions *}
nipkow@15402
  1649
nipkow@15539
  1650
lemma of_nat_id[simp]: "(of_nat n :: nat) = n"
nipkow@15539
  1651
by(induct n, auto)
nipkow@15539
  1652
nipkow@15402
  1653
lemma card_UN_disjoint:
nipkow@15402
  1654
    "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402
  1655
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402
  1656
      card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
nipkow@15539
  1657
  apply (simp add: card_def del: setsum_constant)
nipkow@15402
  1658
  apply (subgoal_tac
nipkow@15402
  1659
           "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
nipkow@15539
  1660
  apply (simp add: setsum_UN_disjoint del: setsum_constant)
nipkow@15539
  1661
  apply (simp cong: setsum_cong)
nipkow@15402
  1662
  done
nipkow@15402
  1663
nipkow@15402
  1664
lemma card_Union_disjoint:
nipkow@15402
  1665
  "finite C ==> (ALL A:C. finite A) ==>
nipkow@15402
  1666
        (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
nipkow@15402
  1667
      card (Union C) = setsum card C"
nipkow@15402
  1668
  apply (frule card_UN_disjoint [of C id])
nipkow@15402
  1669
  apply (unfold Union_def id_def, assumption+)
nipkow@15402
  1670
  done
nipkow@15402
  1671
wenzelm@12396
  1672
subsubsection {* Cardinality of image *}
wenzelm@12396
  1673
paulson@15447
  1674
text{*The image of a finite set can be expressed using @{term fold}.*}
paulson@15447
  1675
lemma image_eq_fold: "finite A ==> f ` A = fold (op Un) (%x. {f x}) {} A"
paulson@15447
  1676
  apply (erule finite_induct, simp)
paulson@15447
  1677
  apply (subst ACf.fold_insert) 
paulson@15447
  1678
  apply (auto simp add: ACf_def) 
paulson@15447
  1679
  done
paulson@15447
  1680
wenzelm@12396
  1681
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
paulson@14208
  1682
  apply (induct set: Finites, simp)
wenzelm@12396
  1683
  apply (simp add: le_SucI finite_imageI card_insert_if)
wenzelm@12396
  1684
  done
wenzelm@12396
  1685
nipkow@15402
  1686
lemma card_image: "inj_on f A ==> card (f ` A) = card A"
nipkow@15539
  1687
by(simp add:card_def setsum_reindex o_def del:setsum_constant)
wenzelm@12396
  1688
wenzelm@12396
  1689
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
wenzelm@12396
  1690
  by (simp add: card_seteq card_image)
wenzelm@12396
  1691
nipkow@15111
  1692
lemma eq_card_imp_inj_on:
nipkow@15111
  1693
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
paulson@15506
  1694
apply (induct rule:finite_induct, simp)
nipkow@15111
  1695
apply(frule card_image_le[where f = f])
nipkow@15111
  1696
apply(simp add:card_insert_if split:if_splits)
nipkow@15111
  1697
done
nipkow@15111
  1698
nipkow@15111
  1699
lemma inj_on_iff_eq_card:
nipkow@15111
  1700
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
nipkow@15111
  1701
by(blast intro: card_image eq_card_imp_inj_on)
nipkow@15111
  1702
wenzelm@12396
  1703
nipkow@15402
  1704
lemma card_inj_on_le:
nipkow@15402
  1705
    "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
nipkow@15402
  1706
apply (subgoal_tac "finite A") 
nipkow@15402
  1707
 apply (force intro: card_mono simp add: card_image [symmetric])
nipkow@15402
  1708
apply (blast intro: finite_imageD dest: finite_subset) 
nipkow@15402
  1709
done
nipkow@15402
  1710
nipkow@15402
  1711
lemma card_bij_eq:
nipkow@15402
  1712
    "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
nipkow@15402
  1713
       finite A; finite B |] ==> card A = card B"
nipkow@15402
  1714
  by (auto intro: le_anti_sym card_inj_on_le)
nipkow@15402
  1715
nipkow@15402
  1716
nipkow@15402
  1717
subsubsection {* Cardinality of products *}
nipkow@15402
  1718
nipkow@15402
  1719
(*
nipkow@15402
  1720
lemma SigmaI_insert: "y \<notin> A ==>
nipkow@15402
  1721
  (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
nipkow@15402
  1722
  by auto
nipkow@15402
  1723
*)
nipkow@15402
  1724
nipkow@15402
  1725
lemma card_SigmaI [simp]:
nipkow@15402
  1726
  "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
nipkow@15402
  1727
  \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
nipkow@15539
  1728
by(simp add:card_def setsum_Sigma del:setsum_constant)
nipkow@15402
  1729
paulson@15409
  1730
lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
paulson@15409
  1731
apply (cases "finite A") 
paulson@15409
  1732
apply (cases "finite B") 
paulson@15409
  1733
apply (auto simp add: card_eq_0_iff
nipkow@15539
  1734
            dest: finite_cartesian_productD1 finite_cartesian_productD2)
paulson@15409
  1735
done
nipkow@15402
  1736
nipkow@15402
  1737
lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
nipkow@15539
  1738
by (simp add: card_cartesian_product)
paulson@15409
  1739
nipkow@15402
  1740
nipkow@15402
  1741
wenzelm@12396
  1742
subsubsection {* Cardinality of the Powerset *}
wenzelm@12396
  1743
wenzelm@12396
  1744
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
wenzelm@12396
  1745
  apply (induct set: Finites)
wenzelm@12396
  1746
   apply (simp_all add: Pow_insert)
paulson@14208
  1747
  apply (subst card_Un_disjoint, blast)
paulson@14208
  1748
    apply (blast intro: finite_imageI, blast)
wenzelm@12396
  1749
  apply (subgoal_tac "inj_on (insert x) (Pow F)")
wenzelm@12396
  1750
   apply (simp add: card_image Pow_insert)
wenzelm@12396
  1751
  apply (unfold inj_on_def)
wenzelm@12396
  1752
  apply (blast elim!: equalityE)
wenzelm@12396
  1753
  done
wenzelm@12396
  1754
nipkow@15392
  1755
text {* Relates to equivalence classes.  Based on a theorem of
nipkow@15392
  1756
F. Kammüller's.  *}
wenzelm@12396
  1757
wenzelm@12396
  1758
lemma dvd_partition:
nipkow@15392
  1759
  "finite (Union C) ==>
wenzelm@12396
  1760
    ALL c : C. k dvd card c ==>
paulson@14430
  1761
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
wenzelm@12396
  1762
  k dvd card (Union C)"
nipkow@15392
  1763
apply(frule finite_UnionD)
nipkow@15392
  1764
apply(rotate_tac -1)
paulson@14208
  1765
  apply (induct set: Finites, simp_all, clarify)
wenzelm@12396
  1766
  apply (subst card_Un_disjoint)
wenzelm@12396
  1767
  apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
wenzelm@12396
  1768
  done
wenzelm@12396
  1769
wenzelm@12396
  1770
nipkow@15392
  1771
subsection{* A fold functional for non-empty sets *}
nipkow@15392
  1772
nipkow@15392
  1773
text{* Does not require start value. *}
wenzelm@12396
  1774
nipkow@15392
  1775
consts
paulson@15506
  1776
  fold1Set :: "('a => 'a => 'a) => ('a set \<times> 'a) set"
nipkow@15392
  1777
paulson@15506
  1778
inductive "fold1Set f"
nipkow@15392
  1779
intros
paulson@15506
  1780
  fold1Set_insertI [intro]:
paulson@15506
  1781
   "\<lbrakk> (A,x) \<in> foldSet f id a; a \<notin> A \<rbrakk> \<Longrightarrow> (insert a A, x) \<in> fold1Set f"
wenzelm@12396
  1782
nipkow@15392
  1783
constdefs
nipkow@15392
  1784
  fold1 :: "('a => 'a => 'a) => 'a set => 'a"
paulson@15506
  1785
  "fold1 f A == THE x. (A, x) : fold1Set f"
paulson@15506
  1786
paulson@15506
  1787
lemma fold1Set_nonempty:
paulson@15506
  1788
 "(A, x) : fold1Set f \<Longrightarrow> A \<noteq> {}"
paulson@15506
  1789
by(erule fold1Set.cases, simp_all) 
paulson@15506
  1790
nipkow@15392
  1791
paulson@15506
  1792
inductive_cases empty_fold1SetE [elim!]: "({}, x) : fold1Set f"
paulson@15506
  1793
paulson@15506
  1794
inductive_cases insert_fold1SetE [elim!]: "(insert a X, x) : fold1Set f"
paulson@15506
  1795
paulson@15506
  1796
paulson@15506
  1797
lemma fold1Set_sing [iff]: "(({a},b) : fold1Set f) = (a = b)"
paulson@15506
  1798
  by (blast intro: foldSet.intros elim: foldSet.cases)
nipkow@15392
  1799
paulson@15508
  1800
lemma fold1_singleton[simp]: "fold1 f {a} = a"
paulson@15508
  1801
  by (unfold fold1_def) blast
wenzelm@12396
  1802
paulson@15508
  1803
lemma finite_nonempty_imp_fold1Set:
paulson@15508
  1804
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. (A, x) : fold1Set f"
paulson@15508
  1805
apply (induct A rule: finite_induct)
paulson@15508
  1806
apply (auto dest: finite_imp_foldSet [of _ f id])  
paulson@15508
  1807
done
paulson@15506
  1808
paulson@15506
  1809
text{*First, some lemmas about @{term foldSet}.*}
nipkow@15392
  1810
paulson@15508
  1811
lemma (in ACf) foldSet_insert_swap:
paulson@15508
  1812
assumes fold: "(A,y) \<in> foldSet f id b"
paulson@15521
  1813
shows "b \<notin> A \<Longrightarrow> (insert b A, z \<cdot> y) \<in> foldSet f id z"
paulson@15508
  1814
using fold
paulson@15508
  1815
proof (induct rule: foldSet.induct)
paulson@15508
  1816
  case emptyI thus ?case by (force simp add: fold_insert_aux commute)
paulson@15508
  1817
next
paulson@15508
  1818
  case (insertI A x y)
paulson@15508
  1819
    have "(insert x (insert b A), x \<cdot> (z \<cdot> y)) \<in> foldSet f (\<lambda>u. u) z"
paulson@15521
  1820
      using insertI by force  --{*how does @{term id} get unfolded?*}
paulson@15508
  1821
    thus ?case by (simp add: insert_commute AC)
paulson@15508
  1822
qed
paulson@15508
  1823
paulson@15508
  1824
lemma (in ACf) foldSet_permute_diff:
paulson@15508
  1825
assumes fold: "(A,x) \<in> foldSet f id b"
paulson@15508
  1826
shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> (insert b (A-{a}), x) \<in> foldSet f id a"
paulson@15508
  1827
using fold
paulson@15508
  1828
proof (induct rule: foldSet.induct)
paulson@15508
  1829
  case emptyI thus ?case by simp
paulson@15508
  1830
next
paulson@15508
  1831
  case (insertI A x y)
paulson@15521
  1832
  have "a = x \<or> a \<in> A" using insertI by simp
paulson@15521
  1833
  thus ?case
paulson@15521
  1834
  proof
paulson@15521
  1835
    assume "a = x"
paulson@15521
  1836
    with insertI show ?thesis
paulson@15521
  1837
      by (simp add: id_def [symmetric], blast intro: foldSet_insert_swap) 
paulson@15521
  1838
  next
paulson@15521
  1839
    assume ainA: "a \<in> A"
paulson@15521
  1840
    hence "(insert x (insert b (A - {a})), x \<cdot> y) \<in> foldSet f id a"
paulson@15521
  1841
      using insertI by (force simp: id_def)
paulson@15521
  1842
    moreover
paulson@15521
  1843
    have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
paulson@15521
  1844
      using ainA insertI by blast
paulson@15521
  1845
    ultimately show ?thesis by (simp add: id_def)
paulson@15508
  1846
  qed
paulson@15508
  1847
qed
paulson@15508
  1848
paulson@15508
  1849
lemma (in ACf) fold1_eq_fold:
paulson@15508
  1850
     "[|finite A; a \<notin> A|] ==> fold1 f (insert a A) = fold f id a A"
paulson@15508
  1851
apply (simp add: fold1_def fold_def) 
paulson@15508
  1852
apply (rule the_equality)
paulson@15508
  1853
apply (best intro: foldSet_determ theI dest: finite_imp_foldSet [of _ f id]) 
paulson@15508
  1854
apply (rule sym, clarify)
paulson@15508
  1855
apply (case_tac "Aa=A")
paulson@15508
  1856
 apply (best intro: the_equality foldSet_determ)  
paulson@15508
  1857
apply (subgoal_tac "(A,x) \<in> foldSet f id a") 
paulson@15508
  1858
 apply (best intro: the_equality foldSet_determ)  
paulson@15508
  1859
apply (subgoal_tac "insert aa (Aa - {a}) = A") 
paulson@15508
  1860
 prefer 2 apply (blast elim: equalityE) 
paulson@15508
  1861
apply (auto dest: foldSet_permute_diff [where a=a]) 
paulson@15508
  1862
done
paulson@15508
  1863
paulson@15521
  1864
lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
paulson@15521
  1865
apply safe
paulson@15521
  1866
apply simp 
paulson@15521
  1867
apply (drule_tac x=x in spec)
paulson@15521
  1868
apply (drule_tac x="A-{x}" in spec, auto) 
paulson@15508
  1869
done
paulson@15508
  1870
paulson@15521
  1871
lemma (in ACf) fold1_insert:
paulson@15521
  1872
  assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
paulson@15521
  1873
  shows "fold1 f (insert x A) = f x (fold1 f A)"
paulson@15521
  1874
proof -
paulson@15521
  1875
  from nonempty obtain a A' where "A = insert a A' & a ~: A'" 
paulson@15521
  1876
    by (auto simp add: nonempty_iff)
paulson@15521
  1877
  with A show ?thesis
paulson@15521
  1878
    by (simp add: insert_commute [of x] fold1_eq_fold eq_commute) 
paulson@15521
  1879
qed
paulson@15521
  1880
paulson@15509
  1881
lemma (in ACIf) fold1_insert_idem [simp]:
paulson@15521
  1882
  assumes nonempty: "A \<noteq> {}" and A: "finite A" 
paulson@15521
  1883
  shows "fold1 f (insert x A) = f x (fold1 f A)"
paulson@15521
  1884
proof -
paulson@15521
  1885
  from nonempty obtain a A' where A': "A = insert a A' & a ~: A'" 
paulson@15521
  1886
    by (auto simp add: nonempty_iff)
paulson@15521
  1887
  show ?thesis
paulson@15521
  1888
  proof cases
paulson@15521
  1889
    assume "a = x"
paulson@15521
  1890
    thus ?thesis 
paulson@15521
  1891
    proof cases
paulson@15521
  1892
      assume "A' = {}"
paulson@15521
  1893
      with prems show ?thesis by (simp add: idem) 
paulson@15521
  1894
    next
paulson@15521
  1895
      assume "A' \<noteq> {}"
paulson@15521
  1896
      with prems show ?thesis
paulson@15521
  1897
	by (simp add: fold1_insert assoc [symmetric] idem) 
paulson@15521
  1898
    qed
paulson@15521
  1899
  next
paulson@15521
  1900
    assume "a \<noteq> x"
paulson@15521
  1901
    with prems show ?thesis
paulson@15521
  1902
      by (simp add: insert_commute fold1_eq_fold fold_insert_idem)
paulson@15521
  1903
  qed
paulson@15521
  1904
qed
paulson@15506
  1905
paulson@15506
  1906
paulson@15508
  1907
text{* Now the recursion rules for definitions: *}
paulson@15508
  1908
paulson@15508
  1909
lemma fold1_singleton_def: "g \<equiv> fold1 f \<Longrightarrow> g {a} = a"
paulson@15508
  1910
by(simp add:fold1_singleton)
paulson@15508
  1911
paulson@15508
  1912
lemma (in ACf) fold1_insert_def:
paulson@15508
  1913
  "\<lbrakk> g \<equiv> fold1 f; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
paulson@15508
  1914
by(simp add:fold1_insert)
paulson@15508
  1915
paulson@15509
  1916
lemma (in ACIf) fold1_insert_idem_def:
paulson@15508
  1917
  "\<lbrakk> g \<equiv> fold1 f; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
paulson@15509
  1918
by(simp add:fold1_insert_idem)
paulson@15508
  1919
paulson@15508
  1920
subsubsection{* Determinacy for @{term fold1Set} *}
paulson@15508
  1921
paulson@15508
  1922
text{*Not actually used!!*}
wenzelm@12396
  1923
paulson@15506
  1924
lemma (in ACf) foldSet_permute:
paulson@15506
  1925
  "[|(insert a A, x) \<in> foldSet f id b; a \<notin> A; b \<notin> A|]
paulson@15506
  1926
   ==> (insert b A, x) \<in> foldSet f id a"
paulson@15506
  1927
apply (case_tac "a=b") 
paulson@15506
  1928
apply (auto dest: foldSet_permute_diff) 
paulson@15506
  1929
done
nipkow@15376
  1930
paulson@15506
  1931
lemma (in ACf) fold1Set_determ:
paulson@15506
  1932
  "(A, x) \<in> fold1Set f ==> (A, y) \<in> fold1Set f ==> y = x"
paulson@15506
  1933
proof (clarify elim!: fold1Set.cases)
paulson@15506
  1934
  fix A x B y a b
paulson@15506
  1935
  assume Ax: "(A, x) \<in> foldSet f id a"
paulson@15506
  1936
  assume By: "(B, y) \<in> foldSet f id b"
paulson@15506
  1937
  assume anotA:  "a \<notin> A"
paulson@15506
  1938
  assume bnotB:  "b \<notin> B"
paulson@15506
  1939
  assume eq: "insert a A = insert b B"
paulson@15506
  1940
  show "y=x"
paulson@15506
  1941
  proof cases
paulson@15506
  1942
    assume same: "a=b"
paulson@15506
  1943
    hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
paulson@15506
  1944
    thus ?thesis using Ax By same by (blast intro: foldSet_determ)
nipkow@15392
  1945
  next
paulson@15506
  1946
    assume diff: "a\<noteq>b"
paulson@15506
  1947
    let ?D = "B - {a}"
paulson@15506
  1948
    have B: "B = insert a ?D" and A: "A = insert b ?D"
paulson@15506
  1949
     and aB: "a \<in> B" and bA: "b \<in> A"
paulson@15506
  1950
      using eq anotA bnotB diff by (blast elim!:equalityE)+
paulson@15506
  1951
    with aB bnotB By
paulson@15506
  1952
    have "(insert b ?D, y) \<in> foldSet f id a" 
paulson@15506
  1953
      by (auto intro: foldSet_permute simp add: insert_absorb)
paulson@15506
  1954
    moreover
paulson@15506
  1955
    have "(insert b ?D, x) \<in> foldSet f id a"
paulson@15506
  1956
      by (simp add: A [symmetric] Ax) 
paulson@15506
  1957
    ultimately show ?thesis by (blast intro: foldSet_determ) 
nipkow@15392
  1958
  qed
wenzelm@12396
  1959
qed
wenzelm@12396
  1960
paulson@15506
  1961
lemma (in ACf) fold1Set_equality: "(A, y) : fold1Set f ==> fold1 f A = y"
paulson@15506
  1962
  by (unfold fold1_def) (blast intro: fold1Set_determ)
paulson@15506
  1963
paulson@15506
  1964
declare
paulson@15506
  1965
  empty_foldSetE [rule del]   foldSet.intros [rule del]
paulson@15506
  1966
  empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
paulson@15506
  1967
  -- {* No more proves involve these relations. *}
nipkow@15376
  1968
nipkow@15497
  1969
subsubsection{* Semi-Lattices *}
nipkow@15497
  1970
nipkow@15497
  1971
locale ACIfSL = ACIf +
nipkow@15500
  1972
  fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubseteq>" 50)
nipkow@15500
  1973
  assumes below_def: "(x \<sqsubseteq> y) = (x\<cdot>y = x)"
nipkow@15497
  1974
nipkow@15497
  1975
locale ACIfSLlin = ACIfSL +
nipkow@15497
  1976
  assumes lin: "x\<cdot>y \<in> {x,y}"
nipkow@15497
  1977
nipkow@15500
  1978
lemma (in ACIfSL) below_refl[simp]: "x \<sqsubseteq> x"
nipkow@15497
  1979
by(simp add: below_def idem)
nipkow@15497
  1980
nipkow@15500
  1981
lemma (in ACIfSL) below_f_conv[simp]: "x \<sqsubseteq> y \<cdot> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
nipkow@15497
  1982
proof
nipkow@15500
  1983
  assume "x \<sqsubseteq> y \<cdot> z"
nipkow@15497
  1984
  hence xyzx: "x \<cdot> (y \<cdot> z) = x"  by(simp add: below_def)
nipkow@15497
  1985
  have "x \<cdot> y = x"
nipkow@15497
  1986
  proof -
nipkow@15497
  1987
    have "x \<cdot> y = (x \<cdot> (y \<cdot> z)) \<cdot> y" by(rule subst[OF xyzx], rule refl)
nipkow@15497
  1988
    also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
nipkow@15497
  1989
    also have "\<dots> = x" by(rule xyzx)
nipkow@15497
  1990
    finally show ?thesis .
nipkow@15497
  1991
  qed
nipkow@15497
  1992
  moreover have "x \<cdot> z = x"
nipkow@15497
  1993
  proof -
nipkow@15497
  1994
    have "x \<cdot> z = (x \<cdot> (y \<cdot> z)) \<cdot> z" by(rule subst[OF xyzx], rule refl)
nipkow@15497
  1995
    also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
nipkow@15497
  1996
    also have "\<dots> = x" by(rule xyzx)
nipkow@15497
  1997
    finally show ?thesis .
nipkow@15497
  1998
  qed
nipkow@15500
  1999
  ultimately show "x \<sqsubseteq> y \<and> x \<sqsubseteq> z" by(simp add: below_def)
nipkow@15497
  2000
next
nipkow@15500
  2001
  assume a: "x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
nipkow@15497
  2002
  hence y: "x \<cdot> y = x" and z: "x \<cdot> z = x" by(simp_all add: below_def)
nipkow@15497
  2003
  have "x \<cdot> (y \<cdot> z) = (x \<cdot> y) \<cdot> z" by(simp add:assoc)
nipkow@15497
  2004
  also have "x \<cdot> y = x" using a by(simp_all add: below_def)
nipkow@15497
  2005
  also have "x \<cdot> z = x" using a by(simp_all add: below_def)
nipkow@15500
  2006
  finally show "x \<sqsubseteq> y \<cdot> z" by(simp_all add: below_def)
nipkow@15497
  2007
qed
nipkow@15497
  2008
nipkow@15497
  2009
lemma (in ACIfSLlin) above_f_conv:
nipkow@15500
  2010
 "x \<cdot> y \<sqsubseteq> z = (x \<sqsubseteq> z \<or> y \<sqsubseteq> z)"
nipkow@15497
  2011
proof
nipkow@15500
  2012
  assume a: "x \<cdot> y \<sqsubseteq> z"
nipkow@15497
  2013
  have "x \<cdot> y = x \<or> x \<cdot> y = y" using lin[of x y] by simp
nipkow@15500
  2014
  thus "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
nipkow@15497
  2015
  proof
nipkow@15500
  2016
    assume "x \<cdot> y = x" hence "x \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
nipkow@15497
  2017
  next
nipkow@15500
  2018
    assume "x \<cdot> y = y" hence "y \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
nipkow@15497
  2019
  qed
nipkow@15497
  2020
next
nipkow@15500
  2021
  assume "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
nipkow@15500
  2022
  thus "x \<cdot> y \<sqsubseteq> z"
nipkow@15497
  2023
  proof
nipkow@15500
  2024
    assume a: "x \<sqsubseteq> z"
nipkow@15497
  2025
    have "(x \<cdot> y) \<cdot> z = (x \<cdot> z) \<cdot> y" by(simp add:ACI)
nipkow@15497
  2026
    also have "x \<cdot> z = x" using a by(simp add:below_def)
nipkow@15500
  2027
    finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
nipkow@15497
  2028
  next
nipkow@15500
  2029
    assume a: "y \<sqsubseteq> z"
nipkow@15497
  2030
    have "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
nipkow@15497
  2031
    also have "y \<cdot> z = y" using a by(simp add:below_def)
nipkow@15500
  2032
    finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
nipkow@15497
  2033
  qed
nipkow@15497
  2034
qed
nipkow@15497
  2035
nipkow@15497
  2036
nipkow@15502
  2037
subsubsection{* Lemmas about @{text fold1} *}
nipkow@15484
  2038
nipkow@15484
  2039
lemma (in ACf) fold1_Un:
nipkow@15484
  2040
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  2041
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
nipkow@15484
  2042
       fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
nipkow@15484
  2043
using A
nipkow@15484
  2044
proof(induct rule:finite_ne_induct)
nipkow@15484
  2045
  case singleton thus ?case by(simp add:fold1_insert)
nipkow@15484
  2046
next
nipkow@15484
  2047
  case insert thus ?case by (simp add:fold1_insert assoc)
nipkow@15484
  2048
qed
nipkow@15484
  2049
nipkow@15484
  2050
lemma (in ACIf) fold1_Un2:
nipkow@15484
  2051
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  2052
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
nipkow@15484
  2053
       fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
nipkow@15484
  2054
using A
nipkow@15484
  2055
proof(induct rule:finite_ne_induct)
paulson@15509
  2056
  case singleton thus ?case by(simp add:fold1_insert_idem)
nipkow@15484
  2057
next
paulson@15509
  2058
  case insert thus ?case by (simp add:fold1_insert_idem assoc)
nipkow@15484
  2059
qed
nipkow@15484
  2060
nipkow@15484
  2061
lemma (in ACf) fold1_in:
nipkow@15484
  2062
  assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x\<cdot>y \<in> {x,y}"
nipkow@15484
  2063
  shows "fold1 f A \<in> A"
nipkow@15484
  2064
using A
nipkow@15484
  2065
proof (induct rule:finite_ne_induct)
paulson@15506
  2066
  case singleton thus ?case by simp
nipkow@15484
  2067
next
nipkow@15484
  2068
  case insert thus ?case using elem by (force simp add:fold1_insert)
nipkow@15484
  2069
qed
nipkow@15484
  2070
nipkow@15497
  2071
lemma (in ACIfSL) below_fold1_iff:
nipkow@15497
  2072
assumes A: "finite A" "A \<noteq> {}"
nipkow@15500
  2073
shows "x \<sqsubseteq> fold1 f A = (\<forall>a\<in>A. x \<sqsubseteq> a)"
nipkow@15497
  2074
using A
nipkow@15497
  2075
by(induct rule:finite_ne_induct) simp_all
nipkow@15497
  2076
nipkow@15497
  2077
lemma (in ACIfSL) fold1_belowI:
nipkow@15497
  2078
assumes A: "finite A" "A \<noteq> {}"
nipkow@15500
  2079
shows "a \<in> A \<Longrightarrow> fold1 f A \<sqsubseteq> a"
nipkow@15484
  2080
using A
nipkow@15484
  2081
proof (induct rule:finite_ne_induct)
nipkow@15497
  2082
  case singleton thus ?case by simp
nipkow@15484
  2083
next
nipkow@15497
  2084
  case (insert x F)
berghofe@15517
  2085
  from insert(5) have "a = x \<or> a \<in> F" by simp
nipkow@15497
  2086
  thus ?case
nipkow@15497
  2087
  proof
nipkow@15497
  2088
    assume "a = x" thus ?thesis using insert by(simp add:below_def ACI)
nipkow@15497
  2089
  next
nipkow@15497
  2090
    assume "a \<in> F"
paulson@15508
  2091
    hence bel: "fold1 f F \<sqsubseteq> a" by(rule insert)
paulson@15508
  2092
    have "fold1 f (insert x F) \<cdot> a = x \<cdot> (fold1 f F \<cdot> a)"
nipkow@15497
  2093
      using insert by(simp add:below_def ACI)
paulson@15508
  2094
    also have "fold1 f F \<cdot> a = fold1 f F"
nipkow@15497
  2095
      using bel  by(simp add:below_def ACI)
paulson@15508
  2096
    also have "x \<cdot> \<dots> = fold1 f (insert x F)"
nipkow@15497
  2097
      using insert by(simp add:below_def ACI)
nipkow@15497
  2098
    finally show ?thesis  by(simp add:below_def)
nipkow@15497
  2099
  qed
nipkow@15484
  2100
qed
nipkow@15484
  2101
nipkow@15497
  2102
lemma (in ACIfSLlin) fold1_below_iff:
nipkow@15497
  2103
assumes A: "finite A" "A \<noteq> {}"
nipkow@15500
  2104
shows "fold1 f A \<sqsubseteq> x = (\<exists>a\<in>A. a \<sqsubseteq> x)"
nipkow@15484
  2105
using A
nipkow@15497
  2106
by(induct rule:finite_ne_induct)(simp_all add:above_f_conv)
nipkow@15484
  2107
nipkow@15512
  2108
nipkow@15500
  2109
subsubsection{* Lattices *}
nipkow@15500
  2110
nipkow@15512
  2111
locale Lattice = lattice +
nipkow@15512
  2112
  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
nipkow@15500
  2113
  and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
nipkow@15500
  2114
  defines "Inf == fold1 inf"  and "Sup == fold1 sup"
nipkow@15500
  2115
nipkow@15512
  2116
locale Distrib_Lattice = distrib_lattice + Lattice
nipkow@15504
  2117
nipkow@15500
  2118
text{* Lattices are semilattices *}
nipkow@15500
  2119
nipkow@15500
  2120
lemma (in Lattice) ACf_inf: "ACf inf"
nipkow@15512
  2121
by(blast intro: ACf.intro inf_commute inf_assoc)
nipkow@15500
  2122
nipkow@15500
  2123
lemma (in Lattice) ACf_sup: "ACf sup"
nipkow@15512
  2124
by(blast intro: ACf.intro sup_commute sup_assoc)
nipkow@15500
  2125
nipkow@15500
  2126
lemma (in Lattice) ACIf_inf: "ACIf inf"
nipkow@15500
  2127
apply(rule ACIf.intro)
nipkow@15500
  2128
apply(rule ACf_inf)
nipkow@15500
  2129
apply(rule ACIf_axioms.intro)
nipkow@15500
  2130
apply(rule inf_idem)
nipkow@15500
  2131
done
nipkow@15500
  2132
nipkow@15500
  2133
lemma (in Lattice) ACIf_sup: "ACIf sup"
nipkow@15500
  2134
apply(rule ACIf.intro)
nipkow@15500
  2135
apply(rule ACf_sup)
nipkow@15500
  2136
apply(rule ACIf_axioms.intro)
nipkow@15500
  2137
apply(rule sup_idem)
nipkow@15500
  2138
done
nipkow@15500
  2139
nipkow@15500
  2140
lemma (in Lattice) ACIfSL_inf: "ACIfSL inf (op \<sqsubseteq>)"
nipkow@15500
  2141
apply(rule ACIfSL.intro)
nipkow@15500
  2142
apply(rule ACf_inf)
nipkow@15500
  2143
apply(rule ACIf.axioms[OF ACIf_inf])
nipkow@15500
  2144
apply(rule ACIfSL_axioms.intro)
nipkow@15500
  2145
apply(rule iffI)
nipkow@15500
  2146
 apply(blast intro: antisym inf_le1 inf_le2 inf_least refl)
nipkow@15500
  2147
apply(erule subst)
nipkow@15500
  2148
apply(rule inf_le2)
nipkow@15500
  2149
done
nipkow@15500
  2150
nipkow@15500
  2151
lemma (in Lattice) ACIfSL_sup: "ACIfSL sup (%x y. y \<sqsubseteq> x)"
nipkow@15500
  2152
apply(rule ACIfSL.intro)
nipkow@15500
  2153
apply(rule ACf_sup)
nipkow@15500
  2154
apply(rule ACIf.axioms[OF ACIf_sup])
nipkow@15500
  2155
apply(rule ACIfSL_axioms.intro)
nipkow@15500
  2156
apply(rule iffI)
nipkow@15500
  2157
 apply(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl)
nipkow@15500
  2158
apply(erule subst)
nipkow@15500
  2159
apply(rule sup_ge2)
nipkow@15500
  2160
done
nipkow@15500
  2161
nipkow@15505
  2162
nipkow@15505
  2163
subsubsection{* Fold laws in lattices *}
nipkow@15500
  2164
nipkow@15780
  2165
lemma (in Lattice) Inf_le_Sup[simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Squnion>A"
nipkow@15500
  2166
apply(unfold Sup_def Inf_def)
nipkow@15500
  2167
apply(subgoal_tac "EX a. a:A")
nipkow@15500
  2168
prefer 2 apply blast
nipkow@15500
  2169
apply(erule exE)
nipkow@15500
  2170
apply(rule trans)
nipkow@15500
  2171
apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_inf])
nipkow@15500
  2172
apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_sup])
nipkow@15500
  2173
done
nipkow@15500
  2174
nipkow@15780
  2175
lemma (in Lattice) sup_Inf_absorb[simp]:
nipkow@15504
  2176
  "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<squnion> \<Sqinter>A) = a"
nipkow@15512
  2177
apply(subst sup_commute)
nipkow@15504
  2178
apply(simp add:Inf_def sup_absorb ACIfSL.fold1_belowI[OF ACIfSL_inf])
nipkow@15504
  2179
done
nipkow@15504
  2180
nipkow@15780
  2181
lemma (in Lattice) inf_Sup_absorb[simp]:
nipkow@15504
  2182
  "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<sqinter> \<Squnion>A) = a"
nipkow@15504
  2183
by(simp add:Sup_def inf_absorb ACIfSL.fold1_belowI[OF ACIfSL_sup])
nipkow@15504
  2184
nipkow@15504
  2185
nipkow@15512
  2186
lemma (in Distrib_Lattice) sup_Inf1_distrib:
nipkow@15500
  2187
assumes A: "finite A" "A \<noteq> {}"
nipkow@15500
  2188
shows "(x \<squnion> \<Sqinter>A) = \<Sqinter>{x \<squnion> a|a. a \<in> A}"
nipkow@15500
  2189
using A
nipkow@15500
  2190
proof (induct rule: finite_ne_induct)
nipkow@15500
  2191
  case singleton thus ?case by(simp add:Inf_def)
nipkow@15500
  2192
next
nipkow@15500
  2193
  case (insert y A)
nipkow@15500
  2194
  have fin: "finite {x \<squnion> a |a. a \<in> A}"
berghofe@15517
  2195
    by(fast intro: finite_surj[where f = "%a. x \<squnion> a", OF insert(1)])
nipkow@15500
  2196
  have "x \<squnion> \<Sqinter> (insert y A) = x \<squnion> (y \<sqinter> \<Sqinter> A)"
nipkow@15500
  2197
    using insert by(simp add:ACf.fold1_insert_def[OF ACf_inf Inf_def])
nipkow@15500
  2198
  also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> \<Sqinter> A)" by(rule sup_inf_distrib1)
nipkow@15500
  2199
  also have "x \<squnion> \<Sqinter> A = \<Sqinter>{x \<squnion> a|a. a \<in> A}" using insert by simp
nipkow@15500
  2200
  also have "(x \<squnion> y) \<sqinter> \<dots> = \<Sqinter> (insert (x \<squnion> y) {x \<squnion> a |a. a \<in> A})"
paulson@15509
  2201
    using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def fin])
nipkow@15500
  2202
  also have "insert (x\<squnion>y) {x\<squnion>a |a. a \<in> A} = {x\<squnion>a |a. a \<in> insert y A}"
nipkow@15500
  2203
    by blast
nipkow@15500
  2204
  finally show ?case .
nipkow@15500
  2205
qed
nipkow@15500
  2206
nipkow@15512
  2207
lemma (in Distrib_Lattice) sup_Inf2_distrib:
nipkow@15500
  2208
assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
nipkow@15500
  2209
shows "(\<Sqinter>A \<squnion> \<Sqinter>B) = \<Sqinter>{a \<squnion> b|a b. a \<in> A \<and> b \<in> B}"
nipkow@15500
  2210
using A
nipkow@15500
  2211
proof (induct rule: finite_ne_induct)
nipkow@15500
  2212
  case singleton thus ?case
nipkow@15500
  2213
    by(simp add: sup_Inf1_distrib[OF B] fold1_singleton_def[OF Inf_def])
nipkow@15500
  2214
next
nipkow@15500
  2215
  case (insert x A)
nipkow@15500
  2216
  have finB: "finite {x \<squnion> b |b. b \<in> B}"
berghofe@15517
  2217
    by(fast intro: finite_surj[where f = "%b. x \<squnion> b", OF B(1)])
nipkow@15500
  2218
  have finAB: "finite {a \<squnion> b |a b. a \<in> A \<and> b \<in> B}"
nipkow@15500
  2219
  proof -
nipkow@15500
  2220
    have "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {a \<squnion> b})"
nipkow@15500
  2221
      by blast
berghofe@15517
  2222
    thus ?thesis by(simp add: insert(1) B(1))
nipkow@15500
  2223
  qed
nipkow@15500
  2224
  have ne: "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
nipkow@15500
  2225
  have "\<Sqinter>(insert x A) \<squnion> \<Sqinter>B = (x \<sqinter> \<Sqinter>A) \<squnion> \<Sqinter>B"
paulson@15509
  2226
    using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def])
nipkow@15500
  2227
  also have "\<dots> = (x \<squnion> \<Sqinter>B) \<sqinter> (\<Sqinter>A \<squnion> \<Sqinter>B)" by(rule sup_inf_distrib2)
nipkow@15500
  2228
  also have "\<dots> = \<Sqinter>{x \<squnion> b|b. b \<in> B} \<sqinter> \<Sqinter>{a \<squnion> b|a b. a \<in> A \<and> b \<in> B}"
nipkow@15500
  2229
    using insert by(simp add:sup_Inf1_distrib[OF B])
nipkow@15500
  2230
  also have "\<dots> = \<Sqinter>({x\<squnion>b |b. b \<in> B} \<union> {a\<squnion>b |a b. a \<in> A \<and> b \<in> B})"
nipkow@15500
  2231
    (is "_ = \<Sqinter>?M")
nipkow@15500
  2232
    using B insert
nipkow@15500
  2233
    by(simp add:Inf_def ACIf.fold1_Un2[OF ACIf_inf finB _ finAB ne])
nipkow@15500
  2234
  also have "?M = {a \<squnion> b |a b. a \<in> insert x A \<and> b \<in> B}"
nipkow@15500
  2235
    by blast
nipkow@15500
  2236
  finally show ?case .
nipkow@15500
  2237
qed
nipkow@15500
  2238
nipkow@15484
  2239
nipkow@15392
  2240
subsection{*Min and Max*}
nipkow@15392
  2241
nipkow@15392
  2242
text{* As an application of @{text fold1} we define the minimal and
nipkow@15497
  2243
maximal element of a (non-empty) set over a linear order. *}
nipkow@15497
  2244
nipkow@15497
  2245
constdefs
nipkow@15497
  2246
  Min :: "('a::linorder)set => 'a"
nipkow@15497
  2247
  "Min  ==  fold1 min"
nipkow@15497
  2248
nipkow@15497
  2249
  Max :: "('a::linorder)set => 'a"
nipkow@15497
  2250
  "Max  ==  fold1 max"
nipkow@15497
  2251
nipkow@15497
  2252
nipkow@15497
  2253
text{* Before we can do anything, we need to show that @{text min} and
nipkow@15497
  2254
@{text max} are ACI and the ordering is linear: *}
nipkow@15392
  2255
ballarin@15837
  2256
interpretation min: ACf ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
nipkow@15392
  2257
apply(rule ACf.intro)
nipkow@15392
  2258
apply(auto simp:min_def)
nipkow@15392
  2259
done
nipkow@15392
  2260
ballarin@15837
  2261
interpretation min: ACIf ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
nipkow@15392
  2262
apply(rule ACIf_axioms.intro)
nipkow@15392
  2263
apply(auto simp:min_def)
nipkow@15376
  2264
done
nipkow@15376
  2265
ballarin@15837
  2266
interpretation max: ACf ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
nipkow@15392
  2267
apply(rule ACf.intro)
nipkow@15392
  2268
apply(auto simp:max_def)
nipkow@15392
  2269
done
nipkow@15392
  2270
ballarin@15837
  2271
interpretation max: ACIf ["max:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
nipkow@15392
  2272
apply(rule ACIf_axioms.intro)
nipkow@15392
  2273
apply(auto simp:max_def)
nipkow@15376
  2274
done
wenzelm@12396
  2275
ballarin@15837
  2276
interpretation min:
nipkow@15780
  2277
  ACIfSL ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "op \<le>"]
nipkow@15497
  2278
apply(rule ACIfSL_axioms.intro)
nipkow@15497
  2279
apply(auto simp:min_def)
nipkow@15497
  2280
done
nipkow@15497
  2281
ballarin@15837
  2282
interpretation min:
nipkow@15780
  2283
  ACIfSLlin ["min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "op \<le>"]
nipkow@15497
  2284
apply(rule ACIfSLlin_axioms.intro)
nipkow@15497
  2285
apply(auto simp:min_def)
nipkow@15497
  2286
done
nipkow@15392
  2287
ballarin@15837
  2288
interpretation max:
nipkow@15780
  2289
  ACIfSL ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "%x y. y\<le>x"]
nipkow@15497
  2290
apply(rule ACIfSL_axioms.intro)
nipkow@15497
  2291
apply(auto simp:max_def)
nipkow@15497
  2292
done
nipkow@15392
  2293
ballarin@15837
  2294
interpretation max:
nipkow@15780
  2295
  ACIfSLlin ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "%x y. y\<le>x"]
nipkow@15497
  2296
apply(rule ACIfSLlin_axioms.intro)
nipkow@15497
  2297
apply(auto simp:max_def)
nipkow@15497
  2298
done
nipkow@15392
  2299
ballarin@15837
  2300
interpretation min_max:
nipkow@15780
  2301
  Lattice ["op \<le>" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max" "Min" "Max"]
nipkow@15780
  2302
apply -
nipkow@15780
  2303
apply(rule Min_def)
nipkow@15780
  2304
apply(rule Max_def)
nipkow@15507
  2305
done
nipkow@15500
  2306
nipkow@15500
  2307
ballarin@15837
  2308
interpretation min_max:
nipkow@15780
  2309
  Distrib_Lattice ["op \<le>" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max" "Min" "Max"]
nipkow@15780
  2310
.
ballarin@15765
  2311
nipkow@15402
  2312
text{* Now we instantiate the recursion equations and declare them
nipkow@15392
  2313
simplification rules: *}
nipkow@15392
  2314
paulson@17085
  2315
(* Making Min or Max a defined parameter of a locale, suitably
paulson@17085
  2316
  extending ACIf, could make the following interpretations more automatic. *)
ballarin@15765
  2317
paulson@17085
  2318
lemmas Min_singleton = fold1_singleton_def [OF Min_def]
paulson@17085
  2319
lemmas Max_singleton = fold1_singleton_def [OF Max_def]
paulson@17085
  2320
lemmas Min_insert = min.fold1_insert_idem_def [OF Min_def]
paulson@17085
  2321
lemmas Max_insert = max.fold1_insert_idem_def [OF Max_def]
paulson@17085
  2322
paulson@17085
  2323
declare Min_singleton [simp]  Max_singleton [simp]
paulson@17085
  2324
declare Min_insert [simp]  Max_insert [simp]
paulson@17085
  2325
nipkow@15392
  2326
nipkow@15484
  2327
text{* Now we instantiate some @{text fold1} properties: *}
nipkow@15392
  2328
nipkow@15392
  2329
lemma Min_in [simp]:
nipkow@15484
  2330
  shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Min A \<in> A"
nipkow@15791
  2331
using min.fold1_in
nipkow@15484
  2332
by(fastsimp simp: Min_def min_def)
nipkow@15392
  2333
nipkow@15392
  2334
lemma Max_in [simp]:
nipkow@15484
  2335
  shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Max A \<in> A"
nipkow@15791
  2336
using max.fold1_in
nipkow@15484
  2337
by(fastsimp simp: Max_def max_def)
nipkow@15392
  2338
nipkow@15484
  2339
lemma Min_le [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> Min A \<le> x"
nipkow@15791
  2340
by(simp add: Min_def min.fold1_belowI)
nipkow@15392
  2341
nipkow@15484
  2342
lemma Max_ge [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> x \<le> Max A"
nipkow@15791
  2343
by(simp add: Max_def max.fold1_belowI)
nipkow@15497
  2344
nipkow@15497
  2345
lemma Min_ge_iff[simp]:
nipkow@15497
  2346
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Min A) = (\<forall>a\<in>A. x \<le> a)"
nipkow@15791
  2347
by(simp add: Min_def min.below_fold1_iff)
nipkow@15497
  2348
nipkow@15497
  2349
lemma Max_le_iff[simp]:
nipkow@15497
  2350
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Max A \<le> x) = (\<forall>a\<in>A. a \<le> x)"
nipkow@15791
  2351
by(simp add: Max_def max.below_fold1_iff)
nipkow@15497
  2352
nipkow@15497
  2353
lemma Min_le_iff:
nipkow@15497
  2354
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Min A \<le> x) = (\<exists>a\<in>A. a \<le> x)"
nipkow@15791
  2355
by(simp add: Min_def min.fold1_below_iff)
nipkow@15497
  2356
nipkow@15497
  2357
lemma Max_ge_iff:
nipkow@15497
  2358
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Max A) = (\<exists>a\<in>A. x \<le> a)"
nipkow@15791
  2359
by(simp add: Max_def max.fold1_below_iff)
wenzelm@12396
  2360
nipkow@17022
  2361
subsection {* Properties of axclass @{text finite} *}
nipkow@17022
  2362
nipkow@17022
  2363
text{* Many of these are by Brian Huffman. *}
nipkow@17022
  2364
nipkow@17022
  2365
lemma finite_set: "finite (A::'a::finite set)"
nipkow@17022
  2366
by (rule finite_subset [OF subset_UNIV finite])
nipkow@17022
  2367
nipkow@17022
  2368
nipkow@17022
  2369
instance unit :: finite
nipkow@17022
  2370
proof
nipkow@17022
  2371
  have "finite {()}" by simp
nipkow@17022
  2372
  also have "{()} = UNIV" by auto
nipkow@17022
  2373
  finally show "finite (UNIV :: unit set)" .
nipkow@17022
  2374
qed
nipkow@17022
  2375
nipkow@17022
  2376
instance bool :: finite
nipkow@17022
  2377
proof
nipkow@17022
  2378
  have "finite {True, False}" by simp
nipkow@17022
  2379
  also have "{True, False} = UNIV" by auto
nipkow@17022
  2380
  finally show "finite (UNIV :: bool set)" .
nipkow@17022
  2381
qed
nipkow@17022
  2382
nipkow@17022
  2383
nipkow@17022
  2384
instance * :: (finite, finite) finite
nipkow@17022
  2385
proof
nipkow@17022
  2386
  show "finite (UNIV :: ('a \<times> 'b) set)"
nipkow@17022
  2387
  proof (rule finite_Prod_UNIV)
nipkow@17022
  2388
    show "finite (UNIV :: 'a set)" by (rule finite)
nipkow@17022
  2389
    show "finite (UNIV :: 'b set)" by (rule finite)
nipkow@17022
  2390
  qed
nipkow@17022
  2391
qed
nipkow@17022
  2392
nipkow@17022
  2393
instance "+" :: (finite, finite) finite
nipkow@17022
  2394
proof
nipkow@17022
  2395
  have a: "finite (UNIV :: 'a set)" by (rule finite)
nipkow@17022
  2396
  have b: "finite (UNIV :: 'b set)" by (rule finite)
nipkow@17022
  2397
  from a b have "finite ((UNIV :: 'a set) <+> (UNIV :: 'b set))"
nipkow@17022
  2398
    by (rule finite_Plus)
nipkow@17022
  2399
  thus "finite (UNIV :: ('a + 'b) set)" by simp
nipkow@17022
  2400
qed
nipkow@17022
  2401
nipkow@17022
  2402
nipkow@17022
  2403
instance set :: (finite) finite
nipkow@17022
  2404
proof
nipkow@17022
  2405
  have "finite (UNIV :: 'a set)" by (rule finite)
nipkow@17022
  2406
  hence "finite (Pow (UNIV :: 'a set))"
nipkow@17022
  2407
    by (rule finite_Pow_iff [THEN iffD2])
nipkow@17022
  2408
  thus "finite (UNIV :: 'a set set)" by simp
nipkow@17022
  2409
qed
nipkow@17022
  2410
nipkow@17022
  2411
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
nipkow@17022
  2412
by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
nipkow@17022
  2413
nipkow@17022
  2414
instance fun :: (finite, finite) finite
nipkow@17022
  2415
proof
nipkow@17022
  2416
  show "finite (UNIV :: ('a => 'b) set)"
nipkow@17022
  2417
  proof (rule finite_imageD)
nipkow@17022
  2418
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
nipkow@17022
  2419
    show "finite (range ?graph)" by (rule finite_set)
nipkow@17022
  2420
    show "inj ?graph" by (rule inj_graph)
nipkow@17022
  2421
  qed
nipkow@17022
  2422
qed
nipkow@17022
  2423
nipkow@15042
  2424
end