src/HOL/Finite_Set.thy
author wenzelm
Tue May 02 20:42:30 2006 +0200 (2006-05-02)
changeset 19535 e4fdeb32eadf
parent 19363 667b5ea637dd
child 19793 14fdd2a3d117
permissions -rw-r--r--
replaced syntax/translations by abbreviation;
tuned proofs;
tuned;
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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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abbreviation
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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_Collect_subset[simp]: "finite A \<Longrightarrow> finite{x \<in> A. P x}"
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using finite_subset[of "{x \<in> A. P x}" "A"] by blast
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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"
paulson@15409
   331
apply (auto simp add: finite_conv_nat_seg_image) 
paulson@15409
   332
apply (drule_tac x=n in spec) 
paulson@15409
   333
apply (drule_tac x="fst o f" in spec) 
paulson@15409
   334
apply (auto simp add: o_def) 
paulson@15409
   335
 prefer 2 apply (force dest!: equalityD2) 
paulson@15409
   336
apply (drule equalityD1) 
paulson@15409
   337
apply (rename_tac y x)
paulson@15409
   338
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
paulson@15409
   339
 prefer 2 apply force
paulson@15409
   340
apply clarify
paulson@15409
   341
apply (rule_tac x=k in image_eqI, auto)
paulson@15409
   342
done
paulson@15409
   343
paulson@15409
   344
lemma finite_cartesian_productD2:
paulson@15409
   345
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
paulson@15409
   346
apply (auto simp add: finite_conv_nat_seg_image) 
paulson@15409
   347
apply (drule_tac x=n in spec) 
paulson@15409
   348
apply (drule_tac x="snd o f" in spec) 
paulson@15409
   349
apply (auto simp add: o_def) 
paulson@15409
   350
 prefer 2 apply (force dest!: equalityD2) 
paulson@15409
   351
apply (drule equalityD1)
paulson@15409
   352
apply (rename_tac x y)
paulson@15409
   353
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
paulson@15409
   354
 prefer 2 apply force
paulson@15409
   355
apply clarify
paulson@15409
   356
apply (rule_tac x=k in image_eqI, auto)
paulson@15409
   357
done
paulson@15409
   358
paulson@15409
   359
nipkow@15392
   360
text {* The powerset of a finite set *}
wenzelm@12396
   361
wenzelm@12396
   362
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
wenzelm@12396
   363
proof
wenzelm@12396
   364
  assume "finite (Pow A)"
wenzelm@12396
   365
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
wenzelm@12396
   366
  thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
wenzelm@12396
   367
next
wenzelm@12396
   368
  assume "finite A"
wenzelm@12396
   369
  thus "finite (Pow A)"
wenzelm@12396
   370
    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
wenzelm@12396
   371
qed
wenzelm@12396
   372
nipkow@15392
   373
nipkow@15392
   374
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
nipkow@15392
   375
by(blast intro: finite_subset[OF subset_Pow_Union])
nipkow@15392
   376
nipkow@15392
   377
wenzelm@12396
   378
lemma finite_converse [iff]: "finite (r^-1) = finite r"
wenzelm@12396
   379
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
wenzelm@12396
   380
   apply simp
wenzelm@12396
   381
   apply (rule iffI)
wenzelm@12396
   382
    apply (erule finite_imageD [unfolded inj_on_def])
wenzelm@12396
   383
    apply (simp split add: split_split)
wenzelm@12396
   384
   apply (erule finite_imageI)
paulson@14208
   385
  apply (simp add: converse_def image_def, auto)
wenzelm@12396
   386
  apply (rule bexI)
wenzelm@12396
   387
   prefer 2 apply assumption
wenzelm@12396
   388
  apply simp
wenzelm@12396
   389
  done
wenzelm@12396
   390
paulson@14430
   391
nipkow@15392
   392
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
nipkow@15392
   393
Ehmety) *}
wenzelm@12396
   394
wenzelm@12396
   395
lemma finite_Field: "finite r ==> finite (Field r)"
wenzelm@12396
   396
  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
wenzelm@12396
   397
  apply (induct set: Finites)
wenzelm@12396
   398
   apply (auto simp add: Field_def Domain_insert Range_insert)
wenzelm@12396
   399
  done
wenzelm@12396
   400
wenzelm@12396
   401
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
wenzelm@12396
   402
  apply clarify
wenzelm@12396
   403
  apply (erule trancl_induct)
wenzelm@12396
   404
   apply (auto simp add: Field_def)
wenzelm@12396
   405
  done
wenzelm@12396
   406
wenzelm@12396
   407
lemma finite_trancl: "finite (r^+) = finite r"
wenzelm@12396
   408
  apply auto
wenzelm@12396
   409
   prefer 2
wenzelm@12396
   410
   apply (rule trancl_subset_Field2 [THEN finite_subset])
wenzelm@12396
   411
   apply (rule finite_SigmaI)
wenzelm@12396
   412
    prefer 3
berghofe@13704
   413
    apply (blast intro: r_into_trancl' finite_subset)
wenzelm@12396
   414
   apply (auto simp add: finite_Field)
wenzelm@12396
   415
  done
wenzelm@12396
   416
wenzelm@12396
   417
nipkow@15392
   418
subsection {* A fold functional for finite sets *}
nipkow@15392
   419
nipkow@15392
   420
text {* The intended behaviour is
nipkow@15480
   421
@{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
   422
if @{text f} is associative-commutative. For an application of @{text fold}
nipkow@15392
   423
se the definitions of sums and products over finite sets.
nipkow@15392
   424
*}
nipkow@15392
   425
nipkow@15392
   426
consts
nipkow@15392
   427
  foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
nipkow@15392
   428
nipkow@15480
   429
inductive "foldSet f g z"
nipkow@15392
   430
intros
nipkow@15480
   431
emptyI [intro]: "({}, z) : foldSet f g z"
paulson@15506
   432
insertI [intro]:
paulson@15506
   433
     "\<lbrakk> x \<notin> A; (A, y) : foldSet f g z \<rbrakk>
paulson@15506
   434
      \<Longrightarrow> (insert x A, f (g x) y) : foldSet f g z"
nipkow@15392
   435
nipkow@15480
   436
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g z"
nipkow@15392
   437
nipkow@15392
   438
constdefs
nipkow@15392
   439
  fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
nipkow@15480
   440
  "fold f g z A == THE x. (A, x) : foldSet f g z"
nipkow@15392
   441
paulson@15498
   442
text{*A tempting alternative for the definiens is
paulson@15498
   443
@{term "if finite A then THE x. (A, x) : foldSet f g e else e"}.
paulson@15498
   444
It allows the removal of finiteness assumptions from the theorems
paulson@15498
   445
@{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}.
paulson@15498
   446
The proofs become ugly, with @{text rule_format}. It is not worth the effort.*}
paulson@15498
   447
paulson@15498
   448
nipkow@15392
   449
lemma Diff1_foldSet:
nipkow@15480
   450
  "(A - {x}, y) : foldSet f g z ==> x: A ==> (A, f (g x) y) : foldSet f g z"
nipkow@15392
   451
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
nipkow@15392
   452
nipkow@15480
   453
lemma foldSet_imp_finite: "(A, x) : foldSet f g z ==> finite A"
nipkow@15392
   454
  by (induct set: foldSet) auto
nipkow@15392
   455
nipkow@15480
   456
lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f g z"
nipkow@15392
   457
  by (induct set: Finites) auto
nipkow@15392
   458
nipkow@15392
   459
nipkow@15392
   460
subsubsection {* Commutative monoids *}
nipkow@15480
   461
nipkow@15392
   462
locale ACf =
nipkow@15392
   463
  fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
nipkow@15392
   464
  assumes commute: "x \<cdot> y = y \<cdot> x"
nipkow@15392
   465
    and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
nipkow@15392
   466
nipkow@15392
   467
locale ACe = ACf +
nipkow@15392
   468
  fixes e :: 'a
nipkow@15392
   469
  assumes ident [simp]: "x \<cdot> e = x"
nipkow@15392
   470
nipkow@15480
   471
locale ACIf = ACf +
nipkow@15480
   472
  assumes idem: "x \<cdot> x = x"
nipkow@15480
   473
nipkow@15392
   474
lemma (in ACf) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
nipkow@15392
   475
proof -
nipkow@15392
   476
  have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
nipkow@15392
   477
  also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
nipkow@15392
   478
  also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
nipkow@15392
   479
  finally show ?thesis .
nipkow@15392
   480
qed
nipkow@15392
   481
nipkow@15392
   482
lemmas (in ACf) AC = assoc commute left_commute
nipkow@15392
   483
nipkow@15392
   484
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
nipkow@15392
   485
proof -
nipkow@15392
   486
  have "x \<cdot> e = x" by (rule ident)
nipkow@15392
   487
  thus ?thesis by (subst commute)
nipkow@15392
   488
qed
nipkow@15392
   489
nipkow@15497
   490
lemma (in ACIf) idem2: "x \<cdot> (x \<cdot> y) = x \<cdot> y"
nipkow@15497
   491
proof -
nipkow@15497
   492
  have "x \<cdot> (x \<cdot> y) = (x \<cdot> x) \<cdot> y" by(simp add:assoc)
nipkow@15497
   493
  also have "\<dots> = x \<cdot> y" by(simp add:idem)
nipkow@15497
   494
  finally show ?thesis .
nipkow@15497
   495
qed
nipkow@15497
   496
nipkow@15497
   497
lemmas (in ACIf) ACI = AC idem idem2
nipkow@15497
   498
ballarin@15765
   499
text{* Interpretation of locales: *}
ballarin@15765
   500
ballarin@15765
   501
interpretation AC_add: ACe ["op +" "0::'a::comm_monoid_add"]
ballarin@15765
   502
by(auto intro: ACf.intro ACe_axioms.intro add_assoc add_commute)
nipkow@15402
   503
ballarin@15765
   504
interpretation AC_mult: ACe ["op *" "1::'a::comm_monoid_mult"]
ballarin@15765
   505
  apply -
nipkow@15780
   506
   apply (fast intro: ACf.intro mult_assoc mult_commute)
nipkow@15780
   507
  apply (fastsimp intro: ACe_axioms.intro mult_assoc mult_commute)
ballarin@15765
   508
  done
ballarin@15765
   509
nipkow@15402
   510
nipkow@15392
   511
subsubsection{*From @{term foldSet} to @{term fold}*}
nipkow@15392
   512
paulson@15510
   513
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
paulson@15510
   514
by (auto simp add: less_Suc_eq) 
paulson@15510
   515
paulson@15510
   516
lemma insert_image_inj_on_eq:
paulson@15510
   517
     "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; 
paulson@15510
   518
        inj_on h {i. i < Suc m}|] 
paulson@15510
   519
      ==> A = h ` {i. i < m}"
paulson@15510
   520
apply (auto simp add: image_less_Suc inj_on_def)
paulson@15510
   521
apply (blast intro: less_trans) 
paulson@15510
   522
done
paulson@15510
   523
paulson@15510
   524
lemma insert_inj_onE:
paulson@15510
   525
  assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" 
paulson@15510
   526
      and inj_on: "inj_on h {i::nat. i<n}"
paulson@15510
   527
  shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"
paulson@15510
   528
proof (cases n)
paulson@15510
   529
  case 0 thus ?thesis using aA by auto
paulson@15510
   530
next
paulson@15510
   531
  case (Suc m)
paulson@15510
   532
  have nSuc: "n = Suc m" . 
paulson@15510
   533
  have mlessn: "m<n" by (simp add: nSuc)
paulson@15532
   534
  from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
paulson@15520
   535
  let ?hm = "swap k m h"
paulson@15520
   536
  have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn 
paulson@15520
   537
    by (simp add: inj_on_swap_iff inj_on)
paulson@15510
   538
  show ?thesis
paulson@15520
   539
  proof (intro exI conjI)
paulson@15520
   540
    show "inj_on ?hm {i. i < m}" using inj_hm
paulson@15510
   541
      by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
paulson@15520
   542
    show "m<n" by (rule mlessn)
paulson@15520
   543
    show "A = ?hm ` {i. i < m}" 
paulson@15520
   544
    proof (rule insert_image_inj_on_eq)
paulson@15520
   545
      show "inj_on (swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
paulson@15520
   546
      show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) 
paulson@15520
   547
      show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
paulson@15520
   548
	using aA hkeq nSuc klessn
paulson@15520
   549
	by (auto simp add: swap_def image_less_Suc fun_upd_image 
paulson@15520
   550
			   less_Suc_eq inj_on_image_set_diff [OF inj_on])
nipkow@15479
   551
    qed
nipkow@15479
   552
  qed
nipkow@15479
   553
qed
nipkow@15479
   554
nipkow@15392
   555
lemma (in ACf) foldSet_determ_aux:
paulson@15510
   556
  "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n}; 
paulson@15510
   557
                (A,x) : foldSet f g z; (A,x') : foldSet f g z \<rbrakk>
nipkow@15392
   558
   \<Longrightarrow> x' = x"
paulson@15510
   559
proof (induct n rule: less_induct)
paulson@15510
   560
  case (less n)
paulson@15510
   561
    have IH: "!!m h A x x'. 
paulson@15510
   562
               \<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m}; 
paulson@15510
   563
                (A,x) \<in> foldSet f g z; (A, x') \<in> foldSet f g z\<rbrakk> \<Longrightarrow> x' = x" .
paulson@15510
   564
    have Afoldx: "(A,x) \<in> foldSet f g z" and Afoldx': "(A,x') \<in> foldSet f g z"
paulson@15510
   565
     and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" .
paulson@15510
   566
    show ?case
paulson@15510
   567
    proof (rule foldSet.cases [OF Afoldx])
paulson@15510
   568
      assume "(A, x) = ({}, z)"
paulson@15510
   569
      with Afoldx' show "x' = x" by blast
nipkow@15392
   570
    next
paulson@15510
   571
      fix B b u
paulson@15510
   572
      assume "(A,x) = (insert b B, g b \<cdot> u)" and notinB: "b \<notin> B"
paulson@15510
   573
         and Bu: "(B,u) \<in> foldSet f g z"
paulson@15510
   574
      hence AbB: "A = insert b B" and x: "x = g b \<cdot> u" by auto
paulson@15510
   575
      show "x'=x" 
paulson@15510
   576
      proof (rule foldSet.cases [OF Afoldx'])
paulson@15510
   577
        assume "(A, x') = ({}, z)"
paulson@15510
   578
        with AbB show "x' = x" by blast
nipkow@15392
   579
      next
paulson@15510
   580
	fix C c v
paulson@15510
   581
	assume "(A,x') = (insert c C, g c \<cdot> v)" and notinC: "c \<notin> C"
paulson@15510
   582
	   and Cv: "(C,v) \<in> foldSet f g z"
paulson@15510
   583
	hence AcC: "A = insert c C" and x': "x' = g c \<cdot> v" by auto
paulson@15510
   584
	from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
paulson@15510
   585
        from insert_inj_onE [OF Beq notinB injh]
paulson@15510
   586
        obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" 
paulson@15510
   587
                     and Beq: "B = hB ` {i. i < mB}"
paulson@15510
   588
                     and lessB: "mB < n" by auto 
paulson@15510
   589
	from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
paulson@15510
   590
        from insert_inj_onE [OF Ceq notinC injh]
paulson@15510
   591
        obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
paulson@15510
   592
                       and Ceq: "C = hC ` {i. i < mC}"
paulson@15510
   593
                       and lessC: "mC < n" by auto 
paulson@15510
   594
	show "x'=x"
nipkow@15392
   595
	proof cases
paulson@15510
   596
          assume "b=c"
paulson@15510
   597
	  then moreover have "B = C" using AbB AcC notinB notinC by auto
paulson@15510
   598
	  ultimately show ?thesis  using Bu Cv x x' IH[OF lessC Ceq inj_onC]
paulson@15510
   599
            by auto
nipkow@15392
   600
	next
nipkow@15392
   601
	  assume diff: "b \<noteq> c"
nipkow@15392
   602
	  let ?D = "B - {c}"
nipkow@15392
   603
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
paulson@15510
   604
	    using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
nipkow@15402
   605
	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
paulson@15510
   606
	  with AbB have "finite ?D" by simp
nipkow@15480
   607
	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g z"
nipkow@17589
   608
	    using finite_imp_foldSet by iprover
paulson@15506
   609
	  moreover have cinB: "c \<in> B" using B by auto
nipkow@15480
   610
	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g z"
nipkow@15392
   611
	    by(rule Diff1_foldSet)
paulson@15510
   612
	  hence "g c \<cdot> d = u" by (rule IH [OF lessB Beq inj_onB Bu]) 
paulson@15510
   613
          moreover have "g b \<cdot> d = v"
paulson@15510
   614
	  proof (rule IH[OF lessC Ceq inj_onC Cv])
paulson@15510
   615
	    show "(C, g b \<cdot> d) \<in> foldSet f g z" using C notinB Dfoldd
nipkow@15392
   616
	      by fastsimp
nipkow@15392
   617
	  qed
paulson@15510
   618
	  ultimately show ?thesis using x x' by (auto simp: AC)
nipkow@15392
   619
	qed
nipkow@15392
   620
      qed
nipkow@15392
   621
    qed
nipkow@15392
   622
  qed
nipkow@15392
   623
nipkow@15392
   624
nipkow@15392
   625
lemma (in ACf) foldSet_determ:
paulson@15510
   626
  "(A,x) : foldSet f g z ==> (A,y) : foldSet f g z ==> y = x"
paulson@15510
   627
apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) 
paulson@15510
   628
apply (blast intro: foldSet_determ_aux [rule_format])
nipkow@15392
   629
done
nipkow@15392
   630
nipkow@15480
   631
lemma (in ACf) fold_equality: "(A, y) : foldSet f g z ==> fold f g z A = y"
nipkow@15392
   632
  by (unfold fold_def) (blast intro: foldSet_determ)
nipkow@15392
   633
nipkow@15392
   634
text{* The base case for @{text fold}: *}
nipkow@15392
   635
nipkow@15480
   636
lemma fold_empty [simp]: "fold f g z {} = z"
nipkow@15392
   637
  by (unfold fold_def) blast
nipkow@15392
   638
nipkow@15392
   639
lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
nipkow@15480
   640
    ((insert x A, v) : foldSet f g z) =
nipkow@15480
   641
    (EX y. (A, y) : foldSet f g z & v = f (g x) y)"
nipkow@15392
   642
  apply auto
nipkow@15392
   643
  apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
nipkow@15392
   644
   apply (fastsimp dest: foldSet_imp_finite)
nipkow@15392
   645
  apply (blast intro: foldSet_determ)
nipkow@15392
   646
  done
nipkow@15392
   647
nipkow@15392
   648
text{* The recursion equation for @{text fold}: *}
nipkow@15392
   649
nipkow@15392
   650
lemma (in ACf) fold_insert[simp]:
nipkow@15480
   651
    "finite A ==> x \<notin> A ==> fold f g z (insert x A) = f (g x) (fold f g z A)"
nipkow@15392
   652
  apply (unfold fold_def)
nipkow@15392
   653
  apply (simp add: fold_insert_aux)
nipkow@15392
   654
  apply (rule the_equality)
nipkow@15392
   655
  apply (auto intro: finite_imp_foldSet
nipkow@15392
   656
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
nipkow@15392
   657
  done
nipkow@15392
   658
nipkow@15535
   659
lemma (in ACf) fold_rec:
nipkow@15535
   660
assumes fin: "finite A" and a: "a:A"
nipkow@15535
   661
shows "fold f g z A = f (g a) (fold f g z (A - {a}))"
nipkow@15535
   662
proof-
nipkow@15535
   663
  have A: "A = insert a (A - {a})" using a by blast
nipkow@15535
   664
  hence "fold f g z A = fold f g z (insert a (A - {a}))" by simp
nipkow@15535
   665
  also have "\<dots> = f (g a) (fold f g z (A - {a}))"
nipkow@15535
   666
    by(rule fold_insert) (simp add:fin)+
nipkow@15535
   667
  finally show ?thesis .
nipkow@15535
   668
qed
nipkow@15535
   669
nipkow@15392
   670
nipkow@15480
   671
text{* A simplified version for idempotent functions: *}
nipkow@15480
   672
paulson@15509
   673
lemma (in ACIf) fold_insert_idem:
nipkow@15480
   674
assumes finA: "finite A"
paulson@15508
   675
shows "fold f g z (insert a A) = g a \<cdot> fold f g z A"
nipkow@15480
   676
proof cases
nipkow@15480
   677
  assume "a \<in> A"
nipkow@15480
   678
  then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
nipkow@15480
   679
    by(blast dest: mk_disjoint_insert)
nipkow@15480
   680
  show ?thesis
nipkow@15480
   681
  proof -
nipkow@15480
   682
    from finA A have finB: "finite B" by(blast intro: finite_subset)
nipkow@15480
   683
    have "fold f g z (insert a A) = fold f g z (insert a B)" using A by simp
nipkow@15480
   684
    also have "\<dots> = (g a) \<cdot> (fold f g z B)"
paulson@15506
   685
      using finB disj by simp
nipkow@15480
   686
    also have "\<dots> = g a \<cdot> fold f g z A"
nipkow@15480
   687
      using A finB disj by(simp add:idem assoc[symmetric])
nipkow@15480
   688
    finally show ?thesis .
nipkow@15480
   689
  qed
nipkow@15480
   690
next
nipkow@15480
   691
  assume "a \<notin> A"
nipkow@15480
   692
  with finA show ?thesis by simp
nipkow@15480
   693
qed
nipkow@15480
   694
nipkow@15484
   695
lemma (in ACIf) foldI_conv_id:
nipkow@15484
   696
  "finite A \<Longrightarrow> fold f g z A = fold f id z (g ` A)"
paulson@15509
   697
by(erule finite_induct)(simp_all add: fold_insert_idem del: fold_insert)
nipkow@15484
   698
nipkow@15392
   699
subsubsection{*Lemmas about @{text fold}*}
nipkow@15392
   700
nipkow@15392
   701
lemma (in ACf) fold_commute:
paulson@15487
   702
  "finite A ==> (!!z. f x (fold f g z A) = fold f g (f x z) A)"
nipkow@15392
   703
  apply (induct set: Finites, simp)
paulson@15487
   704
  apply (simp add: left_commute [of x])
nipkow@15392
   705
  done
nipkow@15392
   706
nipkow@15392
   707
lemma (in ACf) fold_nest_Un_Int:
nipkow@15392
   708
  "finite A ==> finite B
nipkow@15480
   709
    ==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)"
nipkow@15392
   710
  apply (induct set: Finites, simp)
nipkow@15392
   711
  apply (simp add: fold_commute Int_insert_left insert_absorb)
nipkow@15392
   712
  done
nipkow@15392
   713
nipkow@15392
   714
lemma (in ACf) fold_nest_Un_disjoint:
nipkow@15392
   715
  "finite A ==> finite B ==> A Int B = {}
nipkow@15480
   716
    ==> fold f g z (A Un B) = fold f g (fold f g z B) A"
nipkow@15392
   717
  by (simp add: fold_nest_Un_Int)
nipkow@15392
   718
nipkow@15392
   719
lemma (in ACf) fold_reindex:
paulson@15487
   720
assumes fin: "finite A"
paulson@15487
   721
shows "inj_on h A \<Longrightarrow> fold f g z (h ` A) = fold f (g \<circ> h) z A"
paulson@15506
   722
using fin apply induct
nipkow@15392
   723
 apply simp
nipkow@15392
   724
apply simp
nipkow@15392
   725
done
nipkow@15392
   726
nipkow@15392
   727
lemma (in ACe) fold_Un_Int:
nipkow@15392
   728
  "finite A ==> finite B ==>
nipkow@15392
   729
    fold f g e A \<cdot> fold f g e B =
nipkow@15392
   730
    fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
nipkow@15392
   731
  apply (induct set: Finites, simp)
nipkow@15392
   732
  apply (simp add: AC insert_absorb Int_insert_left)
nipkow@15392
   733
  done
nipkow@15392
   734
nipkow@15392
   735
corollary (in ACe) fold_Un_disjoint:
nipkow@15392
   736
  "finite A ==> finite B ==> A Int B = {} ==>
nipkow@15392
   737
    fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B"
nipkow@15392
   738
  by (simp add: fold_Un_Int)
nipkow@15392
   739
nipkow@15392
   740
lemma (in ACe) fold_UN_disjoint:
nipkow@15392
   741
  "\<lbrakk> finite I; ALL i:I. finite (A i);
nipkow@15392
   742
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
nipkow@15392
   743
   \<Longrightarrow> fold f g e (UNION I A) =
nipkow@15392
   744
       fold f (%i. fold f g e (A i)) e I"
nipkow@15392
   745
  apply (induct set: Finites, simp, atomize)
nipkow@15392
   746
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
nipkow@15392
   747
   prefer 2 apply blast
nipkow@15392
   748
  apply (subgoal_tac "A x Int UNION F A = {}")
nipkow@15392
   749
   prefer 2 apply blast
nipkow@15392
   750
  apply (simp add: fold_Un_disjoint)
nipkow@15392
   751
  done
nipkow@15392
   752
paulson@15506
   753
text{*Fusion theorem, as described in
paulson@15506
   754
Graham Hutton's paper,
paulson@15506
   755
A Tutorial on the Universality and Expressiveness of Fold,
paulson@15506
   756
JFP 9:4 (355-372), 1999.*}
paulson@15506
   757
lemma (in ACf) fold_fusion:
paulson@15506
   758
      includes ACf g
paulson@15506
   759
      shows
paulson@15506
   760
	"finite A ==> 
paulson@15506
   761
	 (!!x y. h (g x y) = f x (h y)) ==>
paulson@15506
   762
         h (fold g j w A) = fold f j (h w) A"
paulson@15506
   763
  by (induct set: Finites, simp_all)
paulson@15506
   764
nipkow@15392
   765
lemma (in ACf) fold_cong:
nipkow@15480
   766
  "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A"
nipkow@15480
   767
  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
   768
   apply simp
nipkow@15392
   769
  apply (erule finite_induct, simp)
nipkow@15392
   770
  apply (simp add: subset_insert_iff, clarify)
nipkow@15392
   771
  apply (subgoal_tac "finite C")
nipkow@15392
   772
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
nipkow@15392
   773
  apply (subgoal_tac "C = insert x (C - {x})")
nipkow@15392
   774
   prefer 2 apply blast
nipkow@15392
   775
  apply (erule ssubst)
nipkow@15392
   776
  apply (drule spec)
nipkow@15392
   777
  apply (erule (1) notE impE)
nipkow@15392
   778
  apply (simp add: Ball_def del: insert_Diff_single)
nipkow@15392
   779
  done
nipkow@15392
   780
nipkow@15392
   781
lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@15392
   782
  fold f (%x. fold f (g x) e (B x)) e A =
nipkow@15392
   783
  fold f (split g) e (SIGMA x:A. B x)"
nipkow@15392
   784
apply (subst Sigma_def)
paulson@15506
   785
apply (subst fold_UN_disjoint, assumption, simp)
nipkow@15392
   786
 apply blast
nipkow@15392
   787
apply (erule fold_cong)
paulson@15506
   788
apply (subst fold_UN_disjoint, simp, simp)
nipkow@15392
   789
 apply blast
paulson@15506
   790
apply simp
nipkow@15392
   791
done
nipkow@15392
   792
nipkow@15392
   793
lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
nipkow@15392
   794
   fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
paulson@15506
   795
apply (erule finite_induct, simp)
nipkow@15392
   796
apply (simp add:AC)
nipkow@15392
   797
done
nipkow@15392
   798
nipkow@15392
   799
nipkow@15402
   800
subsection {* Generalized summation over a set *}
nipkow@15402
   801
nipkow@15402
   802
constdefs
nipkow@15402
   803
  setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
nipkow@15402
   804
  "setsum f A == if finite A then fold (op +) f 0 A else 0"
nipkow@15402
   805
wenzelm@19535
   806
abbreviation
wenzelm@19535
   807
  Setsum  ("\<Sum>_" [1000] 999)
wenzelm@19535
   808
  "\<Sum>A == setsum (%x. x) A"
wenzelm@19535
   809
nipkow@15402
   810
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
nipkow@15402
   811
written @{text"\<Sum>x\<in>A. e"}. *}
nipkow@15402
   812
nipkow@15402
   813
syntax
paulson@17189
   814
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
nipkow@15402
   815
syntax (xsymbols)
paulson@17189
   816
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
   817
syntax (HTML output)
paulson@17189
   818
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
   819
nipkow@15402
   820
translations -- {* Beware of argument permutation! *}
nipkow@15402
   821
  "SUM i:A. b" == "setsum (%i. b) A"
nipkow@15402
   822
  "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
nipkow@15402
   823
nipkow@15402
   824
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
nipkow@15402
   825
 @{text"\<Sum>x|P. e"}. *}
nipkow@15402
   826
nipkow@15402
   827
syntax
paulson@17189
   828
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
nipkow@15402
   829
syntax (xsymbols)
paulson@17189
   830
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402
   831
syntax (HTML output)
paulson@17189
   832
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402
   833
nipkow@15402
   834
translations
nipkow@15402
   835
  "SUM x|P. t" => "setsum (%x. t) {x. P}"
nipkow@15402
   836
  "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
nipkow@15402
   837
nipkow@15402
   838
print_translation {*
nipkow@15402
   839
let
wenzelm@19535
   840
  fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
wenzelm@19535
   841
    if x<>y then raise Match
wenzelm@19535
   842
    else let val x' = Syntax.mark_bound x
wenzelm@19535
   843
             val t' = subst_bound(x',t)
wenzelm@19535
   844
             val P' = subst_bound(x',P)
wenzelm@19535
   845
         in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
wenzelm@19535
   846
in [("setsum", setsum_tr')] end
nipkow@15402
   847
*}
nipkow@15402
   848
wenzelm@19535
   849
nipkow@15402
   850
lemma setsum_empty [simp]: "setsum f {} = 0"
nipkow@15402
   851
  by (simp add: setsum_def)
nipkow@15402
   852
nipkow@15402
   853
lemma setsum_insert [simp]:
nipkow@15402
   854
    "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
ballarin@15765
   855
  by (simp add: setsum_def)
nipkow@15402
   856
paulson@15409
   857
lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
paulson@15409
   858
  by (simp add: setsum_def)
paulson@15409
   859
nipkow@15402
   860
lemma setsum_reindex:
nipkow@15402
   861
     "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
ballarin@15765
   862
by(auto simp add: setsum_def AC_add.fold_reindex dest!:finite_imageD)
nipkow@15402
   863
nipkow@15402
   864
lemma setsum_reindex_id:
nipkow@15402
   865
     "inj_on f B ==> setsum f B = setsum id (f ` B)"
nipkow@15402
   866
by (auto simp add: setsum_reindex)
nipkow@15402
   867
nipkow@15402
   868
lemma setsum_cong:
nipkow@15402
   869
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
ballarin@15765
   870
by(fastsimp simp: setsum_def intro: AC_add.fold_cong)
nipkow@15402
   871
nipkow@16733
   872
lemma strong_setsum_cong[cong]:
nipkow@16733
   873
  "A = B ==> (!!x. x:B =simp=> f x = g x)
nipkow@16733
   874
   ==> setsum (%x. f x) A = setsum (%x. g x) B"
berghofe@16632
   875
by(fastsimp simp: simp_implies_def setsum_def intro: AC_add.fold_cong)
berghofe@16632
   876
nipkow@15554
   877
lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A";
nipkow@15554
   878
  by (rule setsum_cong[OF refl], auto);
nipkow@15554
   879
nipkow@15402
   880
lemma setsum_reindex_cong:
nipkow@15554
   881
     "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] 
nipkow@15402
   882
      ==> setsum h B = setsum g A"
nipkow@15402
   883
  by (simp add: setsum_reindex cong: setsum_cong)
nipkow@15402
   884
nipkow@15542
   885
lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
nipkow@15402
   886
apply (clarsimp simp: setsum_def)
ballarin@15765
   887
apply (erule finite_induct, auto)
nipkow@15402
   888
done
nipkow@15402
   889
nipkow@15543
   890
lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
nipkow@15543
   891
by(simp add:setsum_cong)
nipkow@15402
   892
nipkow@15402
   893
lemma setsum_Un_Int: "finite A ==> finite B ==>
nipkow@15402
   894
  setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
nipkow@15402
   895
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
ballarin@15765
   896
by(simp add: setsum_def AC_add.fold_Un_Int [symmetric])
nipkow@15402
   897
nipkow@15402
   898
lemma setsum_Un_disjoint: "finite A ==> finite B
nipkow@15402
   899
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
nipkow@15402
   900
by (subst setsum_Un_Int [symmetric], auto)
nipkow@15402
   901
paulson@15409
   902
(*But we can't get rid of finite I. If infinite, although the rhs is 0, 
paulson@15409
   903
  the lhs need not be, since UNION I A could still be finite.*)
nipkow@15402
   904
lemma setsum_UN_disjoint:
nipkow@15402
   905
    "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402
   906
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402
   907
      setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
ballarin@15765
   908
by(simp add: setsum_def AC_add.fold_UN_disjoint cong: setsum_cong)
nipkow@15402
   909
paulson@15409
   910
text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
paulson@15409
   911
directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
nipkow@15402
   912
lemma setsum_Union_disjoint:
paulson@15409
   913
  "[| (ALL A:C. finite A);
paulson@15409
   914
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
paulson@15409
   915
   ==> setsum f (Union C) = setsum (setsum f) C"
paulson@15409
   916
apply (cases "finite C") 
paulson@15409
   917
 prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
nipkow@15402
   918
  apply (frule setsum_UN_disjoint [of C id f])
paulson@15409
   919
 apply (unfold Union_def id_def, assumption+)
paulson@15409
   920
done
nipkow@15402
   921
paulson@15409
   922
(*But we can't get rid of finite A. If infinite, although the lhs is 0, 
paulson@15409
   923
  the rhs need not be, since SIGMA A B could still be finite.*)
nipkow@15402
   924
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
paulson@17189
   925
    (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
ballarin@15765
   926
by(simp add:setsum_def AC_add.fold_Sigma split_def cong:setsum_cong)
nipkow@15402
   927
paulson@15409
   928
text{*Here we can eliminate the finiteness assumptions, by cases.*}
paulson@15409
   929
lemma setsum_cartesian_product: 
paulson@17189
   930
   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
paulson@15409
   931
apply (cases "finite A") 
paulson@15409
   932
 apply (cases "finite B") 
paulson@15409
   933
  apply (simp add: setsum_Sigma)
paulson@15409
   934
 apply (cases "A={}", simp)
nipkow@15543
   935
 apply (simp) 
paulson@15409
   936
apply (auto simp add: setsum_def
paulson@15409
   937
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
paulson@15409
   938
done
nipkow@15402
   939
nipkow@15402
   940
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
ballarin@15765
   941
by(simp add:setsum_def AC_add.fold_distrib)
nipkow@15402
   942
nipkow@15402
   943
nipkow@15402
   944
subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402
   945
nipkow@15402
   946
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
nipkow@15402
   947
  apply (case_tac "finite A")
nipkow@15402
   948
   prefer 2 apply (simp add: setsum_def)
nipkow@15402
   949
  apply (erule rev_mp)
nipkow@15402
   950
  apply (erule finite_induct, auto)
nipkow@15402
   951
  done
nipkow@15402
   952
nipkow@15402
   953
lemma setsum_eq_0_iff [simp]:
nipkow@15402
   954
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
nipkow@15402
   955
  by (induct set: Finites) auto
nipkow@15402
   956
nipkow@15402
   957
lemma setsum_Un_nat: "finite A ==> finite B ==>
nipkow@15402
   958
    (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
nipkow@15402
   959
  -- {* For the natural numbers, we have subtraction. *}
nipkow@15402
   960
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
nipkow@15402
   961
nipkow@15402
   962
lemma setsum_Un: "finite A ==> finite B ==>
nipkow@15402
   963
    (setsum f (A Un B) :: 'a :: ab_group_add) =
nipkow@15402
   964
      setsum f A + setsum f B - setsum f (A Int B)"
nipkow@15402
   965
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
nipkow@15402
   966
nipkow@15402
   967
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
nipkow@15402
   968
    (if a:A then setsum f A - f a else setsum f A)"
nipkow@15402
   969
  apply (case_tac "finite A")
nipkow@15402
   970
   prefer 2 apply (simp add: setsum_def)
nipkow@15402
   971
  apply (erule finite_induct)
nipkow@15402
   972
   apply (auto simp add: insert_Diff_if)
nipkow@15402
   973
  apply (drule_tac a = a in mk_disjoint_insert, auto)
nipkow@15402
   974
  done
nipkow@15402
   975
nipkow@15402
   976
lemma setsum_diff1: "finite A \<Longrightarrow>
nipkow@15402
   977
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
nipkow@15402
   978
  (if a:A then setsum f A - f a else setsum f A)"
nipkow@15402
   979
  by (erule finite_induct) (auto simp add: insert_Diff_if)
nipkow@15402
   980
obua@15552
   981
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
   982
  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
   983
  apply (auto simp add: insert_Diff_if add_ac)
obua@15552
   984
  done
obua@15552
   985
nipkow@15402
   986
(* By Jeremy Siek: *)
nipkow@15402
   987
nipkow@15402
   988
lemma setsum_diff_nat: 
wenzelm@19535
   989
  assumes "finite B"
wenzelm@19535
   990
    and "B \<subseteq> A"
wenzelm@19535
   991
  shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
wenzelm@19535
   992
  using prems
wenzelm@19535
   993
proof induct
nipkow@15402
   994
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
nipkow@15402
   995
next
nipkow@15402
   996
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
nipkow@15402
   997
    and xFinA: "insert x F \<subseteq> A"
nipkow@15402
   998
    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
nipkow@15402
   999
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
nipkow@15402
  1000
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
nipkow@15402
  1001
    by (simp add: setsum_diff1_nat)
nipkow@15402
  1002
  from xFinA have "F \<subseteq> A" by simp
nipkow@15402
  1003
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
nipkow@15402
  1004
  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
nipkow@15402
  1005
    by simp
nipkow@15402
  1006
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
nipkow@15402
  1007
  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
nipkow@15402
  1008
    by simp
nipkow@15402
  1009
  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
nipkow@15402
  1010
  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
nipkow@15402
  1011
    by simp
nipkow@15402
  1012
  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
nipkow@15402
  1013
qed
nipkow@15402
  1014
nipkow@15402
  1015
lemma setsum_diff:
nipkow@15402
  1016
  assumes le: "finite A" "B \<subseteq> A"
nipkow@15402
  1017
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
nipkow@15402
  1018
proof -
nipkow@15402
  1019
  from le have finiteB: "finite B" using finite_subset by auto
nipkow@15402
  1020
  show ?thesis using finiteB le
wenzelm@19535
  1021
  proof (induct)
wenzelm@19535
  1022
    case empty
wenzelm@19535
  1023
    thus ?case by auto
wenzelm@19535
  1024
  next
wenzelm@19535
  1025
    case (insert x F)
wenzelm@19535
  1026
    thus ?case using le finiteB 
wenzelm@19535
  1027
      by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
nipkow@15402
  1028
  qed
wenzelm@19535
  1029
qed
nipkow@15402
  1030
nipkow@15402
  1031
lemma setsum_mono:
nipkow@15402
  1032
  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
nipkow@15402
  1033
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
nipkow@15402
  1034
proof (cases "finite K")
nipkow@15402
  1035
  case True
nipkow@15402
  1036
  thus ?thesis using le
wenzelm@19535
  1037
  proof induct
nipkow@15402
  1038
    case empty
nipkow@15402
  1039
    thus ?case by simp
nipkow@15402
  1040
  next
nipkow@15402
  1041
    case insert
wenzelm@19535
  1042
    thus ?case using add_mono by fastsimp
nipkow@15402
  1043
  qed
nipkow@15402
  1044
next
nipkow@15402
  1045
  case False
nipkow@15402
  1046
  thus ?thesis
nipkow@15402
  1047
    by (simp add: setsum_def)
nipkow@15402
  1048
qed
nipkow@15402
  1049
nipkow@15554
  1050
lemma setsum_strict_mono:
wenzelm@19535
  1051
  fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
wenzelm@19535
  1052
  assumes "finite A"  "A \<noteq> {}"
wenzelm@19535
  1053
    and "!!x. x:A \<Longrightarrow> f x < g x"
wenzelm@19535
  1054
  shows "setsum f A < setsum g A"
wenzelm@19535
  1055
  using prems
nipkow@15554
  1056
proof (induct rule: finite_ne_induct)
nipkow@15554
  1057
  case singleton thus ?case by simp
nipkow@15554
  1058
next
nipkow@15554
  1059
  case insert thus ?case by (auto simp: add_strict_mono)
nipkow@15554
  1060
qed
nipkow@15554
  1061
nipkow@15535
  1062
lemma setsum_negf:
wenzelm@19535
  1063
  "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
nipkow@15535
  1064
proof (cases "finite A")
nipkow@15535
  1065
  case True thus ?thesis by (induct set: Finites, auto)
nipkow@15535
  1066
next
nipkow@15535
  1067
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15535
  1068
qed
nipkow@15402
  1069
nipkow@15535
  1070
lemma setsum_subtractf:
wenzelm@19535
  1071
  "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
wenzelm@19535
  1072
    setsum f A - setsum g A"
nipkow@15535
  1073
proof (cases "finite A")
nipkow@15535
  1074
  case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
nipkow@15535
  1075
next
nipkow@15535
  1076
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15535
  1077
qed
nipkow@15402
  1078
nipkow@15535
  1079
lemma setsum_nonneg:
wenzelm@19535
  1080
  assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
wenzelm@19535
  1081
  shows "0 \<le> setsum f A"
nipkow@15535
  1082
proof (cases "finite A")
nipkow@15535
  1083
  case True thus ?thesis using nn
wenzelm@19535
  1084
  proof (induct)
wenzelm@19535
  1085
    case empty then show ?case by simp
wenzelm@19535
  1086
  next
wenzelm@19535
  1087
    case (insert x F)
wenzelm@19535
  1088
    then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
wenzelm@19535
  1089
    with insert show ?case by simp
wenzelm@19535
  1090
  qed
nipkow@15535
  1091
next
nipkow@15535
  1092
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15535
  1093
qed
nipkow@15402
  1094
nipkow@15535
  1095
lemma setsum_nonpos:
wenzelm@19535
  1096
  assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
wenzelm@19535
  1097
  shows "setsum f A \<le> 0"
nipkow@15535
  1098
proof (cases "finite A")
nipkow@15535
  1099
  case True thus ?thesis using np
wenzelm@19535
  1100
  proof (induct)
wenzelm@19535
  1101
    case empty then show ?case by simp
wenzelm@19535
  1102
  next
wenzelm@19535
  1103
    case (insert x F)
wenzelm@19535
  1104
    then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
wenzelm@19535
  1105
    with insert show ?case by simp
wenzelm@19535
  1106
  qed
nipkow@15535
  1107
next
nipkow@15535
  1108
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15535
  1109
qed
nipkow@15402
  1110
nipkow@15539
  1111
lemma setsum_mono2:
nipkow@15539
  1112
fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}"
nipkow@15539
  1113
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
nipkow@15539
  1114
shows "setsum f A \<le> setsum f B"
nipkow@15539
  1115
proof -
nipkow@15539
  1116
  have "setsum f A \<le> setsum f A + setsum f (B-A)"
nipkow@15539
  1117
    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
nipkow@15539
  1118
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
nipkow@15539
  1119
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
nipkow@15539
  1120
  also have "A \<union> (B-A) = B" using sub by blast
nipkow@15539
  1121
  finally show ?thesis .
nipkow@15539
  1122
qed
nipkow@15542
  1123
avigad@16775
  1124
lemma setsum_mono3: "finite B ==> A <= B ==> 
avigad@16775
  1125
    ALL x: B - A. 
avigad@16775
  1126
      0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==>
avigad@16775
  1127
        setsum f A <= setsum f B"
avigad@16775
  1128
  apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
avigad@16775
  1129
  apply (erule ssubst)
avigad@16775
  1130
  apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
avigad@16775
  1131
  apply simp
avigad@16775
  1132
  apply (rule add_left_mono)
avigad@16775
  1133
  apply (erule setsum_nonneg)
avigad@16775
  1134
  apply (subst setsum_Un_disjoint [THEN sym])
avigad@16775
  1135
  apply (erule finite_subset, assumption)
avigad@16775
  1136
  apply (rule finite_subset)
avigad@16775
  1137
  prefer 2
avigad@16775
  1138
  apply assumption
avigad@16775
  1139
  apply auto
avigad@16775
  1140
  apply (rule setsum_cong)
avigad@16775
  1141
  apply auto
avigad@16775
  1142
done
avigad@16775
  1143
ballarin@19279
  1144
lemma setsum_right_distrib: 
nipkow@15402
  1145
  fixes f :: "'a => ('b::semiring_0_cancel)"
nipkow@15402
  1146
  shows "r * setsum f A = setsum (%n. r * f n) A"
nipkow@15402
  1147
proof (cases "finite A")
nipkow@15402
  1148
  case True
nipkow@15402
  1149
  thus ?thesis
nipkow@15402
  1150
  proof (induct)
nipkow@15402
  1151
    case empty thus ?case by simp
nipkow@15402
  1152
  next
nipkow@15402
  1153
    case (insert x A) thus ?case by (simp add: right_distrib)
nipkow@15402
  1154
  qed
nipkow@15402
  1155
next
nipkow@15402
  1156
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15402
  1157
qed
nipkow@15402
  1158
ballarin@17149
  1159
lemma setsum_left_distrib:
ballarin@17149
  1160
  "setsum f A * (r::'a::semiring_0_cancel) = (\<Sum>n\<in>A. f n * r)"
ballarin@17149
  1161
proof (cases "finite A")
ballarin@17149
  1162
  case True
ballarin@17149
  1163
  then show ?thesis
ballarin@17149
  1164
  proof induct
ballarin@17149
  1165
    case empty thus ?case by simp
ballarin@17149
  1166
  next
ballarin@17149
  1167
    case (insert x A) thus ?case by (simp add: left_distrib)
ballarin@17149
  1168
  qed
ballarin@17149
  1169
next
ballarin@17149
  1170
  case False thus ?thesis by (simp add: setsum_def)
ballarin@17149
  1171
qed
ballarin@17149
  1172
ballarin@17149
  1173
lemma setsum_divide_distrib:
ballarin@17149
  1174
  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
ballarin@17149
  1175
proof (cases "finite A")
ballarin@17149
  1176
  case True
ballarin@17149
  1177
  then show ?thesis
ballarin@17149
  1178
  proof induct
ballarin@17149
  1179
    case empty thus ?case by simp
ballarin@17149
  1180
  next
ballarin@17149
  1181
    case (insert x A) thus ?case by (simp add: add_divide_distrib)
ballarin@17149
  1182
  qed
ballarin@17149
  1183
next
ballarin@17149
  1184
  case False thus ?thesis by (simp add: setsum_def)
ballarin@17149
  1185
qed
ballarin@17149
  1186
nipkow@15535
  1187
lemma setsum_abs[iff]: 
nipkow@15402
  1188
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
nipkow@15402
  1189
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
nipkow@15535
  1190
proof (cases "finite A")
nipkow@15535
  1191
  case True
nipkow@15535
  1192
  thus ?thesis
nipkow@15535
  1193
  proof (induct)
nipkow@15535
  1194
    case empty thus ?case by simp
nipkow@15535
  1195
  next
nipkow@15535
  1196
    case (insert x A)
nipkow@15535
  1197
    thus ?case by (auto intro: abs_triangle_ineq order_trans)
nipkow@15535
  1198
  qed
nipkow@15402
  1199
next
nipkow@15535
  1200
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15402
  1201
qed
nipkow@15402
  1202
nipkow@15535
  1203
lemma setsum_abs_ge_zero[iff]: 
nipkow@15402
  1204
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
nipkow@15402
  1205
  shows "0 \<le> setsum (%i. abs(f i)) A"
nipkow@15535
  1206
proof (cases "finite A")
nipkow@15535
  1207
  case True
nipkow@15535
  1208
  thus ?thesis
nipkow@15535
  1209
  proof (induct)
nipkow@15535
  1210
    case empty thus ?case by simp
nipkow@15535
  1211
  next
nipkow@15535
  1212
    case (insert x A) thus ?case by (auto intro: order_trans)
nipkow@15535
  1213
  qed
nipkow@15402
  1214
next
nipkow@15535
  1215
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15402
  1216
qed
nipkow@15402
  1217
nipkow@15539
  1218
lemma abs_setsum_abs[simp]: 
nipkow@15539
  1219
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
nipkow@15539
  1220
  shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
nipkow@15539
  1221
proof (cases "finite A")
nipkow@15539
  1222
  case True
nipkow@15539
  1223
  thus ?thesis
nipkow@15539
  1224
  proof (induct)
nipkow@15539
  1225
    case empty thus ?case by simp
nipkow@15539
  1226
  next
nipkow@15539
  1227
    case (insert a A)
nipkow@15539
  1228
    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
  1229
    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
  1230
    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
avigad@16775
  1231
      by (simp del: abs_of_nonneg)
nipkow@15539
  1232
    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
nipkow@15539
  1233
    finally show ?case .
nipkow@15539
  1234
  qed
nipkow@15539
  1235
next
nipkow@15539
  1236
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15539
  1237
qed
nipkow@15539
  1238
nipkow@15402
  1239
ballarin@17149
  1240
text {* Commuting outer and inner summation *}
ballarin@17149
  1241
ballarin@17149
  1242
lemma swap_inj_on:
ballarin@17149
  1243
  "inj_on (%(i, j). (j, i)) (A \<times> B)"
ballarin@17149
  1244
  by (unfold inj_on_def) fast
ballarin@17149
  1245
ballarin@17149
  1246
lemma swap_product:
ballarin@17149
  1247
  "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
ballarin@17149
  1248
  by (simp add: split_def image_def) blast
ballarin@17149
  1249
ballarin@17149
  1250
lemma setsum_commute:
ballarin@17149
  1251
  "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
ballarin@17149
  1252
proof (simp add: setsum_cartesian_product)
paulson@17189
  1253
  have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
paulson@17189
  1254
    (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
ballarin@17149
  1255
    (is "?s = _")
ballarin@17149
  1256
    apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
ballarin@17149
  1257
    apply (simp add: split_def)
ballarin@17149
  1258
    done
paulson@17189
  1259
  also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
ballarin@17149
  1260
    (is "_ = ?t")
ballarin@17149
  1261
    apply (simp add: swap_product)
ballarin@17149
  1262
    done
ballarin@17149
  1263
  finally show "?s = ?t" .
ballarin@17149
  1264
qed
ballarin@17149
  1265
ballarin@19279
  1266
lemma setsum_product:
ballarin@19279
  1267
  fixes f :: "nat => ('a::semiring_0_cancel)"
ballarin@19279
  1268
  shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
ballarin@19279
  1269
  by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
ballarin@19279
  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
wenzelm@19535
  1278
abbreviation
wenzelm@19535
  1279
  Setprod  ("\<Prod>_" [1000] 999)
wenzelm@19535
  1280
  "\<Prod>A == setprod (%x. x) A"
wenzelm@19535
  1281
nipkow@15402
  1282
syntax
paulson@17189
  1283
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
nipkow@15402
  1284
syntax (xsymbols)
paulson@17189
  1285
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402
  1286
syntax (HTML output)
paulson@17189
  1287
  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@16550
  1288
nipkow@16550
  1289
translations -- {* Beware of argument permutation! *}
nipkow@16550
  1290
  "PROD i:A. b" == "setprod (%i. b) A" 
nipkow@16550
  1291
  "\<Prod>i\<in>A. b" == "setprod (%i. b) A" 
nipkow@16550
  1292
nipkow@16550
  1293
text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
nipkow@16550
  1294
 @{text"\<Prod>x|P. e"}. *}
nipkow@16550
  1295
nipkow@16550
  1296
syntax
paulson@17189
  1297
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
nipkow@16550
  1298
syntax (xsymbols)
paulson@17189
  1299
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
nipkow@16550
  1300
syntax (HTML output)
paulson@17189
  1301
  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
nipkow@16550
  1302
nipkow@15402
  1303
translations
nipkow@16550
  1304
  "PROD x|P. t" => "setprod (%x. t) {x. P}"
nipkow@16550
  1305
  "\<Prod>x|P. t" => "setprod (%x. t) {x. P}"
nipkow@16550
  1306
nipkow@15402
  1307
nipkow@15402
  1308
lemma setprod_empty [simp]: "setprod f {} = 1"
nipkow@15402
  1309
  by (auto simp add: setprod_def)
nipkow@15402
  1310
nipkow@15402
  1311
lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
nipkow@15402
  1312
    setprod f (insert a A) = f a * setprod f A"
ballarin@15765
  1313
by (simp add: setprod_def)
nipkow@15402
  1314
paulson@15409
  1315
lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
paulson@15409
  1316
  by (simp add: setprod_def)
paulson@15409
  1317
nipkow@15402
  1318
lemma setprod_reindex:
nipkow@15402
  1319
     "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
ballarin@15765
  1320
by(auto simp: setprod_def AC_mult.fold_reindex dest!:finite_imageD)
nipkow@15402
  1321
nipkow@15402
  1322
lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
nipkow@15402
  1323
by (auto simp add: setprod_reindex)
nipkow@15402
  1324
nipkow@15402
  1325
lemma setprod_cong:
nipkow@15402
  1326
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
ballarin@15765
  1327
by(fastsimp simp: setprod_def intro: AC_mult.fold_cong)
nipkow@15402
  1328
berghofe@16632
  1329
lemma strong_setprod_cong:
berghofe@16632
  1330
  "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
berghofe@16632
  1331
by(fastsimp simp: simp_implies_def setprod_def intro: AC_mult.fold_cong)
berghofe@16632
  1332
nipkow@15402
  1333
lemma setprod_reindex_cong: "inj_on f A ==>
nipkow@15402
  1334
    B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
nipkow@15402
  1335
  by (frule setprod_reindex, simp)
nipkow@15402
  1336
nipkow@15402
  1337
nipkow@15402
  1338
lemma setprod_1: "setprod (%i. 1) A = 1"
nipkow@15402
  1339
  apply (case_tac "finite A")
nipkow@15402
  1340
  apply (erule finite_induct, auto simp add: mult_ac)
nipkow@15402
  1341
  done
nipkow@15402
  1342
nipkow@15402
  1343
lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
nipkow@15402
  1344
  apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
nipkow@15402
  1345
  apply (erule ssubst, rule setprod_1)
nipkow@15402
  1346
  apply (rule setprod_cong, auto)
nipkow@15402
  1347
  done
nipkow@15402
  1348
nipkow@15402
  1349
lemma setprod_Un_Int: "finite A ==> finite B
nipkow@15402
  1350
    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
ballarin@15765
  1351
by(simp add: setprod_def AC_mult.fold_Un_Int[symmetric])
nipkow@15402
  1352
nipkow@15402
  1353
lemma setprod_Un_disjoint: "finite A ==> finite B
nipkow@15402
  1354
  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
nipkow@15402
  1355
by (subst setprod_Un_Int [symmetric], auto)
nipkow@15402
  1356
nipkow@15402
  1357
lemma setprod_UN_disjoint:
nipkow@15402
  1358
    "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402
  1359
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402
  1360
      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
ballarin@15765
  1361
by(simp add: setprod_def AC_mult.fold_UN_disjoint cong: setprod_cong)
nipkow@15402
  1362
nipkow@15402
  1363
lemma setprod_Union_disjoint:
paulson@15409
  1364
  "[| (ALL A:C. finite A);
paulson@15409
  1365
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |] 
paulson@15409
  1366
   ==> setprod f (Union C) = setprod (setprod f) C"
paulson@15409
  1367
apply (cases "finite C") 
paulson@15409
  1368
 prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
nipkow@15402
  1369
  apply (frule setprod_UN_disjoint [of C id f])
paulson@15409
  1370
 apply (unfold Union_def id_def, assumption+)
paulson@15409
  1371
done
nipkow@15402
  1372
nipkow@15402
  1373
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@16550
  1374
    (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
paulson@17189
  1375
    (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
ballarin@15765
  1376
by(simp add:setprod_def AC_mult.fold_Sigma split_def cong:setprod_cong)
nipkow@15402
  1377
paulson@15409
  1378
text{*Here we can eliminate the finiteness assumptions, by cases.*}
paulson@15409
  1379
lemma setprod_cartesian_product: 
paulson@17189
  1380
     "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
paulson@15409
  1381
apply (cases "finite A") 
paulson@15409
  1382
 apply (cases "finite B") 
paulson@15409
  1383
  apply (simp add: setprod_Sigma)
paulson@15409
  1384
 apply (cases "A={}", simp)
paulson@15409
  1385
 apply (simp add: setprod_1) 
paulson@15409
  1386
apply (auto simp add: setprod_def
paulson@15409
  1387
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
paulson@15409
  1388
done
nipkow@15402
  1389
nipkow@15402
  1390
lemma setprod_timesf:
paulson@15409
  1391
     "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
ballarin@15765
  1392
by(simp add:setprod_def AC_mult.fold_distrib)
nipkow@15402
  1393
nipkow@15402
  1394
nipkow@15402
  1395
subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402
  1396
nipkow@15402
  1397
lemma setprod_eq_1_iff [simp]:
nipkow@15402
  1398
    "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
nipkow@15402
  1399
  by (induct set: Finites) auto
nipkow@15402
  1400
nipkow@15402
  1401
lemma setprod_zero:
nipkow@15402
  1402
     "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
nipkow@15402
  1403
  apply (induct set: Finites, force, clarsimp)
nipkow@15402
  1404
  apply (erule disjE, auto)
nipkow@15402
  1405
  done
nipkow@15402
  1406
nipkow@15402
  1407
lemma setprod_nonneg [rule_format]:
nipkow@15402
  1408
     "(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
nipkow@15402
  1409
  apply (case_tac "finite A")
nipkow@15402
  1410
  apply (induct set: Finites, force, clarsimp)
nipkow@15402
  1411
  apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
nipkow@15402
  1412
  apply (rule mult_mono, assumption+)
nipkow@15402
  1413
  apply (auto simp add: setprod_def)
nipkow@15402
  1414
  done
nipkow@15402
  1415
nipkow@15402
  1416
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
nipkow@15402
  1417
     --> 0 < setprod f A"
nipkow@15402
  1418
  apply (case_tac "finite A")
nipkow@15402
  1419
  apply (induct set: Finites, force, clarsimp)
nipkow@15402
  1420
  apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
nipkow@15402
  1421
  apply (rule mult_strict_mono, assumption+)
nipkow@15402
  1422
  apply (auto simp add: setprod_def)
nipkow@15402
  1423
  done
nipkow@15402
  1424
nipkow@15402
  1425
lemma setprod_nonzero [rule_format]:
nipkow@15402
  1426
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
nipkow@15402
  1427
      finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
nipkow@15402
  1428
  apply (erule finite_induct, auto)
nipkow@15402
  1429
  done
nipkow@15402
  1430
nipkow@15402
  1431
lemma setprod_zero_eq:
nipkow@15402
  1432
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
nipkow@15402
  1433
     finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
nipkow@15402
  1434
  apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
nipkow@15402
  1435
  done
nipkow@15402
  1436
nipkow@15402
  1437
lemma setprod_nonzero_field:
nipkow@15402
  1438
    "finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0"
nipkow@15402
  1439
  apply (rule setprod_nonzero, auto)
nipkow@15402
  1440
  done
nipkow@15402
  1441
nipkow@15402
  1442
lemma setprod_zero_eq_field:
nipkow@15402
  1443
    "finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)"
nipkow@15402
  1444
  apply (rule setprod_zero_eq, auto)
nipkow@15402
  1445
  done
nipkow@15402
  1446
nipkow@15402
  1447
lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
nipkow@15402
  1448
    (setprod f (A Un B) :: 'a ::{field})
nipkow@15402
  1449
      = setprod f A * setprod f B / setprod f (A Int B)"
nipkow@15402
  1450
  apply (subst setprod_Un_Int [symmetric], auto)
nipkow@15402
  1451
  apply (subgoal_tac "finite (A Int B)")
nipkow@15402
  1452
  apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
nipkow@15402
  1453
  apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self)
nipkow@15402
  1454
  done
nipkow@15402
  1455
nipkow@15402
  1456
lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
nipkow@15402
  1457
    (setprod f (A - {a}) :: 'a :: {field}) =
nipkow@15402
  1458
      (if a:A then setprod f A / f a else setprod f A)"
nipkow@15402
  1459
  apply (erule finite_induct)
nipkow@15402
  1460
   apply (auto simp add: insert_Diff_if)
nipkow@15402
  1461
  apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a")
nipkow@15402
  1462
  apply (erule ssubst)
nipkow@15402
  1463
  apply (subst times_divide_eq_right [THEN sym])
nipkow@15402
  1464
  apply (auto simp add: mult_ac times_divide_eq_right divide_self)
nipkow@15402
  1465
  done
nipkow@15402
  1466
nipkow@15402
  1467
lemma setprod_inversef: "finite A ==>
nipkow@15402
  1468
    ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
nipkow@15402
  1469
      setprod (inverse \<circ> f) A = inverse (setprod f A)"
nipkow@15402
  1470
  apply (erule finite_induct)
nipkow@15402
  1471
  apply (simp, simp)
nipkow@15402
  1472
  done
nipkow@15402
  1473
nipkow@15402
  1474
lemma setprod_dividef:
nipkow@15402
  1475
     "[|finite A;
nipkow@15402
  1476
        \<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
nipkow@15402
  1477
      ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
nipkow@15402
  1478
  apply (subgoal_tac
nipkow@15402
  1479
         "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
nipkow@15402
  1480
  apply (erule ssubst)
nipkow@15402
  1481
  apply (subst divide_inverse)
nipkow@15402
  1482
  apply (subst setprod_timesf)
nipkow@15402
  1483
  apply (subst setprod_inversef, assumption+, rule refl)
nipkow@15402
  1484
  apply (rule setprod_cong, rule refl)
nipkow@15402
  1485
  apply (subst divide_inverse, auto)
nipkow@15402
  1486
  done
nipkow@15402
  1487
wenzelm@12396
  1488
subsection {* Finite cardinality *}
wenzelm@12396
  1489
nipkow@15402
  1490
text {* This definition, although traditional, is ugly to work with:
nipkow@15402
  1491
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
nipkow@15402
  1492
But now that we have @{text setsum} things are easy:
wenzelm@12396
  1493
*}
wenzelm@12396
  1494
wenzelm@12396
  1495
constdefs
wenzelm@12396
  1496
  card :: "'a set => nat"
nipkow@15402
  1497
  "card A == setsum (%x. 1::nat) A"
wenzelm@12396
  1498
wenzelm@12396
  1499
lemma card_empty [simp]: "card {} = 0"
nipkow@15402
  1500
  by (simp add: card_def)
nipkow@15402
  1501
paulson@15409
  1502
lemma card_infinite [simp]: "~ finite A ==> card A = 0"
paulson@15409
  1503
  by (simp add: card_def)
paulson@15409
  1504
nipkow@15402
  1505
lemma card_eq_setsum: "card A = setsum (%x. 1) A"
nipkow@15402
  1506
by (simp add: card_def)
wenzelm@12396
  1507
wenzelm@12396
  1508
lemma card_insert_disjoint [simp]:
wenzelm@12396
  1509
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
ballarin@15765
  1510
by(simp add: card_def)
nipkow@15402
  1511
nipkow@15402
  1512
lemma card_insert_if:
nipkow@15402
  1513
    "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
nipkow@15402
  1514
  by (simp add: insert_absorb)
wenzelm@12396
  1515
wenzelm@12396
  1516
lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
wenzelm@12396
  1517
  apply auto
paulson@15506
  1518
  apply (drule_tac a = x in mk_disjoint_insert, clarify, auto)
wenzelm@12396
  1519
  done
wenzelm@12396
  1520
paulson@15409
  1521
lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
paulson@15409
  1522
by auto
paulson@15409
  1523
wenzelm@12396
  1524
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
nipkow@14302
  1525
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
nipkow@14302
  1526
apply(simp del:insert_Diff_single)
nipkow@14302
  1527
done
wenzelm@12396
  1528
wenzelm@12396
  1529
lemma card_Diff_singleton:
wenzelm@12396
  1530
    "finite A ==> x: A ==> card (A - {x}) = card A - 1"
wenzelm@12396
  1531
  by (simp add: card_Suc_Diff1 [symmetric])
wenzelm@12396
  1532
wenzelm@12396
  1533
lemma card_Diff_singleton_if:
wenzelm@12396
  1534
    "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
wenzelm@12396
  1535
  by (simp add: card_Diff_singleton)
wenzelm@12396
  1536
wenzelm@12396
  1537
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
wenzelm@12396
  1538
  by (simp add: card_insert_if card_Suc_Diff1)
wenzelm@12396
  1539
wenzelm@12396
  1540
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
wenzelm@12396
  1541
  by (simp add: card_insert_if)
wenzelm@12396
  1542
nipkow@15402
  1543
lemma card_mono: "\<lbrakk> finite B; A \<subseteq> B \<rbrakk> \<Longrightarrow> card A \<le> card B"
nipkow@15539
  1544
by (simp add: card_def setsum_mono2)
nipkow@15402
  1545
wenzelm@12396
  1546
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
paulson@14208
  1547
  apply (induct set: Finites, simp, clarify)
wenzelm@12396
  1548
  apply (subgoal_tac "finite A & A - {x} <= F")
paulson@14208
  1549
   prefer 2 apply (blast intro: finite_subset, atomize)
wenzelm@12396
  1550
  apply (drule_tac x = "A - {x}" in spec)
wenzelm@12396
  1551
  apply (simp add: card_Diff_singleton_if split add: split_if_asm)
paulson@14208
  1552
  apply (case_tac "card A", auto)
wenzelm@12396
  1553
  done
wenzelm@12396
  1554
wenzelm@12396
  1555
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
wenzelm@12396
  1556
  apply (simp add: psubset_def linorder_not_le [symmetric])
wenzelm@12396
  1557
  apply (blast dest: card_seteq)
wenzelm@12396
  1558
  done
wenzelm@12396
  1559
wenzelm@12396
  1560
lemma card_Un_Int: "finite A ==> finite B
wenzelm@12396
  1561
    ==> card A + card B = card (A Un B) + card (A Int B)"
nipkow@15402
  1562
by(simp add:card_def setsum_Un_Int)
wenzelm@12396
  1563
wenzelm@12396
  1564
lemma card_Un_disjoint: "finite A ==> finite B
wenzelm@12396
  1565
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
wenzelm@12396
  1566
  by (simp add: card_Un_Int)
wenzelm@12396
  1567
wenzelm@12396
  1568
lemma card_Diff_subset:
nipkow@15402
  1569
  "finite B ==> B <= A ==> card (A - B) = card A - card B"
nipkow@15402
  1570
by(simp add:card_def setsum_diff_nat)
wenzelm@12396
  1571
wenzelm@12396
  1572
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
wenzelm@12396
  1573
  apply (rule Suc_less_SucD)
wenzelm@12396
  1574
  apply (simp add: card_Suc_Diff1)
wenzelm@12396
  1575
  done
wenzelm@12396
  1576
wenzelm@12396
  1577
lemma card_Diff2_less:
wenzelm@12396
  1578
    "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
wenzelm@12396
  1579
  apply (case_tac "x = y")
wenzelm@12396
  1580
   apply (simp add: card_Diff1_less)
wenzelm@12396
  1581
  apply (rule less_trans)
wenzelm@12396
  1582
   prefer 2 apply (auto intro!: card_Diff1_less)
wenzelm@12396
  1583
  done
wenzelm@12396
  1584
wenzelm@12396
  1585
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
wenzelm@12396
  1586
  apply (case_tac "x : A")
wenzelm@12396
  1587
   apply (simp_all add: card_Diff1_less less_imp_le)
wenzelm@12396
  1588
  done
wenzelm@12396
  1589
wenzelm@12396
  1590
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
paulson@14208
  1591
by (erule psubsetI, blast)
wenzelm@12396
  1592
paulson@14889
  1593
lemma insert_partition:
nipkow@15402
  1594
  "\<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
  1595
  \<Longrightarrow> x \<inter> \<Union> F = {}"
paulson@14889
  1596
by auto
paulson@14889
  1597
paulson@14889
  1598
(* main cardinality theorem *)
paulson@14889
  1599
lemma card_partition [rule_format]:
paulson@14889
  1600
     "finite C ==>  
paulson@14889
  1601
        finite (\<Union> C) -->  
paulson@14889
  1602
        (\<forall>c\<in>C. card c = k) -->   
paulson@14889
  1603
        (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->  
paulson@14889
  1604
        k * card(C) = card (\<Union> C)"
paulson@14889
  1605
apply (erule finite_induct, simp)
paulson@14889
  1606
apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition 
paulson@14889
  1607
       finite_subset [of _ "\<Union> (insert x F)"])
paulson@14889
  1608
done
paulson@14889
  1609
wenzelm@12396
  1610
nipkow@15539
  1611
lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
nipkow@15539
  1612
apply (cases "finite A")
nipkow@15539
  1613
apply (erule finite_induct)
nipkow@15539
  1614
apply (auto simp add: ring_distrib add_ac)
paulson@15409
  1615
done
nipkow@15402
  1616
nipkow@15539
  1617
nipkow@16550
  1618
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::recpower)) = y^(card A)"
nipkow@15402
  1619
  apply (erule finite_induct)
nipkow@15402
  1620
  apply (auto simp add: power_Suc)
nipkow@15402
  1621
  done
nipkow@15402
  1622
nipkow@15542
  1623
lemma setsum_bounded:
nipkow@15542
  1624
  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{comm_semiring_1_cancel, pordered_ab_semigroup_add})"
nipkow@15542
  1625
  shows "setsum f A \<le> of_nat(card A) * K"
nipkow@15542
  1626
proof (cases "finite A")
nipkow@15542
  1627
  case True
nipkow@15542
  1628
  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
nipkow@15542
  1629
next
nipkow@15542
  1630
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15542
  1631
qed
nipkow@15542
  1632
nipkow@15402
  1633
nipkow@15402
  1634
subsubsection {* Cardinality of unions *}
nipkow@15402
  1635
nipkow@15539
  1636
lemma of_nat_id[simp]: "(of_nat n :: nat) = n"
nipkow@15539
  1637
by(induct n, auto)
nipkow@15539
  1638
nipkow@15402
  1639
lemma card_UN_disjoint:
nipkow@15402
  1640
    "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402
  1641
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402
  1642
      card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
nipkow@15539
  1643
  apply (simp add: card_def del: setsum_constant)
nipkow@15402
  1644
  apply (subgoal_tac
nipkow@15402
  1645
           "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
nipkow@15539
  1646
  apply (simp add: setsum_UN_disjoint del: setsum_constant)
nipkow@15539
  1647
  apply (simp cong: setsum_cong)
nipkow@15402
  1648
  done
nipkow@15402
  1649
nipkow@15402
  1650
lemma card_Union_disjoint:
nipkow@15402
  1651
  "finite C ==> (ALL A:C. finite A) ==>
nipkow@15402
  1652
        (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
nipkow@15402
  1653
      card (Union C) = setsum card C"
nipkow@15402
  1654
  apply (frule card_UN_disjoint [of C id])
nipkow@15402
  1655
  apply (unfold Union_def id_def, assumption+)
nipkow@15402
  1656
  done
nipkow@15402
  1657
wenzelm@12396
  1658
subsubsection {* Cardinality of image *}
wenzelm@12396
  1659
paulson@15447
  1660
text{*The image of a finite set can be expressed using @{term fold}.*}
paulson@15447
  1661
lemma image_eq_fold: "finite A ==> f ` A = fold (op Un) (%x. {f x}) {} A"
paulson@15447
  1662
  apply (erule finite_induct, simp)
paulson@15447
  1663
  apply (subst ACf.fold_insert) 
paulson@15447
  1664
  apply (auto simp add: ACf_def) 
paulson@15447
  1665
  done
paulson@15447
  1666
wenzelm@12396
  1667
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
paulson@14208
  1668
  apply (induct set: Finites, simp)
wenzelm@12396
  1669
  apply (simp add: le_SucI finite_imageI card_insert_if)
wenzelm@12396
  1670
  done
wenzelm@12396
  1671
nipkow@15402
  1672
lemma card_image: "inj_on f A ==> card (f ` A) = card A"
nipkow@15539
  1673
by(simp add:card_def setsum_reindex o_def del:setsum_constant)
wenzelm@12396
  1674
wenzelm@12396
  1675
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
wenzelm@12396
  1676
  by (simp add: card_seteq card_image)
wenzelm@12396
  1677
nipkow@15111
  1678
lemma eq_card_imp_inj_on:
nipkow@15111
  1679
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
paulson@15506
  1680
apply (induct rule:finite_induct, simp)
nipkow@15111
  1681
apply(frule card_image_le[where f = f])
nipkow@15111
  1682
apply(simp add:card_insert_if split:if_splits)
nipkow@15111
  1683
done
nipkow@15111
  1684
nipkow@15111
  1685
lemma inj_on_iff_eq_card:
nipkow@15111
  1686
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
nipkow@15111
  1687
by(blast intro: card_image eq_card_imp_inj_on)
nipkow@15111
  1688
wenzelm@12396
  1689
nipkow@15402
  1690
lemma card_inj_on_le:
nipkow@15402
  1691
    "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
nipkow@15402
  1692
apply (subgoal_tac "finite A") 
nipkow@15402
  1693
 apply (force intro: card_mono simp add: card_image [symmetric])
nipkow@15402
  1694
apply (blast intro: finite_imageD dest: finite_subset) 
nipkow@15402
  1695
done
nipkow@15402
  1696
nipkow@15402
  1697
lemma card_bij_eq:
nipkow@15402
  1698
    "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
nipkow@15402
  1699
       finite A; finite B |] ==> card A = card B"
nipkow@15402
  1700
  by (auto intro: le_anti_sym card_inj_on_le)
nipkow@15402
  1701
nipkow@15402
  1702
nipkow@15402
  1703
subsubsection {* Cardinality of products *}
nipkow@15402
  1704
nipkow@15402
  1705
(*
nipkow@15402
  1706
lemma SigmaI_insert: "y \<notin> A ==>
nipkow@15402
  1707
  (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
nipkow@15402
  1708
  by auto
nipkow@15402
  1709
*)
nipkow@15402
  1710
nipkow@15402
  1711
lemma card_SigmaI [simp]:
nipkow@15402
  1712
  "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
nipkow@15402
  1713
  \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
nipkow@15539
  1714
by(simp add:card_def setsum_Sigma del:setsum_constant)
nipkow@15402
  1715
paulson@15409
  1716
lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
paulson@15409
  1717
apply (cases "finite A") 
paulson@15409
  1718
apply (cases "finite B") 
paulson@15409
  1719
apply (auto simp add: card_eq_0_iff
nipkow@15539
  1720
            dest: finite_cartesian_productD1 finite_cartesian_productD2)
paulson@15409
  1721
done
nipkow@15402
  1722
nipkow@15402
  1723
lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
nipkow@15539
  1724
by (simp add: card_cartesian_product)
paulson@15409
  1725
nipkow@15402
  1726
nipkow@15402
  1727
wenzelm@12396
  1728
subsubsection {* Cardinality of the Powerset *}
wenzelm@12396
  1729
wenzelm@12396
  1730
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
wenzelm@12396
  1731
  apply (induct set: Finites)
wenzelm@12396
  1732
   apply (simp_all add: Pow_insert)
paulson@14208
  1733
  apply (subst card_Un_disjoint, blast)
paulson@14208
  1734
    apply (blast intro: finite_imageI, blast)
wenzelm@12396
  1735
  apply (subgoal_tac "inj_on (insert x) (Pow F)")
wenzelm@12396
  1736
   apply (simp add: card_image Pow_insert)
wenzelm@12396
  1737
  apply (unfold inj_on_def)
wenzelm@12396
  1738
  apply (blast elim!: equalityE)
wenzelm@12396
  1739
  done
wenzelm@12396
  1740
nipkow@15392
  1741
text {* Relates to equivalence classes.  Based on a theorem of
nipkow@15392
  1742
F. Kammüller's.  *}
wenzelm@12396
  1743
wenzelm@12396
  1744
lemma dvd_partition:
nipkow@15392
  1745
  "finite (Union C) ==>
wenzelm@12396
  1746
    ALL c : C. k dvd card c ==>
paulson@14430
  1747
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
wenzelm@12396
  1748
  k dvd card (Union C)"
nipkow@15392
  1749
apply(frule finite_UnionD)
nipkow@15392
  1750
apply(rotate_tac -1)
paulson@14208
  1751
  apply (induct set: Finites, simp_all, clarify)
wenzelm@12396
  1752
  apply (subst card_Un_disjoint)
wenzelm@12396
  1753
  apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
wenzelm@12396
  1754
  done
wenzelm@12396
  1755
wenzelm@12396
  1756
nipkow@15392
  1757
subsection{* A fold functional for non-empty sets *}
nipkow@15392
  1758
nipkow@15392
  1759
text{* Does not require start value. *}
wenzelm@12396
  1760
nipkow@15392
  1761
consts
paulson@15506
  1762
  fold1Set :: "('a => 'a => 'a) => ('a set \<times> 'a) set"
nipkow@15392
  1763
paulson@15506
  1764
inductive "fold1Set f"
nipkow@15392
  1765
intros
paulson@15506
  1766
  fold1Set_insertI [intro]:
paulson@15506
  1767
   "\<lbrakk> (A,x) \<in> foldSet f id a; a \<notin> A \<rbrakk> \<Longrightarrow> (insert a A, x) \<in> fold1Set f"
wenzelm@12396
  1768
nipkow@15392
  1769
constdefs
nipkow@15392
  1770
  fold1 :: "('a => 'a => 'a) => 'a set => 'a"
paulson@15506
  1771
  "fold1 f A == THE x. (A, x) : fold1Set f"
paulson@15506
  1772
paulson@15506
  1773
lemma fold1Set_nonempty:
paulson@15506
  1774
 "(A, x) : fold1Set f \<Longrightarrow> A \<noteq> {}"
paulson@15506
  1775
by(erule fold1Set.cases, simp_all) 
paulson@15506
  1776
nipkow@15392
  1777
paulson@15506
  1778
inductive_cases empty_fold1SetE [elim!]: "({}, x) : fold1Set f"
paulson@15506
  1779
paulson@15506
  1780
inductive_cases insert_fold1SetE [elim!]: "(insert a X, x) : fold1Set f"
paulson@15506
  1781
paulson@15506
  1782
paulson@15506
  1783
lemma fold1Set_sing [iff]: "(({a},b) : fold1Set f) = (a = b)"
paulson@15506
  1784
  by (blast intro: foldSet.intros elim: foldSet.cases)
nipkow@15392
  1785
paulson@15508
  1786
lemma fold1_singleton[simp]: "fold1 f {a} = a"
paulson@15508
  1787
  by (unfold fold1_def) blast
wenzelm@12396
  1788
paulson@15508
  1789
lemma finite_nonempty_imp_fold1Set:
paulson@15508
  1790
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. (A, x) : fold1Set f"
paulson@15508
  1791
apply (induct A rule: finite_induct)
paulson@15508
  1792
apply (auto dest: finite_imp_foldSet [of _ f id])  
paulson@15508
  1793
done
paulson@15506
  1794
paulson@15506
  1795
text{*First, some lemmas about @{term foldSet}.*}
nipkow@15392
  1796
paulson@15508
  1797
lemma (in ACf) foldSet_insert_swap:
paulson@15508
  1798
assumes fold: "(A,y) \<in> foldSet f id b"
paulson@15521
  1799
shows "b \<notin> A \<Longrightarrow> (insert b A, z \<cdot> y) \<in> foldSet f id z"
paulson@15508
  1800
using fold
paulson@15508
  1801
proof (induct rule: foldSet.induct)
paulson@15508
  1802
  case emptyI thus ?case by (force simp add: fold_insert_aux commute)
paulson@15508
  1803
next
paulson@15508
  1804
  case (insertI A x y)
paulson@15508
  1805
    have "(insert x (insert b A), x \<cdot> (z \<cdot> y)) \<in> foldSet f (\<lambda>u. u) z"
paulson@15521
  1806
      using insertI by force  --{*how does @{term id} get unfolded?*}
paulson@15508
  1807
    thus ?case by (simp add: insert_commute AC)
paulson@15508
  1808
qed
paulson@15508
  1809
paulson@15508
  1810
lemma (in ACf) foldSet_permute_diff:
paulson@15508
  1811
assumes fold: "(A,x) \<in> foldSet f id b"
paulson@15508
  1812
shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> (insert b (A-{a}), x) \<in> foldSet f id a"
paulson@15508
  1813
using fold
paulson@15508
  1814
proof (induct rule: foldSet.induct)
paulson@15508
  1815
  case emptyI thus ?case by simp
paulson@15508
  1816
next
paulson@15508
  1817
  case (insertI A x y)
paulson@15521
  1818
  have "a = x \<or> a \<in> A" using insertI by simp
paulson@15521
  1819
  thus ?case
paulson@15521
  1820
  proof
paulson@15521
  1821
    assume "a = x"
paulson@15521
  1822
    with insertI show ?thesis
paulson@15521
  1823
      by (simp add: id_def [symmetric], blast intro: foldSet_insert_swap) 
paulson@15521
  1824
  next
paulson@15521
  1825
    assume ainA: "a \<in> A"
paulson@15521
  1826
    hence "(insert x (insert b (A - {a})), x \<cdot> y) \<in> foldSet f id a"
paulson@15521
  1827
      using insertI by (force simp: id_def)
paulson@15521
  1828
    moreover
paulson@15521
  1829
    have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
paulson@15521
  1830
      using ainA insertI by blast
paulson@15521
  1831
    ultimately show ?thesis by (simp add: id_def)
paulson@15508
  1832
  qed
paulson@15508
  1833
qed
paulson@15508
  1834
paulson@15508
  1835
lemma (in ACf) fold1_eq_fold:
paulson@15508
  1836
     "[|finite A; a \<notin> A|] ==> fold1 f (insert a A) = fold f id a A"
paulson@15508
  1837
apply (simp add: fold1_def fold_def) 
paulson@15508
  1838
apply (rule the_equality)
paulson@15508
  1839
apply (best intro: foldSet_determ theI dest: finite_imp_foldSet [of _ f id]) 
paulson@15508
  1840
apply (rule sym, clarify)
paulson@15508
  1841
apply (case_tac "Aa=A")
paulson@15508
  1842
 apply (best intro: the_equality foldSet_determ)  
paulson@15508
  1843
apply (subgoal_tac "(A,x) \<in> foldSet f id a") 
paulson@15508
  1844
 apply (best intro: the_equality foldSet_determ)  
paulson@15508
  1845
apply (subgoal_tac "insert aa (Aa - {a}) = A") 
paulson@15508
  1846
 prefer 2 apply (blast elim: equalityE) 
paulson@15508
  1847
apply (auto dest: foldSet_permute_diff [where a=a]) 
paulson@15508
  1848
done
paulson@15508
  1849
paulson@15521
  1850
lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
paulson@15521
  1851
apply safe
paulson@15521
  1852
apply simp 
paulson@15521
  1853
apply (drule_tac x=x in spec)
paulson@15521
  1854
apply (drule_tac x="A-{x}" in spec, auto) 
paulson@15508
  1855
done
paulson@15508
  1856
paulson@15521
  1857
lemma (in ACf) fold1_insert:
paulson@15521
  1858
  assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
paulson@15521
  1859
  shows "fold1 f (insert x A) = f x (fold1 f A)"
paulson@15521
  1860
proof -
paulson@15521
  1861
  from nonempty obtain a A' where "A = insert a A' & a ~: A'" 
paulson@15521
  1862
    by (auto simp add: nonempty_iff)
paulson@15521
  1863
  with A show ?thesis
paulson@15521
  1864
    by (simp add: insert_commute [of x] fold1_eq_fold eq_commute) 
paulson@15521
  1865
qed
paulson@15521
  1866
paulson@15509
  1867
lemma (in ACIf) fold1_insert_idem [simp]:
paulson@15521
  1868
  assumes nonempty: "A \<noteq> {}" and A: "finite A" 
paulson@15521
  1869
  shows "fold1 f (insert x A) = f x (fold1 f A)"
paulson@15521
  1870
proof -
paulson@15521
  1871
  from nonempty obtain a A' where A': "A = insert a A' & a ~: A'" 
paulson@15521
  1872
    by (auto simp add: nonempty_iff)
paulson@15521
  1873
  show ?thesis
paulson@15521
  1874
  proof cases
paulson@15521
  1875
    assume "a = x"
paulson@15521
  1876
    thus ?thesis 
paulson@15521
  1877
    proof cases
paulson@15521
  1878
      assume "A' = {}"
paulson@15521
  1879
      with prems show ?thesis by (simp add: idem) 
paulson@15521
  1880
    next
paulson@15521
  1881
      assume "A' \<noteq> {}"
paulson@15521
  1882
      with prems show ?thesis
paulson@15521
  1883
	by (simp add: fold1_insert assoc [symmetric] idem) 
paulson@15521
  1884
    qed
paulson@15521
  1885
  next
paulson@15521
  1886
    assume "a \<noteq> x"
paulson@15521
  1887
    with prems show ?thesis
paulson@15521
  1888
      by (simp add: insert_commute fold1_eq_fold fold_insert_idem)
paulson@15521
  1889
  qed
paulson@15521
  1890
qed
paulson@15506
  1891
paulson@15506
  1892
paulson@15508
  1893
text{* Now the recursion rules for definitions: *}
paulson@15508
  1894
paulson@15508
  1895
lemma fold1_singleton_def: "g \<equiv> fold1 f \<Longrightarrow> g {a} = a"
paulson@15508
  1896
by(simp add:fold1_singleton)
paulson@15508
  1897
paulson@15508
  1898
lemma (in ACf) fold1_insert_def:
paulson@15508
  1899
  "\<lbrakk> g \<equiv> fold1 f; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
paulson@15508
  1900
by(simp add:fold1_insert)
paulson@15508
  1901
paulson@15509
  1902
lemma (in ACIf) fold1_insert_idem_def:
paulson@15508
  1903
  "\<lbrakk> g \<equiv> fold1 f; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
paulson@15509
  1904
by(simp add:fold1_insert_idem)
paulson@15508
  1905
paulson@15508
  1906
subsubsection{* Determinacy for @{term fold1Set} *}
paulson@15508
  1907
paulson@15508
  1908
text{*Not actually used!!*}
wenzelm@12396
  1909
paulson@15506
  1910
lemma (in ACf) foldSet_permute:
paulson@15506
  1911
  "[|(insert a A, x) \<in> foldSet f id b; a \<notin> A; b \<notin> A|]
paulson@15506
  1912
   ==> (insert b A, x) \<in> foldSet f id a"
paulson@15506
  1913
apply (case_tac "a=b") 
paulson@15506
  1914
apply (auto dest: foldSet_permute_diff) 
paulson@15506
  1915
done
nipkow@15376
  1916
paulson@15506
  1917
lemma (in ACf) fold1Set_determ:
paulson@15506
  1918
  "(A, x) \<in> fold1Set f ==> (A, y) \<in> fold1Set f ==> y = x"
paulson@15506
  1919
proof (clarify elim!: fold1Set.cases)
paulson@15506
  1920
  fix A x B y a b
paulson@15506
  1921
  assume Ax: "(A, x) \<in> foldSet f id a"
paulson@15506
  1922
  assume By: "(B, y) \<in> foldSet f id b"
paulson@15506
  1923
  assume anotA:  "a \<notin> A"
paulson@15506
  1924
  assume bnotB:  "b \<notin> B"
paulson@15506
  1925
  assume eq: "insert a A = insert b B"
paulson@15506
  1926
  show "y=x"
paulson@15506
  1927
  proof cases
paulson@15506
  1928
    assume same: "a=b"
paulson@15506
  1929
    hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
paulson@15506
  1930
    thus ?thesis using Ax By same by (blast intro: foldSet_determ)
nipkow@15392
  1931
  next
paulson@15506
  1932
    assume diff: "a\<noteq>b"
paulson@15506
  1933
    let ?D = "B - {a}"
paulson@15506
  1934
    have B: "B = insert a ?D" and A: "A = insert b ?D"
paulson@15506
  1935
     and aB: "a \<in> B" and bA: "b \<in> A"
paulson@15506
  1936
      using eq anotA bnotB diff by (blast elim!:equalityE)+
paulson@15506
  1937
    with aB bnotB By
paulson@15506
  1938
    have "(insert b ?D, y) \<in> foldSet f id a" 
paulson@15506
  1939
      by (auto intro: foldSet_permute simp add: insert_absorb)
paulson@15506
  1940
    moreover
paulson@15506
  1941
    have "(insert b ?D, x) \<in> foldSet f id a"
paulson@15506
  1942
      by (simp add: A [symmetric] Ax) 
paulson@15506
  1943
    ultimately show ?thesis by (blast intro: foldSet_determ) 
nipkow@15392
  1944
  qed
wenzelm@12396
  1945
qed
wenzelm@12396
  1946
paulson@15506
  1947
lemma (in ACf) fold1Set_equality: "(A, y) : fold1Set f ==> fold1 f A = y"
paulson@15506
  1948
  by (unfold fold1_def) (blast intro: fold1Set_determ)
paulson@15506
  1949
paulson@15506
  1950
declare
paulson@15506
  1951
  empty_foldSetE [rule del]   foldSet.intros [rule del]
paulson@15506
  1952
  empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
paulson@15506
  1953
  -- {* No more proves involve these relations. *}
nipkow@15376
  1954
nipkow@15497
  1955
subsubsection{* Semi-Lattices *}
nipkow@15497
  1956
nipkow@15497
  1957
locale ACIfSL = ACIf +
nipkow@15500
  1958
  fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubseteq>" 50)
nipkow@18493
  1959
  and strict_below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubset>" 50)
nipkow@15500
  1960
  assumes below_def: "(x \<sqsubseteq> y) = (x\<cdot>y = x)"
nipkow@18493
  1961
  defines strict_below_def:  "(x \<sqsubset> y) \<equiv> (x \<sqsubseteq> y \<and> x \<noteq> y)"
nipkow@15497
  1962
nipkow@15497
  1963
locale ACIfSLlin = ACIfSL +
nipkow@15497
  1964
  assumes lin: "x\<cdot>y \<in> {x,y}"
nipkow@15497
  1965
nipkow@15500
  1966
lemma (in ACIfSL) below_refl[simp]: "x \<sqsubseteq> x"
nipkow@15497
  1967
by(simp add: below_def idem)
nipkow@15497
  1968
nipkow@15500
  1969
lemma (in ACIfSL) below_f_conv[simp]: "x \<sqsubseteq> y \<cdot> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
nipkow@15497
  1970
proof
nipkow@15500
  1971
  assume "x \<sqsubseteq> y \<cdot> z"
nipkow@15497
  1972
  hence xyzx: "x \<cdot> (y \<cdot> z) = x"  by(simp add: below_def)
nipkow@15497
  1973
  have "x \<cdot> y = x"
nipkow@15497
  1974
  proof -
nipkow@15497
  1975
    have "x \<cdot> y = (x \<cdot> (y \<cdot> z)) \<cdot> y" by(rule subst[OF xyzx], rule refl)
nipkow@15497
  1976
    also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
nipkow@15497
  1977
    also have "\<dots> = x" by(rule xyzx)
nipkow@15497
  1978
    finally show ?thesis .
nipkow@15497
  1979
  qed
nipkow@15497
  1980
  moreover have "x \<cdot> z = x"
nipkow@15497
  1981
  proof -
nipkow@15497
  1982
    have "x \<cdot> z = (x \<cdot> (y \<cdot> z)) \<cdot> z" by(rule subst[OF xyzx], rule refl)
nipkow@15497
  1983
    also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
nipkow@15497
  1984
    also have "\<dots> = x" by(rule xyzx)
nipkow@15497
  1985
    finally show ?thesis .
nipkow@15497
  1986
  qed
nipkow@15500
  1987
  ultimately show "x \<sqsubseteq> y \<and> x \<sqsubseteq> z" by(simp add: below_def)
nipkow@15497
  1988
next
nipkow@15500
  1989
  assume a: "x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
nipkow@15497
  1990
  hence y: "x \<cdot> y = x" and z: "x \<cdot> z = x" by(simp_all add: below_def)
nipkow@15497
  1991
  have "x \<cdot> (y \<cdot> z) = (x \<cdot> y) \<cdot> z" by(simp add:assoc)
nipkow@15497
  1992
  also have "x \<cdot> y = x" using a by(simp_all add: below_def)
nipkow@15497
  1993
  also have "x \<cdot> z = x" using a by(simp_all add: below_def)
nipkow@15500
  1994
  finally show "x \<sqsubseteq> y \<cdot> z" by(simp_all add: below_def)
nipkow@15497
  1995
qed
nipkow@15497
  1996
nipkow@15497
  1997
lemma (in ACIfSLlin) above_f_conv:
nipkow@15500
  1998
 "x \<cdot> y \<sqsubseteq> z = (x \<sqsubseteq> z \<or> y \<sqsubseteq> z)"
nipkow@15497
  1999
proof
nipkow@15500
  2000
  assume a: "x \<cdot> y \<sqsubseteq> z"
nipkow@15497
  2001
  have "x \<cdot> y = x \<or> x \<cdot> y = y" using lin[of x y] by simp
nipkow@15500
  2002
  thus "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
nipkow@15497
  2003
  proof
nipkow@15500
  2004
    assume "x \<cdot> y = x" hence "x \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
nipkow@15497
  2005
  next
nipkow@15500
  2006
    assume "x \<cdot> y = y" hence "y \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
nipkow@15497
  2007
  qed
nipkow@15497
  2008
next
nipkow@15500
  2009
  assume "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
nipkow@15500
  2010
  thus "x \<cdot> y \<sqsubseteq> z"
nipkow@15497
  2011
  proof
nipkow@15500
  2012
    assume a: "x \<sqsubseteq> z"
nipkow@15497
  2013
    have "(x \<cdot> y) \<cdot> z = (x \<cdot> z) \<cdot> y" by(simp add:ACI)
nipkow@15497
  2014
    also have "x \<cdot> z = x" using a by(simp add:below_def)
nipkow@15500
  2015
    finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
nipkow@15497
  2016
  next
nipkow@15500
  2017
    assume a: "y \<sqsubseteq> z"
nipkow@15497
  2018
    have "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
nipkow@15497
  2019
    also have "y \<cdot> z = y" using a by(simp add:below_def)
nipkow@15500
  2020
    finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
nipkow@15497
  2021
  qed
nipkow@15497
  2022
qed
nipkow@15497
  2023
nipkow@15497
  2024
nipkow@18493
  2025
lemma (in ACIfSLlin) strict_below_f_conv[simp]: "x \<sqsubset> y \<cdot> z = (x \<sqsubset> y \<and> x \<sqsubset> z)"
nipkow@18493
  2026
apply(simp add: strict_below_def)
nipkow@18493
  2027
using lin[of y z] by (auto simp:below_def ACI)
nipkow@18493
  2028
nipkow@18493
  2029
nipkow@18493
  2030
lemma (in ACIfSLlin) strict_above_f_conv:
nipkow@18493
  2031
 "x \<cdot> y \<sqsubset> z = (x \<sqsubset> z \<or> y \<sqsubset> z)"
nipkow@18493
  2032
apply(simp add: strict_below_def above_f_conv)
nipkow@18493
  2033
using lin[of y z] lin[of x z] by (auto simp:below_def ACI)
nipkow@18493
  2034
nipkow@18493
  2035
nipkow@15502
  2036
subsubsection{* Lemmas about @{text fold1} *}
nipkow@15484
  2037
nipkow@15484
  2038
lemma (in ACf) fold1_Un:
nipkow@15484
  2039
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  2040
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
nipkow@15484
  2041
       fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
nipkow@15484
  2042
using A
nipkow@15484
  2043
proof(induct rule:finite_ne_induct)
nipkow@15484
  2044
  case singleton thus ?case by(simp add:fold1_insert)
nipkow@15484
  2045
next
nipkow@15484
  2046
  case insert thus ?case by (simp add:fold1_insert assoc)
nipkow@15484
  2047
qed
nipkow@15484
  2048
nipkow@15484
  2049
lemma (in ACIf) fold1_Un2:
nipkow@15484
  2050
assumes A: "finite A" "A \<noteq> {}"
nipkow@15484
  2051
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
nipkow@15484
  2052
       fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
nipkow@15484
  2053
using A
nipkow@15484
  2054
proof(induct rule:finite_ne_induct)
paulson@15509
  2055
  case singleton thus ?case by(simp add:fold1_insert_idem)
nipkow@15484
  2056
next
paulson@15509
  2057
  case insert thus ?case by (simp add:fold1_insert_idem assoc)
nipkow@15484
  2058
qed
nipkow@15484
  2059
nipkow@15484
  2060
lemma (in ACf) fold1_in:
nipkow@15484
  2061
  assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x\<cdot>y \<in> {x,y}"
nipkow@15484
  2062
  shows "fold1 f A \<in> A"
nipkow@15484
  2063
using A
nipkow@15484
  2064
proof (induct rule:finite_ne_induct)
paulson@15506
  2065
  case singleton thus ?case by simp
nipkow@15484
  2066
next
nipkow@15484
  2067
  case insert thus ?case using elem by (force simp add:fold1_insert)
nipkow@15484
  2068
qed
nipkow@15484
  2069
nipkow@15497
  2070
lemma (in ACIfSL) below_fold1_iff:
nipkow@15497
  2071
assumes A: "finite A" "A \<noteq> {}"
nipkow@15500
  2072
shows "x \<sqsubseteq> fold1 f A = (\<forall>a\<in>A. x \<sqsubseteq> a)"
nipkow@15497
  2073
using A
nipkow@15497
  2074
by(induct rule:finite_ne_induct) simp_all
nipkow@15497
  2075
nipkow@18493
  2076
lemma (in ACIfSLlin) strict_below_fold1_iff:
nipkow@18493
  2077
  "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> x \<sqsubset> fold1 f A = (\<forall>a\<in>A. x \<sqsubset> a)"
nipkow@18493
  2078
by(induct rule:finite_ne_induct) simp_all
nipkow@18493
  2079
nipkow@18493
  2080
nipkow@15497
  2081
lemma (in ACIfSL) fold1_belowI:
nipkow@15497
  2082
assumes A: "finite A" "A \<noteq> {}"
nipkow@15500
  2083
shows "a \<in> A \<Longrightarrow> fold1 f A \<sqsubseteq> a"
nipkow@15484
  2084
using A
nipkow@15484
  2085
proof (induct rule:finite_ne_induct)
nipkow@15497
  2086
  case singleton thus ?case by simp
nipkow@15484
  2087
next
nipkow@15497
  2088
  case (insert x F)
berghofe@15517
  2089
  from insert(5) have "a = x \<or> a \<in> F" by simp
nipkow@15497
  2090
  thus ?case
nipkow@15497
  2091
  proof
nipkow@15497
  2092
    assume "a = x" thus ?thesis using insert by(simp add:below_def ACI)
nipkow@15497
  2093
  next
nipkow@15497
  2094
    assume "a \<in> F"
paulson@15508
  2095
    hence bel: "fold1 f F \<sqsubseteq> a" by(rule insert)
paulson@15508
  2096
    have "fold1 f (insert x F) \<cdot> a = x \<cdot> (fold1 f F \<cdot> a)"
nipkow@15497
  2097
      using insert by(simp add:below_def ACI)
paulson@15508
  2098
    also have "fold1 f F \<cdot> a = fold1 f F"
nipkow@15497
  2099
      using bel  by(simp add:below_def ACI)
paulson@15508
  2100
    also have "x \<cdot> \<dots> = fold1 f (insert x F)"
nipkow@15497
  2101
      using insert by(simp add:below_def ACI)
nipkow@15497
  2102
    finally show ?thesis  by(simp add:below_def)
nipkow@15497
  2103
  qed
nipkow@15484
  2104
qed
nipkow@15484
  2105
nipkow@18493
  2106
nipkow@15497
  2107
lemma (in ACIfSLlin) fold1_below_iff:
nipkow@15497
  2108
assumes A: "finite A" "A \<noteq> {}"
nipkow@15500
  2109
shows "fold1 f A \<sqsubseteq> x = (\<exists>a\<in>A. a \<sqsubseteq> x)"
nipkow@15484
  2110
using A
nipkow@15497
  2111
by(induct rule:finite_ne_induct)(simp_all add:above_f_conv)
nipkow@15484
  2112
nipkow@18493
  2113
lemma (in ACIfSLlin) fold1_strict_below_iff:
nipkow@18493
  2114
assumes A: "finite A" "A \<noteq> {}"
nipkow@18493
  2115
shows "fold1 f A \<sqsubset> x = (\<exists>a\<in>A. a \<sqsubset> x)"
nipkow@18493
  2116
using A
nipkow@18493
  2117
by(induct rule:finite_ne_induct)(simp_all add:strict_above_f_conv)
nipkow@18493
  2118
nipkow@15512
  2119
nipkow@18423
  2120
lemma (in ACIfSLlin) fold1_antimono:
nipkow@18423
  2121
assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B"
nipkow@18423
  2122
shows "fold1 f B \<sqsubseteq> fold1 f A"
nipkow@18423
  2123
proof(cases)
nipkow@18423
  2124
  assume "A = B" thus ?thesis by simp
nipkow@18423
  2125
next
nipkow@18423
  2126
  assume "A \<noteq> B"
nipkow@18423
  2127
  have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast
nipkow@18423
  2128
  have "fold1 f B = fold1 f (A \<union> (B-A))" by(subst B)(rule refl)
nipkow@18423
  2129
  also have "\<dots> = f (fold1 f A) (fold1 f (B-A))"
nipkow@18423
  2130
  proof -
nipkow@18423
  2131
    have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])
nipkow@18493
  2132
    moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`]) (* by(blast intro:finite_Diff prems) fails *)
nipkow@18423
  2133
    moreover have "(B-A) \<noteq> {}" using prems by blast
nipkow@18423
  2134
    moreover have "A Int (B-A) = {}" using prems by blast
nipkow@18423
  2135
    ultimately show ?thesis using `A \<noteq> {}` by(rule_tac fold1_Un)
nipkow@18423
  2136
  qed
nipkow@18423
  2137
  also have "\<dots> \<sqsubseteq> fold1 f A" by(simp add: above_f_conv)
nipkow@18423
  2138
  finally show ?thesis .
nipkow@18423
  2139
qed
nipkow@18423
  2140
nipkow@18423
  2141
nipkow@18493
  2142
nipkow@15500
  2143
subsubsection{* Lattices *}
nipkow@15500
  2144
nipkow@15512
  2145
locale Lattice = lattice +
nipkow@15512
  2146
  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
nipkow@15500
  2147
  and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
nipkow@15500
  2148
  defines "Inf == fold1 inf"  and "Sup == fold1 sup"
nipkow@15500
  2149
nipkow@15512
  2150
locale Distrib_Lattice = distrib_lattice + Lattice
nipkow@15504
  2151
nipkow@15500
  2152
text{* Lattices are semilattices *}
nipkow@15500
  2153
nipkow@15500
  2154
lemma (in Lattice) ACf_inf: "ACf inf"
nipkow@15512
  2155
by(blast intro: ACf.intro inf_commute inf_assoc)
nipkow@15500
  2156
nipkow@15500
  2157
lemma (in Lattice) ACf_sup: "ACf sup"
nipkow@15512
  2158
by(blast intro: ACf.intro sup_commute sup_assoc)
nipkow@15500
  2159
nipkow@15500
  2160
lemma (in Lattice) ACIf_inf: "ACIf inf"
nipkow@15500
  2161
apply(rule ACIf.intro)
nipkow@15500
  2162
apply(rule ACf_inf)
nipkow@15500
  2163
apply(rule ACIf_axioms.intro)
nipkow@15500
  2164
apply(rule inf_idem)
nipkow@15500
  2165
done
nipkow@15500
  2166
nipkow@15500
  2167
lemma (in Lattice) ACIf_sup: "ACIf sup"
nipkow@15500
  2168
apply(rule ACIf.intro)
nipkow@15500
  2169
apply(rule ACf_sup)
nipkow@15500
  2170
apply(rule ACIf_axioms.intro)
nipkow@15500
  2171
apply(rule sup_idem)
nipkow@15500
  2172
done
nipkow@15500
  2173
nipkow@15500
  2174
lemma (in Lattice) ACIfSL_inf: "ACIfSL inf (op \<sqsubseteq>)"
nipkow@15500
  2175
apply(rule ACIfSL.intro)
nipkow@15500
  2176
apply(rule ACf_inf)
nipkow@15500
  2177
apply(rule ACIf.axioms[OF ACIf_inf])
nipkow@15500
  2178
apply(rule ACIfSL_axioms.intro)
nipkow@15500
  2179
apply(rule iffI)
nipkow@15500
  2180
 apply(blast intro: antisym inf_le1 inf_le2 inf_least refl)
nipkow@15500
  2181
apply(erule subst)
nipkow@15500
  2182
apply(rule inf_le2)
nipkow@15500
  2183
done
nipkow@15500
  2184
nipkow@15500
  2185
lemma (in Lattice) ACIfSL_sup: "ACIfSL sup (%x y. y \<sqsubseteq> x)"
nipkow@15500
  2186
apply(rule ACIfSL.intro)
nipkow@15500
  2187
apply(rule ACf_sup)
nipkow@15500
  2188
apply(rule ACIf.axioms[OF ACIf_sup])
nipkow@15500
  2189
apply(rule ACIfSL_axioms.intro)
nipkow@15500
  2190
apply(rule iffI)
nipkow@15500
  2191
 apply(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl)
nipkow@15500
  2192
apply(erule subst)
nipkow@15500
  2193
apply(rule sup_ge2)
nipkow@15500
  2194
done
nipkow@15500
  2195
nipkow@15505
  2196
nipkow@15505
  2197
subsubsection{* Fold laws in lattices *}
nipkow@15500
  2198
nipkow@15780
  2199
lemma (in Lattice) Inf_le_Sup[simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Squnion>A"
nipkow@15500
  2200
apply(unfold Sup_def Inf_def)
nipkow@15500
  2201
apply(subgoal_tac "EX a. a:A")
nipkow@15500
  2202
prefer 2 apply blast
nipkow@15500
  2203
apply(erule exE)
nipkow@15500
  2204
apply(rule trans)
nipkow@15500
  2205
apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_inf])
nipkow@15500
  2206
apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_sup])
nipkow@15500
  2207
done
nipkow@15500
  2208
nipkow@15780
  2209
lemma (in Lattice) sup_Inf_absorb[simp]:
nipkow@15504
  2210
  "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<squnion> \<Sqinter>A) = a"
nipkow@15512
  2211
apply(subst sup_commute)
nipkow@15504
  2212
apply(simp add:Inf_def sup_absorb ACIfSL.fold1_belowI[OF ACIfSL_inf])
nipkow@15504
  2213
done
nipkow@15504
  2214
nipkow@15780
  2215
lemma (in Lattice) inf_Sup_absorb[simp]:
nipkow@15504
  2216
  "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<sqinter> \<Squnion>A) = a"
nipkow@15504
  2217
by(simp add:Sup_def inf_absorb ACIfSL.fold1_belowI[OF ACIfSL_sup])
nipkow@15504
  2218
nipkow@15504
  2219
nipkow@18423
  2220
lemma (in ACIf) hom_fold1_commute:
nipkow@18423
  2221
assumes hom: "!!x y. h(f x y) = f (h x) (h y)"
nipkow@18423
  2222
and N: "finite N" "N \<noteq> {}" shows "h(fold1 f N) = fold1 f (h ` N)"
nipkow@18423
  2223
using N proof (induct rule: finite_ne_induct)
nipkow@18423
  2224
  case singleton thus ?case by simp
nipkow@15500
  2225
next
nipkow@18423
  2226
  case (insert n N)
nipkow@18423
  2227
  have "h(fold1 f (insert n N)) = h(f n (fold1 f N))" using insert by(simp)
nipkow@18423
  2228
  also have "\<dots> = f (h n) (h(fold1 f N))" by(rule hom)
nipkow@18423
  2229
  also have "h(fold1 f N) = fold1 f (h ` N)" by(rule insert)
nipkow@18423
  2230
  also have "f (h n) \<dots> = fold1 f (insert (h n) (h ` N))"
nipkow@18423
  2231
    using insert by(simp)
nipkow@18423
  2232
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
nipkow@15500
  2233
  finally show ?case .
nipkow@15500
  2234
qed
nipkow@15500
  2235
nipkow@18423
  2236
lemma (in Distrib_Lattice) sup_Inf1_distrib:
nipkow@18423
  2237
 "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> (x \<squnion> \<Sqinter>A) = \<Sqinter>{x \<squnion> a|a. a \<in> A}"
nipkow@18423
  2238
apply(simp add:Inf_def image_def
nipkow@18423
  2239
  ACIf.hom_fold1_commute[OF ACIf_inf, where h="sup x", OF sup_inf_distrib1])
nipkow@18423
  2240
apply(rule arg_cong, blast)
nipkow@18423
  2241
done
nipkow@18423
  2242
nipkow@18423
  2243
nipkow@15512
  2244
lemma (in Distrib_Lattice) sup_Inf2_distrib:
nipkow@15500
  2245
assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
nipkow@15500
  2246
shows "(\<Sqinter>A \<squnion> \<Sqinter>B) = \<Sqinter>{a \<squnion> b|a b. a \<in> A \<and> b \<in> B}"
nipkow@15500
  2247
using A
nipkow@15500
  2248
proof (induct rule: finite_ne_induct)
nipkow@15500
  2249
  case singleton thus ?case
nipkow@15500
  2250
    by(simp add: sup_Inf1_distrib[OF B] fold1_singleton_def[OF Inf_def])
nipkow@15500
  2251
next
nipkow@15500
  2252
  case (insert x A)
nipkow@15500
  2253
  have finB: "finite {x \<squnion> b |b. b \<in> B}"
berghofe@15517
  2254
    by(fast intro: finite_surj[where f = "%b. x \<squnion> b", OF B(1)])
nipkow@15500
  2255
  have finAB: "finite {a \<squnion> b |a b. a \<in> A \<and> b \<in> B}"
nipkow@15500
  2256
  proof -
nipkow@15500
  2257
    have "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {a \<squnion> b})"
nipkow@15500
  2258
      by blast
berghofe@15517
  2259
    thus ?thesis by(simp add: insert(1) B(1))
nipkow@15500
  2260
  qed
nipkow@15500
  2261
  have ne: "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
nipkow@15500
  2262
  have "\<Sqinter>(insert x A) \<squnion> \<Sqinter>B = (x \<sqinter> \<Sqinter>A) \<squnion> \<Sqinter>B"
paulson@15509
  2263
    using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def])
nipkow@15500
  2264
  also have "\<dots> = (x \<squnion> \<Sqinter>B) \<sqinter> (\<Sqinter>A \<squnion> \<Sqinter>B)" by(rule sup_inf_distrib2)
nipkow@15500
  2265
  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
  2266
    using insert by(simp add:sup_Inf1_distrib[OF B])
nipkow@15500
  2267
  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
  2268
    (is "_ = \<Sqinter>?M")
nipkow@15500
  2269
    using B insert
nipkow@15500
  2270
    by(simp add:Inf_def ACIf.fold1_Un2[OF ACIf_inf finB _ finAB ne])
nipkow@15500
  2271
  also have "?M = {a \<squnion> b |a b. a \<in> insert x A \<and> b \<in> B}"
nipkow@15500
  2272
    by blast
nipkow@15500
  2273
  finally show ?case .
nipkow@15500
  2274
qed
nipkow@15500
  2275
nipkow@15484
  2276
nipkow@18423
  2277
lemma (in Distrib_Lattice) inf_Sup1_distrib:
nipkow@18423
  2278
 "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> (x \<sqinter> \<Squnion>A) = \<Squnion>{x \<sqinter> a|a. a \<in> A}"
nipkow@18423
  2279
apply(simp add:Sup_def image_def
nipkow@18423
  2280
  ACIf.hom_fold1_commute[OF ACIf_sup, where h="inf x", OF inf_sup_distrib1])
nipkow@18423
  2281
apply(rule arg_cong, blast)
nipkow@18423
  2282
done
nipkow@18423
  2283
nipkow@18423
  2284
nipkow@18423
  2285
lemma (in Distrib_Lattice) inf_Sup2_distrib:
nipkow@18423
  2286
assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
nipkow@18423
  2287
shows "(\<Squnion>A \<sqinter> \<Squnion>B) = \<Squnion>{a \<sqinter> b|a b. a \<in> A \<and> b \<in> B}"
nipkow@18423
  2288
using A
nipkow@18423
  2289
proof (induct rule: finite_ne_induct)
nipkow@18423
  2290
  case singleton thus ?case
nipkow@18423
  2291
    by(simp add: inf_Sup1_distrib[OF B] fold1_singleton_def[OF Sup_def])
nipkow@18423
  2292
next
nipkow@18423
  2293
  case (insert x A)
nipkow@18423
  2294
  have finB: "finite {x \<sqinter> b |b. b \<in> B}"
nipkow@18423
  2295
    by(fast intro: finite_surj[where f = "%b. x \<sqinter> b", OF B(1)])
nipkow@18423
  2296
  have finAB: "finite {a \<sqinter> b |a b. a \<in> A \<and> b \<in> B}"
nipkow@18423
  2297
  proof -
nipkow@18423
  2298
    have "{a \<sqinter> b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {a \<sqinter> b})"
nipkow@18423
  2299
      by blast
nipkow@18423
  2300
    thus ?thesis by(simp add: insert(1) B(1))
nipkow@18423
  2301
  qed
nipkow@18423
  2302
  have ne: "{a \<sqinter> b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
nipkow@18423
  2303
  have "\<Squnion>(insert x A) \<sqinter> \<Squnion>B = (x \<squnion> \<Squnion>A) \<sqinter> \<Squnion>B"
nipkow@18423
  2304
    using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_sup Sup_def])
nipkow@18423
  2305
  also have "\<dots> = (x \<sqinter> \<Squnion>B) \<squnion> (\<Squnion>A \<sqinter> \<Squnion>B)" by(rule inf_sup_distrib2)
nipkow@18423
  2306
  also have "\<dots> = \<Squnion>{x \<sqinter> b|b. b \<in> B} \<squnion> \<Squnion>{a \<sqinter> b|a b. a \<in> A \<and> b \<in> B}"
nipkow@18423
  2307
    using insert by(simp add:inf_Sup1_distrib[OF B])
nipkow@18423
  2308
  also have "\<dots> = \<Squnion>({x\<sqinter>b |b. b \<in> B} \<union> {a\<sqinter>b |a b. a \<in> A \<and> b \<in> B})"
nipkow@18423
  2309
    (is "_ = \<Squnion>?M")
nipkow@18423
  2310
    using B insert
nipkow@18423
  2311
    by(simp add:Sup_def ACIf.fold1_Un2[OF ACIf_sup finB _ finAB ne])
nipkow@18423
  2312
  also have "?M = {a \<sqinter> b |a b. a \<in> insert x A \<and> b \<in> B}"
nipkow@18423
  2313
    by blast
nipkow@18423
  2314
  finally show ?case .
nipkow@18423
  2315
qed
nipkow@18423
  2316
nipkow@18423
  2317
nipkow@15392
  2318
subsection{*Min and Max*}
nipkow@15392
  2319
nipkow@15392
  2320
text{* As an application of @{text fold1} we define the minimal and
nipkow@15497
  2321
maximal element of a (non-empty) set over a linear order. *}
nipkow@15497
  2322
nipkow@15497
  2323
constdefs
nipkow@15497
  2324
  Min :: "('a::linorder)set => 'a"
nipkow@15497
  2325
  "Min  ==  fold1 min"
nipkow@15497
  2326
nipkow@15497
  2327
  Max :: "('a::linorder)set => 'a"
nipkow@15497
  2328
  "Max  ==  fold1 max"
nipkow@15497
  2329
nipkow@15497
  2330
nipkow@15497
  2331
text{* Before we can do anything, we need to show that @{text min} and
nipkow@15497
  2332
@{text max} are ACI and the ordering is linear: *}
nipkow@15392
  2333
ballarin@15837
  2334
interpretation min: ACf ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
nipkow@15392
  2335
apply(rule ACf.intro)
nipkow@15392
  2336
apply(auto simp:min_def)
nipkow@15392
  2337
done
nipkow@15392
  2338
ballarin@15837
  2339
interpretation min: ACIf ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
nipkow@15392
  2340
apply(rule ACIf_axioms.intro)
nipkow@15392
  2341
apply(auto simp:min_def)
nipkow@15376
  2342
done
nipkow@15376
  2343
ballarin@15837
  2344
interpretation max: ACf ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
nipkow@15392
  2345
apply(rule ACf.intro)
nipkow@15392
  2346
apply(auto simp:max_def)
nipkow@15392
  2347
done
nipkow@15392
  2348
ballarin@15837
  2349
interpretation max: ACIf ["max:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
nipkow@15392
  2350
apply(rule ACIf_axioms.intro)
nipkow@15392
  2351
apply(auto simp:max_def)
nipkow@15376
  2352
done
wenzelm@12396
  2353
ballarin@15837
  2354
interpretation min:
nipkow@18493
  2355
  ACIfSL ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "op \<le>" "op <"]
nipkow@18493
  2356
apply(simp add:order_less_le)
nipkow@15497
  2357
apply(rule ACIfSL_axioms.intro)
nipkow@15497
  2358
apply(auto simp:min_def)
nipkow@15497
  2359
done
nipkow@15497
  2360
ballarin@15837
  2361
interpretation min:
nipkow@18493
  2362
  ACIfSLlin ["min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "op \<le>" "op <"]
nipkow@15497
  2363
apply(rule ACIfSLlin_axioms.intro)
nipkow@15497
  2364
apply(auto simp:min_def)
nipkow@15497
  2365
done
nipkow@15392
  2366
ballarin@15837
  2367
interpretation max:
nipkow@18493
  2368
  ACIfSL ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "%x y. y\<le>x" "%x y. y<x"]
nipkow@18493
  2369
apply(simp add:order_less_le eq_sym_conv)
nipkow@15497
  2370
apply(rule ACIfSL_axioms.intro)
nipkow@15497
  2371
apply(auto simp:max_def)
nipkow@15497
  2372
done
nipkow@15392
  2373
ballarin@15837
  2374
interpretation max:
nipkow@18493
  2375
  ACIfSLlin ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "%x y. y\<le>x" "%x y. y<x"]
nipkow@15497
  2376
apply(rule ACIfSLlin_axioms.intro)
nipkow@15497
  2377
apply(auto simp:max_def)
nipkow@15497
  2378
done
nipkow@15392
  2379
ballarin@15837
  2380
interpretation min_max:
nipkow@15780
  2381
  Lattice ["op \<le>" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max" "Min" "Max"]
nipkow@15780
  2382
apply -
nipkow@15780
  2383
apply(rule Min_def)
nipkow@15780
  2384
apply(rule Max_def)
nipkow@15507
  2385
done
nipkow@15500
  2386
nipkow@15500
  2387
ballarin@15837
  2388
interpretation min_max:
nipkow@15780
  2389
  Distrib_Lattice ["op \<le>" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max" "Min" "Max"]
nipkow@15780
  2390
.
ballarin@15765
  2391
nipkow@15402
  2392
text{* Now we instantiate the recursion equations and declare them
nipkow@15392
  2393
simplification rules: *}
nipkow@15392
  2394
paulson@17085
  2395
(* Making Min or Max a defined parameter of a locale, suitably
paulson@17085
  2396
  extending ACIf, could make the following interpretations more automatic. *)
ballarin@15765
  2397
paulson@17085
  2398
lemmas Min_singleton = fold1_singleton_def [OF Min_def]
paulson@17085
  2399
lemmas Max_singleton = fold1_singleton_def [OF Max_def]
paulson@17085
  2400
lemmas Min_insert = min.fold1_insert_idem_def [OF Min_def]
paulson@17085
  2401
lemmas Max_insert = max.fold1_insert_idem_def [OF Max_def]
paulson@17085
  2402
paulson@17085
  2403
declare Min_singleton [simp]  Max_singleton [simp]
paulson@17085
  2404
declare Min_insert [simp]  Max_insert [simp]
paulson@17085
  2405
nipkow@15392
  2406
nipkow@15484
  2407
text{* Now we instantiate some @{text fold1} properties: *}
nipkow@15392
  2408
nipkow@15392
  2409
lemma Min_in [simp]:
nipkow@15484
  2410
  shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Min A \<in> A"
nipkow@15791
  2411
using min.fold1_in
nipkow@15484
  2412
by(fastsimp simp: Min_def min_def)
nipkow@15392
  2413
nipkow@15392
  2414
lemma Max_in [simp]:
nipkow@15484
  2415
  shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Max A \<in> A"
nipkow@15791
  2416
using max.fold1_in
nipkow@15484
  2417
by(fastsimp simp: Max_def max_def)
nipkow@15392
  2418
nipkow@18423
  2419
lemma Min_antimono: "\<lbrakk> M \<subseteq> N; M \<noteq> {}; finite N \<rbrakk> \<Longrightarrow> Min N \<le> Min M"
nipkow@18423
  2420
by(simp add:Finite_Set.Min_def min.fold1_antimono)
nipkow@18423
  2421
nipkow@18423
  2422
lemma Max_mono: "\<lbrakk> M \<subseteq> N; M \<noteq> {}; finite N \<rbrakk> \<Longrightarrow> Max M \<le> Max N"
nipkow@18423
  2423
by(simp add:Max_def max.fold1_antimono)
nipkow@18423
  2424
nipkow@15484
  2425
lemma Min_le [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> Min A \<le> x"
nipkow@15791
  2426
by(simp add: Min_def min.fold1_belowI)
nipkow@15392
  2427
nipkow@15484
  2428
lemma Max_ge [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> x \<le> Max A"
nipkow@15791
  2429
by(simp add: Max_def max.fold1_belowI)
nipkow@15497
  2430
nipkow@15497
  2431
lemma Min_ge_iff[simp]:
nipkow@15497
  2432
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Min A) = (\<forall>a\<in>A. x \<le> a)"
nipkow@15791
  2433
by(simp add: Min_def min.below_fold1_iff)
nipkow@15497
  2434
nipkow@15497
  2435
lemma Max_le_iff[simp]:
nipkow@15497
  2436
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Max A \<le> x) = (\<forall>a\<in>A. a \<le> x)"
nipkow@15791
  2437
by(simp add: Max_def max.below_fold1_iff)
nipkow@15497
  2438
nipkow@18493
  2439
lemma Min_gr_iff[simp]:
nipkow@18493
  2440
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x < Min A) = (\<forall>a\<in>A. x < a)"
nipkow@18493
  2441
by(simp add: Min_def min.strict_below_fold1_iff)
nipkow@18493
  2442
nipkow@18493
  2443
lemma Max_less_iff[simp]:
nipkow@18493
  2444
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Max A < x) = (\<forall>a\<in>A. a < x)"
nipkow@18493
  2445
by(simp add: Max_def max.strict_below_fold1_iff)
nipkow@18493
  2446
nipkow@15497
  2447
lemma Min_le_iff:
nipkow@15497
  2448
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Min A \<le> x) = (\<exists>a\<in>A. a \<le> x)"
nipkow@15791
  2449
by(simp add: Min_def min.fold1_below_iff)
nipkow@15497
  2450
nipkow@15497
  2451
lemma Max_ge_iff:
nipkow@15497
  2452
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Max A) = (\<exists>a\<in>A. x \<le> a)"
nipkow@15791
  2453
by(simp add: Max_def max.fold1_below_iff)
wenzelm@12396
  2454
nipkow@18493
  2455
lemma Min_le_iff:
nipkow@18493
  2456
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Min A < x) = (\<exists>a\<in>A. a < x)"
nipkow@18493
  2457
by(simp add: Min_def min.fold1_strict_below_iff)
nipkow@18493
  2458
nipkow@18493
  2459
lemma Max_ge_iff:
nipkow@18493
  2460
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x < Max A) = (\<exists>a\<in>A. x < a)"
nipkow@18493
  2461
by(simp add: Max_def max.fold1_strict_below_iff)
nipkow@18493
  2462
nipkow@18423
  2463
lemma Min_Un: "\<lbrakk>finite A; A \<noteq> {}; finite B; B \<noteq> {}\<rbrakk>
nipkow@18423
  2464
  \<Longrightarrow> Min (A \<union> B) = min (Min A) (Min B)"
nipkow@18423
  2465
by(simp add:Min_def min.f.fold1_Un2)
nipkow@18423
  2466
nipkow@18423
  2467
lemma Max_Un: "\<lbrakk>finite A; A \<noteq> {}; finite B; B \<noteq> {}\<rbrakk>
nipkow@18423
  2468
  \<Longrightarrow> Max (A \<union> B) = max (Max A) (Max B)"
nipkow@18423
  2469
by(simp add:Max_def max.f.fold1_Un2)
nipkow@18423
  2470
nipkow@18423
  2471
nipkow@18423
  2472
lemma hom_Min_commute:
nipkow@18423
  2473
 "(!!x y::'a::linorder. h(min x y) = min (h x) (h y::'a))
nipkow@18423
  2474
  \<Longrightarrow> finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> h(Min N) = Min(h ` N)"
nipkow@18423
  2475
by(simp add:Finite_Set.Min_def min.hom_fold1_commute)
nipkow@18423
  2476
nipkow@18423
  2477
lemma hom_Max_commute:
nipkow@18423
  2478
 "(!!x y::'a::linorder. h(max x y) = max (h x) (h y::'a))
nipkow@18423
  2479
  \<Longrightarrow> finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> h(Max N) = Max(h ` N)"
nipkow@18423
  2480
by(simp add:Max_def max.hom_fold1_commute)
nipkow@18423
  2481
nipkow@18423
  2482
nipkow@18423
  2483
lemma add_Min_commute: fixes k::"'a::{pordered_ab_semigroup_add,linorder}"
nipkow@18423
  2484
 shows "finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> k + Min N = Min {k+m|m. m \<in> N}"
nipkow@18423
  2485
apply(subgoal_tac "!!x y. k + min x y = min (k + x) (k + y)")
nipkow@18423
  2486
using hom_Min_commute[of "op + k" N]
nipkow@18423
  2487
apply simp apply(rule arg_cong[where f = Min]) apply blast
nipkow@18423
  2488
apply(simp add:min_def linorder_not_le)
nipkow@18423
  2489
apply(blast intro:order.antisym order_less_imp_le add_left_mono)
nipkow@18423
  2490
done
nipkow@18423
  2491
nipkow@18423
  2492
lemma add_Max_commute: fixes k::"'a::{pordered_ab_semigroup_add,linorder}"
nipkow@18423
  2493
 shows "finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> k + Max N = Max {k+m|m. m \<in> N}"
nipkow@18423
  2494
apply(subgoal_tac "!!x y. k + max x y = max (k + x) (k + y)")
nipkow@18423
  2495
using hom_Max_commute[of "op + k" N]
nipkow@18423
  2496
apply simp apply(rule arg_cong[where f = Max]) apply blast
nipkow@18423
  2497
apply(simp add:max_def linorder_not_le)
nipkow@18423
  2498
apply(blast intro:order.antisym order_less_imp_le add_left_mono)
nipkow@18423
  2499
done
nipkow@18423
  2500
nipkow@18423
  2501
nipkow@18423
  2502
nipkow@17022
  2503
subsection {* Properties of axclass @{text finite} *}
nipkow@17022
  2504
nipkow@17022
  2505
text{* Many of these are by Brian Huffman. *}
nipkow@17022
  2506
nipkow@17022
  2507
lemma finite_set: "finite (A::'a::finite set)"
nipkow@17022
  2508
by (rule finite_subset [OF subset_UNIV finite])
nipkow@17022
  2509
nipkow@17022
  2510
nipkow@17022
  2511
instance unit :: finite
nipkow@17022
  2512
proof
nipkow@17022
  2513
  have "finite {()}" by simp
nipkow@17022
  2514
  also have "{()} = UNIV" by auto
nipkow@17022
  2515
  finally show "finite (UNIV :: unit set)" .
nipkow@17022
  2516
qed
nipkow@17022
  2517
nipkow@17022
  2518
instance bool :: finite
nipkow@17022
  2519
proof
nipkow@17022
  2520
  have "finite {True, False}" by simp
nipkow@17022
  2521
  also have "{True, False} = UNIV" by auto
nipkow@17022
  2522
  finally show "finite (UNIV :: bool set)" .
nipkow@17022
  2523
qed
nipkow@17022
  2524
nipkow@17022
  2525
nipkow@17022
  2526
instance * :: (finite, finite) finite
nipkow@17022
  2527
proof
nipkow@17022
  2528
  show "finite (UNIV :: ('a \<times> 'b) set)"
nipkow@17022
  2529
  proof (rule finite_Prod_UNIV)
nipkow@17022
  2530
    show "finite (UNIV :: 'a set)" by (rule finite)
nipkow@17022
  2531
    show "finite (UNIV :: 'b set)" by (rule finite)
nipkow@17022
  2532
  qed
nipkow@17022
  2533
qed
nipkow@17022
  2534
nipkow@17022
  2535
instance "+" :: (finite, finite) finite
nipkow@17022
  2536
proof
nipkow@17022
  2537
  have a: "finite (UNIV :: 'a set)" by (rule finite)
nipkow@17022
  2538
  have b: "finite (UNIV :: 'b set)" by (rule finite)
nipkow@17022
  2539
  from a b have "finite ((UNIV :: 'a set) <+> (UNIV :: 'b set))"
nipkow@17022
  2540
    by (rule finite_Plus)
nipkow@17022
  2541
  thus "finite (UNIV :: ('a + 'b) set)" by simp
nipkow@17022
  2542
qed
nipkow@17022
  2543
nipkow@17022
  2544
nipkow@17022
  2545
instance set :: (finite) finite
nipkow@17022
  2546
proof
nipkow@17022
  2547
  have "finite (UNIV :: 'a set)" by (rule finite)
nipkow@17022
  2548
  hence "finite (Pow (UNIV :: 'a set))"
nipkow@17022
  2549
    by (rule finite_Pow_iff [THEN iffD2])
nipkow@17022
  2550
  thus "finite (UNIV :: 'a set set)" by simp
nipkow@17022
  2551
qed
nipkow@17022
  2552
nipkow@17022
  2553
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
nipkow@17022
  2554
by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
nipkow@17022
  2555
nipkow@17022
  2556
instance fun :: (finite, finite) finite
nipkow@17022
  2557
proof
nipkow@17022
  2558
  show "finite (UNIV :: ('a => 'b) set)"
nipkow@17022
  2559
  proof (rule finite_imageD)
nipkow@17022
  2560
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
nipkow@17022
  2561
    show "finite (range ?graph)" by (rule finite_set)
nipkow@17022
  2562
    show "inj ?graph" by (rule inj_graph)
nipkow@17022
  2563
  qed
nipkow@17022
  2564
qed
nipkow@17022
  2565
nipkow@15042
  2566
end