src/HOL/Finite_Set.thy
author nipkow
Fri, 04 Feb 2005 17:14:42 +0100
changeset 15497 53bca254719a
parent 15487 55497029b255
child 15498 3988e90613d4
permissions -rw-r--r--
Added semi-lattice locales and reorganized fold1 lemmas
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(*  Title:      HOL/Finite_Set.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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                Additions by Jeremy Avigad in Feb 2004
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*)
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header {* Finite sets *}
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theory Finite_Set
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imports Divides Power Inductive
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begin
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subsection {* Definition and basic properties *}
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consts Finites :: "'a set set"
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syntax
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  finite :: "'a set => bool"
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translations
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  "finite A" == "A : Finites"
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inductive Finites
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  intros
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    emptyI [simp, intro!]: "{} : Finites"
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    insertI [simp, intro!]: "A : Finites ==> insert a A : Finites"
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axclass finite \<subseteq> type
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  finite: "finite UNIV"
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lemma ex_new_if_finite: -- "does not depend on def of finite at all"
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  assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
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  shows "\<exists>a::'a. a \<notin> A"
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proof -
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  from prems have "A \<noteq> UNIV" by blast
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  thus ?thesis by blast
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qed
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lemma finite_induct [case_names empty insert, induct set: Finites]:
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  "finite F ==>
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    P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
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  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
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proof -
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  assume "P {}" and
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    insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
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  assume "finite F"
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  thus "P F"
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  proof induct
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    show "P {}" .
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    fix x F assume F: "finite F" and P: "P F"
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    show "P (insert x F)"
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    proof cases
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      assume "x \<in> F"
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      hence "insert x F = F" by (rule insert_absorb)
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      with P show ?thesis by (simp only:)
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    next
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      assume "x \<notin> F"
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      from F this P show ?thesis by (rule insert)
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    qed
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  qed
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qed
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lemma finite_ne_induct[case_names singleton insert, consumes 2]:
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assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>
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 \<lbrakk> \<And>x. P{x};
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   \<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk>
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 \<Longrightarrow> P F"
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using fin
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proof induct
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  case empty thus ?case by simp
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next
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  case (insert x F)
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  show ?case
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  proof cases
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    assume "F = {}" thus ?thesis using insert(4) by simp
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  next
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    assume "F \<noteq> {}" thus ?thesis using insert by blast
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  qed
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qed
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lemma finite_subset_induct [consumes 2, case_names empty insert]:
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  "finite F ==> F \<subseteq> A ==>
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    P {} ==> (!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==>
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    P F"
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proof -
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  assume "P {}" and insert:
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    "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
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  assume "finite F"
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  thus "F \<subseteq> A ==> P F"
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  proof induct
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    show "P {}" .
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    fix x F assume "finite F" and "x \<notin> F"
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      and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
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    show "P (insert x F)"
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    proof (rule insert)
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      from i show "x \<in> A" by blast
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      from i have "F \<subseteq> A" by blast
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      with P show "P F" .
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    qed
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  qed
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qed
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text{* Finite sets are the images of initial segments of natural numbers: *}
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lemma finite_imp_nat_seg_image:
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assumes fin: "finite A" shows "\<exists> (n::nat) f. A = f ` {i::nat. i<n}"
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using fin
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proof induct
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  case empty
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  show ?case
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  proof show "\<exists>f. {} = f ` {i::nat. i < 0}" by(simp add:image_def) qed
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next
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  case (insert a A)
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  from insert.hyps obtain n f where "A = f ` {i::nat. i < n}" by blast
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  hence "insert a A = (%i. if i<n then f i else a) ` {i. i < n+1}"
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    by (auto simp add:image_def Ball_def)
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  thus ?case by blast
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qed
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lemma nat_seg_image_imp_finite:
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  "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
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proof (induct n)
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  case 0 thus ?case by simp
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next
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  case (Suc n)
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  let ?B = "f ` {i. i < n}"
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  have finB: "finite ?B" by(rule Suc.hyps[OF refl])
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  show ?case
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  proof cases
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    assume "\<exists>k<n. f n = f k"
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    hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  next
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    assume "\<not>(\<exists> k<n. f n = f k)"
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    hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  qed
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qed
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lemma finite_conv_nat_seg_image:
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  "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
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by(blast intro: finite_imp_nat_seg_image nat_seg_image_imp_finite)
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subsubsection{* Finiteness and set theoretic constructions *}
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lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
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  -- {* The union of two finite sets is finite. *}
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  by (induct set: Finites) simp_all
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lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
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  -- {* Every subset of a finite set is finite. *}
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proof -
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  assume "finite B"
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  thus "!!A. A \<subseteq> B ==> finite A"
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  proof induct
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    case empty
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    thus ?case by simp
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  next
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    case (insert x F A)
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    have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" .
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    show "finite A"
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   160
    proof cases
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   161
      assume x: "x \<in> A"
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   162
      with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
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wenzelm
parents:
diff changeset
   163
      with r have "finite (A - {x})" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   164
      hence "finite (insert x (A - {x}))" ..
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   165
      also have "insert x (A - {x}) = A" by (rule insert_Diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   166
      finally show ?thesis .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   167
    next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   168
      show "A \<subseteq> F ==> ?thesis" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   169
      assume "x \<notin> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   170
      with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
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parents:
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   171
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   172
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   173
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   174
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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   175
lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
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parents:
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   176
  by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
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parents:
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   177
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   178
lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
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   179
  -- {* The converse obviously fails. *}
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wenzelm
parents:
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   180
  by (blast intro: finite_subset)
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wenzelm
parents:
diff changeset
   181
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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   182
lemma finite_insert [simp]: "finite (insert a A) = finite A"
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parents:
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   183
  apply (subst insert_is_Un)
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   184
  apply (simp only: finite_Un, blast)
12396
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   185
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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diff changeset
   186
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   187
lemma finite_Union[simp, intro]:
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   188
 "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
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   189
by (induct rule:finite_induct) simp_all
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   190
12396
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   191
lemma finite_empty_induct:
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wenzelm
parents:
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   192
  "finite A ==>
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wenzelm
parents:
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   193
  P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   194
proof -
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wenzelm
parents:
diff changeset
   195
  assume "finite A"
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wenzelm
parents:
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   196
    and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   197
  have "P (A - A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   198
  proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   199
    fix c b :: "'a set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   200
    presume c: "finite c" and b: "finite b"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   201
      and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   202
    from c show "c \<subseteq> b ==> P (b - c)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   203
    proof induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   204
      case empty
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   205
      from P1 show ?case by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   206
    next
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0230a10582d3 changed the order of !!-quantifiers in finite set induction.
nipkow
parents: 15318
diff changeset
   207
      case (insert x F)
12396
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wenzelm
parents:
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   208
      have "P (b - F - {x})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   209
      proof (rule P2)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   210
        from _ b show "finite (b - F)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   211
        from insert show "x \<in> b - F" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   212
        from insert show "P (b - F)" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   213
      qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   214
      also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   215
      finally show ?case .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   216
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   217
  next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   218
    show "A \<subseteq> A" ..
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   219
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   220
  thus "P {}" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   221
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   222
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   223
lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   224
  by (rule Diff_subset [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   225
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   226
lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   227
  apply (subst Diff_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   228
  apply (case_tac "a : A - B")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   229
   apply (rule finite_insert [symmetric, THEN trans])
14208
144f45277d5a misc tidying
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diff changeset
   230
   apply (subst insert_Diff, simp_all)
12396
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wenzelm
parents:
diff changeset
   231
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   232
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   233
15392
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   234
text {* Image and Inverse Image over Finite Sets *}
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   235
ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   236
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
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parents: 13737
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   237
  -- {* The image of a finite set is finite. *}
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paulson
parents: 13737
diff changeset
   238
  by (induct set: Finites) simp_all
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   239
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
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diff changeset
   240
lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
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paulson
parents: 14331
diff changeset
   241
  apply (frule finite_imageI)
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paulson
parents: 14331
diff changeset
   242
  apply (erule finite_subset, assumption)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   243
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   244
13825
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parents: 13737
diff changeset
   245
lemma finite_range_imageI:
ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   246
    "finite (range g) ==> finite (range (%x. f (g x)))"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   247
  apply (drule finite_imageI, simp)
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   248
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   249
12396
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wenzelm
parents:
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   250
lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   251
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   252
  have aux: "!!A. finite (A - {}) = finite A" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   253
  fix B :: "'a set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   254
  assume "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   255
  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   256
    apply induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   257
     apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   258
    apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   259
     apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   260
     apply (simp (no_asm_use) add: inj_on_def)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   261
     apply (blast dest!: aux [THEN iffD1], atomize)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   262
    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   263
    apply (frule subsetD [OF equalityD2 insertI1], clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   264
    apply (rule_tac x = xa in bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   265
     apply (simp_all add: inj_on_image_set_diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   266
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   267
qed (rule refl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   268
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   269
13825
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paulson
parents: 13737
diff changeset
   270
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   271
  -- {* The inverse image of a singleton under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
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   272
         is included in a singleton. *}
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
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   273
  apply (auto simp add: inj_on_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   274
  apply (blast intro: the_equality [symmetric])
13825
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paulson
parents: 13737
diff changeset
   275
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   276
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   277
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
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   278
  -- {* The inverse image of a finite set under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   279
         is finite. *}
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   280
  apply (induct set: Finites, simp_all)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   281
  apply (subst vimage_insert)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   282
  apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   283
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   284
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   285
15392
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parents: 15376
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   286
text {* The finite UNION of finite sets *}
12396
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wenzelm
parents:
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   287
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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diff changeset
   288
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   289
  by (induct set: Finites) simp_all
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   290
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   291
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   292
  Strengthen RHS to
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   293
  @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
12396
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wenzelm
parents:
diff changeset
   294
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   295
  We'd need to prove
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   296
  @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
12396
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wenzelm
parents:
diff changeset
   297
  by induction. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   298
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   299
lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   300
  by (blast intro: finite_UN_I finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   301
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   302
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   303
text {* Sigma of finite sets *}
12396
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wenzelm
parents:
diff changeset
   304
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   305
lemma finite_SigmaI [simp]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   306
    "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   307
  by (unfold Sigma_def) (blast intro!: finite_UN_I)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   308
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   309
lemma finite_cartesian_product: "[| finite A; finite B |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   310
    finite (A <*> B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   311
  by (rule finite_SigmaI)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   312
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   313
lemma finite_Prod_UNIV:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   314
    "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   315
  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
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wenzelm
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   316
   apply (erule ssubst)
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paulson
parents: 13825
diff changeset
   317
   apply (erule finite_SigmaI, auto)
12396
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   318
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
diff changeset
   319
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
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parents: 15402
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   320
lemma finite_cartesian_productD1:
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   321
     "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   322
apply (auto simp add: finite_conv_nat_seg_image) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   323
apply (drule_tac x=n in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   324
apply (drule_tac x="fst o f" in spec) 
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paulson
parents: 15402
diff changeset
   325
apply (auto simp add: o_def) 
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paulson
parents: 15402
diff changeset
   326
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   327
apply (drule equalityD1) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   328
apply (rename_tac y x)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   329
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   330
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   331
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   332
apply (rule_tac x=k in image_eqI, auto)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   333
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   334
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   335
lemma finite_cartesian_productD2:
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   336
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   337
apply (auto simp add: finite_conv_nat_seg_image) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   338
apply (drule_tac x=n in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   339
apply (drule_tac x="snd o f" in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   340
apply (auto simp add: o_def) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   341
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   342
apply (drule equalityD1)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   343
apply (rename_tac x y)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   344
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   345
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   346
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   347
apply (rule_tac x=k in image_eqI, auto)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   348
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   349
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   350
12396
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wenzelm
parents:
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   351
instance unit :: finite
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wenzelm
parents:
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   352
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   353
  have "finite {()}" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   354
  also have "{()} = UNIV" by auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   355
  finally show "finite (UNIV :: unit set)" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   356
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   357
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   358
instance * :: (finite, finite) finite
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   359
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   360
  show "finite (UNIV :: ('a \<times> 'b) set)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   361
  proof (rule finite_Prod_UNIV)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   362
    show "finite (UNIV :: 'a set)" by (rule finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   363
    show "finite (UNIV :: 'b set)" by (rule finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   364
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   365
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   366
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   367
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   368
text {* The powerset of a finite set *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   369
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   370
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   371
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   372
  assume "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   373
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   374
  thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   375
next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   376
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   377
  thus "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   378
    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   379
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   380
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   381
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   382
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   383
by(blast intro: finite_subset[OF subset_Pow_Union])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   384
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   385
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   386
lemma finite_converse [iff]: "finite (r^-1) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   387
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   388
   apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   389
   apply (rule iffI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   390
    apply (erule finite_imageD [unfolded inj_on_def])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   391
    apply (simp split add: split_split)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   392
   apply (erule finite_imageI)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   393
  apply (simp add: converse_def image_def, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   394
  apply (rule bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   395
   prefer 2 apply assumption
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   396
  apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   397
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   398
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   399
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   400
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   401
Ehmety) *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   402
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   403
lemma finite_Field: "finite r ==> finite (Field r)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   404
  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   405
  apply (induct set: Finites)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   406
   apply (auto simp add: Field_def Domain_insert Range_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   407
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   408
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   409
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   410
  apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   411
  apply (erule trancl_induct)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   412
   apply (auto simp add: Field_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   413
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   414
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   415
lemma finite_trancl: "finite (r^+) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   416
  apply auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   417
   prefer 2
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   418
   apply (rule trancl_subset_Field2 [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   419
   apply (rule finite_SigmaI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   420
    prefer 3
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 13595
diff changeset
   421
    apply (blast intro: r_into_trancl' finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   422
   apply (auto simp add: finite_Field)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   423
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   424
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   425
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   426
subsection {* A fold functional for finite sets *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   427
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   428
text {* The intended behaviour is
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   429
@{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<dots>)"}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   430
if @{text f} is associative-commutative. For an application of @{text fold}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   431
se the definitions of sums and products over finite sets.
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   432
*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   433
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   434
consts
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   435
  foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   436
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   437
inductive "foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   438
intros
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   439
emptyI [intro]: "({}, z) : foldSet f g z"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   440
insertI [intro]: "\<lbrakk> x \<notin> A; (A, y) : foldSet f g z \<rbrakk>
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   441
 \<Longrightarrow> (insert x A, f (g x) y) : foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   442
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   443
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   444
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   445
constdefs
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   446
  fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   447
  "fold f g z A == THE x. (A, x) : foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   448
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   449
lemma Diff1_foldSet:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   450
  "(A - {x}, y) : foldSet f g z ==> x: A ==> (A, f (g x) y) : foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   451
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   452
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   453
lemma foldSet_imp_finite: "(A, x) : foldSet f g z ==> finite A"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   454
  by (induct set: foldSet) auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   455
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   456
lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   457
  by (induct set: Finites) auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   458
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   459
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   460
subsubsection {* Commutative monoids *}
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   461
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   462
locale ACf =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   463
  fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   464
  assumes commute: "x \<cdot> y = y \<cdot> x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   465
    and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   466
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   467
locale ACe = ACf +
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   468
  fixes e :: 'a
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   469
  assumes ident [simp]: "x \<cdot> e = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   470
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   471
locale ACIf = ACf +
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   472
  assumes idem: "x \<cdot> x = x"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   473
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   474
lemma (in ACf) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   475
proof -
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   476
  have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   477
  also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   478
  also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   479
  finally show ?thesis .
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   480
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   481
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   482
lemmas (in ACf) AC = assoc commute left_commute
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   483
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   484
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   485
proof -
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   486
  have "x \<cdot> e = x" by (rule ident)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   487
  thus ?thesis by (subst commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   488
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   489
15497
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   490
lemma (in ACIf) idem2: "x \<cdot> (x \<cdot> y) = x \<cdot> y"
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   491
proof -
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   492
  have "x \<cdot> (x \<cdot> y) = (x \<cdot> x) \<cdot> y" by(simp add:assoc)
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   493
  also have "\<dots> = x \<cdot> y" by(simp add:idem)
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   494
  finally show ?thesis .
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   495
qed
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   496
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   497
lemmas (in ACIf) ACI = AC idem idem2
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   498
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   499
text{* Instantiation of locales: *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   500
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   501
lemma ACf_add: "ACf (op + :: 'a::comm_monoid_add \<Rightarrow> 'a \<Rightarrow> 'a)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   502
by(fastsimp intro: ACf.intro add_assoc add_commute)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   503
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   504
lemma ACe_add: "ACe (op +) (0::'a::comm_monoid_add)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   505
by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_add)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   506
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   507
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   508
lemma ACf_mult: "ACf (op * :: 'a::comm_monoid_mult \<Rightarrow> 'a \<Rightarrow> 'a)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   509
by(fast intro: ACf.intro mult_assoc ab_semigroup_mult.mult_commute)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   510
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   511
lemma ACe_mult: "ACe (op *) (1::'a::comm_monoid_mult)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   512
by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_mult)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   513
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   514
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   515
subsubsection{*From @{term foldSet} to @{term fold}*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   516
15479
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   517
(* only used in the next lemma, but in there twice *)
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   518
lemma card_lemma: assumes A1: "A = insert b B" and notinB: "b \<notin> B" and
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   519
  card: "A = h`{i. i<Suc n}" and new: "\<not>(EX k<n. h n = h k)"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   520
shows "EX h. B = h`{i. i<n}" (is "EX h. ?P h")
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   521
proof
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   522
  let ?h = "%i. if h i = b then h n else h i"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   523
  show "B = ?h`{i. i<n}" (is "_ = ?r")
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   524
  proof
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   525
    show "B \<subseteq> ?r"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   526
    proof
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   527
      fix u assume "u \<in> B"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   528
      hence uinA: "u \<in> A" and unotb: "u \<noteq> b" using A1 notinB by blast+
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   529
      then obtain i\<^isub>u where below: "i\<^isub>u < Suc n" and [simp]: "u = h i\<^isub>u"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   530
	using card by(auto simp:image_def)
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   531
      show "u \<in> ?r"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   532
      proof cases
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   533
	assume "i\<^isub>u < n"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   534
	thus ?thesis using unotb by(fastsimp)
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   535
      next
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   536
	assume "\<not> i\<^isub>u < n"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   537
	with below have [simp]: "i\<^isub>u = n" by arith
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   538
	obtain i\<^isub>k where i\<^isub>k: "i\<^isub>k < Suc n" and [simp]: "b = h i\<^isub>k"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   539
	  using A1 card by blast
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   540
	have "i\<^isub>k < n"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   541
	proof (rule ccontr)
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   542
	  assume "\<not> i\<^isub>k < n"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   543
	  hence "i\<^isub>k = n" using i\<^isub>k by arith
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   544
	  thus False using unotb by simp
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   545
	qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   546
	thus ?thesis by(auto simp add:image_def)
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   547
      qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   548
    qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   549
  next
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   550
    show "?r \<subseteq> B"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   551
    proof
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   552
      fix u assume "u \<in> ?r"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   553
      then obtain i\<^isub>u where below: "i\<^isub>u < n" and
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   554
        or: "b = h i\<^isub>u \<and> u = h n \<or> h i\<^isub>u \<noteq> b \<and> h i\<^isub>u = u"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   555
	by(auto simp:image_def)
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   556
      from or show "u \<in> B"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   557
      proof
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   558
	assume [simp]: "b = h i\<^isub>u \<and> u = h n"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   559
	have "u \<in> A" using card by auto
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   560
        moreover have "u \<noteq> b" using new below by auto
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   561
	ultimately show "u \<in> B" using A1 by blast
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   562
      next
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   563
	assume "h i\<^isub>u \<noteq> b \<and> h i\<^isub>u = u"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   564
	moreover hence "u \<in> A" using card below by auto
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   565
	ultimately show "u \<in> B" using A1 by blast
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   566
      qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   567
    qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   568
  qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   569
qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   570
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   571
lemma (in ACf) foldSet_determ_aux:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   572
  "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; (A,x) : foldSet f g z; (A,x') : foldSet f g z \<rbrakk>
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   573
   \<Longrightarrow> x' = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   574
proof (induct n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   575
  case 0 thus ?case by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   576
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   577
  case (Suc n)
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   578
  have IH: "!!A x x' h. \<lbrakk>A = h`{i::nat. i<n}; (A,x) \<in> foldSet f g z; (A,x') \<in> foldSet f g z\<rbrakk>
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   579
           \<Longrightarrow> x' = x" and card: "A = h`{i. i<Suc n}"
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   580
  and Afoldx: "(A, x) \<in> foldSet f g z" and Afoldy: "(A,x') \<in> foldSet f g z" .
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   581
  show ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   582
  proof cases
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   583
    assume "EX k<n. h n = h k" 
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   584
      --{*@{term h} is not injective, so the cardinality has not increased*}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   585
    hence card': "A = h ` {i. i < n}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   586
      using card by (auto simp:image_def less_Suc_eq)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   587
    show ?thesis by(rule IH[OF card' Afoldx Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   588
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   589
    assume new: "\<not>(EX k<n. h n = h k)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   590
    show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   591
    proof (rule foldSet.cases[OF Afoldx])
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   592
      assume "(A, x) = ({}, z)"  --{*fold of a singleton set*}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   593
      thus "x' = x" using Afoldy by (auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   594
    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   595
      fix B b y
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   596
      assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   597
	and y: "(B,y) \<in> foldSet f g z" and notinB: "b \<notin> B"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   598
      hence A1: "A = insert b B" and x: "x = g b \<cdot> y" by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   599
      show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   600
      proof (rule foldSet.cases[OF Afoldy])
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   601
	assume "(A,x') = ({}, z)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   602
	thus ?thesis using A1 by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   603
      next
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   604
	fix C c r
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   605
	assume eq2: "(A,x') = (insert c C, g c \<cdot> r)"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   606
	  and r: "(C,r) \<in> foldSet f g z" and notinC: "c \<notin> C"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   607
	hence A2: "A = insert c C" and x': "x' = g c \<cdot> r" by auto
15479
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   608
	obtain hB where lessB: "B = hB ` {i. i<n}"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   609
	  using card_lemma[OF A1 notinB card new] by auto
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   610
	obtain hC where lessC: "C = hC ` {i. i<n}"
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   611
	  using card_lemma[OF A2 notinC card new] by auto
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   612
	show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   613
	proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   614
	  assume "b = c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   615
	  then moreover have "B = C" using A1 A2 notinB notinC by auto
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   616
	  ultimately show ?thesis using IH[OF lessB] y r x x' by auto
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   617
	next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   618
	  assume diff: "b \<noteq> c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   619
	  let ?D = "B - {c}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   620
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   621
	    using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   622
	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   623
	  with A1 have "finite ?D" by simp
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   624
	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   625
	    using finite_imp_foldSet by rules
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   626
	  moreover have cinB: "c \<in> B" using B by(auto)
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   627
	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   628
	    by(rule Diff1_foldSet)
15479
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   629
	  hence "g c \<cdot> d = y" by(rule IH[OF lessB y])
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   630
          moreover have "g b \<cdot> d = r"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   631
	  proof (rule IH[OF lessC r])
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   632
	    show "(C,g b \<cdot> d) \<in> foldSet f g z" using C notinB Dfoldd
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   633
	      by fastsimp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   634
	  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   635
	  ultimately show ?thesis using x x' by(auto simp:AC)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   636
	qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   637
      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   638
    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   639
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   640
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   641
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   642
(* The same proof, but using card 
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   643
lemma (in ACf) foldSet_determ_aux:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   644
  "!!A x x'. \<lbrakk> card A < n; (A,x) : foldSet f g e; (A,x') : foldSet f g e \<rbrakk>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   645
   \<Longrightarrow> x' = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   646
proof (induct n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   647
  case 0 thus ?case by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   648
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   649
  case (Suc n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   650
  have IH: "!!A x x'. \<lbrakk>card A < n; (A,x) \<in> foldSet f g e; (A,x') \<in> foldSet f g e\<rbrakk>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   651
           \<Longrightarrow> x' = x" and card: "card A < Suc n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   652
  and Afoldx: "(A, x) \<in> foldSet f g e" and Afoldy: "(A,x') \<in> foldSet f g e" .
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   653
  from card have "card A < n \<or> card A = n" by arith
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   654
  thus ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   655
  proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   656
    assume less: "card A < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   657
    show ?thesis by(rule IH[OF less Afoldx Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   658
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   659
    assume cardA: "card A = n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   660
    show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   661
    proof (rule foldSet.cases[OF Afoldx])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   662
      assume "(A, x) = ({}, e)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   663
      thus "x' = x" using Afoldy by (auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   664
    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   665
      fix B b y
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   666
      assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   667
	and y: "(B,y) \<in> foldSet f g e" and notinB: "b \<notin> B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   668
      hence A1: "A = insert b B" and x: "x = g b \<cdot> y" by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   669
      show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   670
      proof (rule foldSet.cases[OF Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   671
	assume "(A,x') = ({}, e)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   672
	thus ?thesis using A1 by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   673
      next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   674
	fix C c z
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   675
	assume eq2: "(A,x') = (insert c C, g c \<cdot> z)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   676
	  and z: "(C,z) \<in> foldSet f g e" and notinC: "c \<notin> C"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   677
	hence A2: "A = insert c C" and x': "x' = g c \<cdot> z" by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   678
	have finA: "finite A" by(rule foldSet_imp_finite[OF Afoldx])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   679
	with cardA A1 notinB have less: "card B < n" by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   680
	show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   681
	proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   682
	  assume "b = c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   683
	  then moreover have "B = C" using A1 A2 notinB notinC by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   684
	  ultimately show ?thesis using IH[OF less] y z x x' by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   685
	next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   686
	  assume diff: "b \<noteq> c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   687
	  let ?D = "B - {c}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   688
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   689
	    using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   690
	  have "finite ?D" using finA A1 by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   691
	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   692
	    using finite_imp_foldSet by rules
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   693
	  moreover have cinB: "c \<in> B" using B by(auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   694
	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   695
	    by(rule Diff1_foldSet)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   696
	  hence "g c \<cdot> d = y" by(rule IH[OF less y])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   697
          moreover have "g b \<cdot> d = z"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   698
	  proof (rule IH[OF _ z])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   699
	    show "card C < n" using C cardA A1 notinB finA cinB
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   700
	      by(auto simp:card_Diff1_less)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   701
	  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   702
	    show "(C,g b \<cdot> d) \<in> foldSet f g e" using C notinB Dfoldd
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   703
	      by fastsimp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   704
	  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   705
	  ultimately show ?thesis using x x' by(auto simp:AC)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   706
	qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   707
      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   708
    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   709
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   710
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   711
*)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   712
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   713
lemma (in ACf) foldSet_determ:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   714
  "(A, x) : foldSet f g z ==> (A, y) : foldSet f g z ==> y = x"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   715
apply(frule foldSet_imp_finite)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   716
apply(simp add:finite_conv_nat_seg_image)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   717
apply(blast intro: foldSet_determ_aux [rule_format])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   718
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   719
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   720
lemma (in ACf) fold_equality: "(A, y) : foldSet f g z ==> fold f g z A = y"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   721
  by (unfold fold_def) (blast intro: foldSet_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   722
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   723
text{* The base case for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   724
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   725
lemma fold_empty [simp]: "fold f g z {} = z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   726
  by (unfold fold_def) blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   727
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   728
lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   729
    ((insert x A, v) : foldSet f g z) =
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   730
    (EX y. (A, y) : foldSet f g z & v = f (g x) y)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   731
  apply auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   732
  apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   733
   apply (fastsimp dest: foldSet_imp_finite)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   734
  apply (blast intro: foldSet_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   735
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   736
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   737
text{* The recursion equation for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   738
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   739
lemma (in ACf) fold_insert[simp]:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   740
    "finite A ==> x \<notin> A ==> fold f g z (insert x A) = f (g x) (fold f g z A)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   741
  apply (unfold fold_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   742
  apply (simp add: fold_insert_aux)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   743
  apply (rule the_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   744
  apply (auto intro: finite_imp_foldSet
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   745
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   746
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   747
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   748
declare
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   749
  empty_foldSetE [rule del]  foldSet.intros [rule del]
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   750
  -- {* Delete rules to do with @{text foldSet} relation. *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   751
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   752
text{* A simplified version for idempotent functions: *}
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   753
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   754
lemma (in ACIf) fold_insert2:
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   755
assumes finA: "finite A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   756
shows "fold (op \<cdot>) g z (insert a A) = g a \<cdot> fold f g z A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   757
proof cases
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   758
  assume "a \<in> A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   759
  then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   760
    by(blast dest: mk_disjoint_insert)
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   761
  show ?thesis
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   762
  proof -
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   763
    from finA A have finB: "finite B" by(blast intro: finite_subset)
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   764
    have "fold f g z (insert a A) = fold f g z (insert a B)" using A by simp
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   765
    also have "\<dots> = (g a) \<cdot> (fold f g z B)"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   766
      using finB disj by(simp)
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   767
    also have "\<dots> = g a \<cdot> fold f g z A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   768
      using A finB disj by(simp add:idem assoc[symmetric])
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   769
    finally show ?thesis .
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   770
  qed
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   771
next
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   772
  assume "a \<notin> A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   773
  with finA show ?thesis by simp
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   774
qed
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   775
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   776
lemma (in ACIf) foldI_conv_id:
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   777
  "finite A \<Longrightarrow> fold f g z A = fold f id z (g ` A)"
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   778
by(erule finite_induct)(simp_all add: fold_insert2 del: fold_insert)
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   779
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   780
subsubsection{*Lemmas about @{text fold}*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   781
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   782
lemma (in ACf) fold_commute:
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   783
  "finite A ==> (!!z. f x (fold f g z A) = fold f g (f x z) A)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   784
  apply (induct set: Finites, simp)
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   785
  apply (simp add: left_commute [of x])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   786
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   787
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   788
lemma (in ACf) fold_nest_Un_Int:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   789
  "finite A ==> finite B
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   790
    ==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   791
  apply (induct set: Finites, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   792
  apply (simp add: fold_commute Int_insert_left insert_absorb)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   793
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   794
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   795
lemma (in ACf) fold_nest_Un_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   796
  "finite A ==> finite B ==> A Int B = {}
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   797
    ==> fold f g z (A Un B) = fold f g (fold f g z B) A"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   798
  by (simp add: fold_nest_Un_Int)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   799
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   800
lemma (in ACf) fold_reindex:
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   801
assumes fin: "finite A"
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   802
shows "inj_on h A \<Longrightarrow> fold f g z (h ` A) = fold f (g \<circ> h) z A"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   803
using fin apply (induct)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   804
 apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   805
apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   806
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   807
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   808
lemma (in ACe) fold_Un_Int:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   809
  "finite A ==> finite B ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   810
    fold f g e A \<cdot> fold f g e B =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   811
    fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   812
  apply (induct set: Finites, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   813
  apply (simp add: AC insert_absorb Int_insert_left)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   814
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   815
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   816
corollary (in ACe) fold_Un_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   817
  "finite A ==> finite B ==> A Int B = {} ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   818
    fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   819
  by (simp add: fold_Un_Int)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   820
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   821
lemma (in ACe) fold_UN_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   822
  "\<lbrakk> finite I; ALL i:I. finite (A i);
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   823
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   824
   \<Longrightarrow> fold f g e (UNION I A) =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   825
       fold f (%i. fold f g e (A i)) e I"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   826
  apply (induct set: Finites, simp, atomize)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   827
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   828
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   829
  apply (subgoal_tac "A x Int UNION F A = {}")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   830
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   831
  apply (simp add: fold_Un_disjoint)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   832
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   833
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   834
lemma (in ACf) fold_cong:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   835
  "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   836
  apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g z C = fold f h z C")
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   837
   apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   838
  apply (erule finite_induct, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   839
  apply (simp add: subset_insert_iff, clarify)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   840
  apply (subgoal_tac "finite C")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   841
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   842
  apply (subgoal_tac "C = insert x (C - {x})")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   843
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   844
  apply (erule ssubst)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   845
  apply (drule spec)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   846
  apply (erule (1) notE impE)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   847
  apply (simp add: Ball_def del: insert_Diff_single)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   848
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   849
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   850
lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   851
  fold f (%x. fold f (g x) e (B x)) e A =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   852
  fold f (split g) e (SIGMA x:A. B x)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   853
apply (subst Sigma_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   854
apply (subst fold_UN_disjoint)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   855
   apply assumption
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   856
  apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   857
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   858
apply (erule fold_cong)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   859
apply (subst fold_UN_disjoint)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   860
   apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   861
  apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   862
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   863
apply (simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   864
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   865
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   866
lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   867
   fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   868
apply (erule finite_induct)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   869
 apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   870
apply (simp add:AC)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   871
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   872
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   873
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   874
subsection {* Generalized summation over a set *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   875
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   876
constdefs
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   877
  setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   878
  "setsum f A == if finite A then fold (op +) f 0 A else 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   879
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   880
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   881
written @{text"\<Sum>x\<in>A. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   882
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   883
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   884
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   885
syntax (xsymbols)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   886
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   887
syntax (HTML output)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   888
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   889
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   890
translations -- {* Beware of argument permutation! *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   891
  "SUM i:A. b" == "setsum (%i. b) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   892
  "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   893
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   894
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   895
 @{text"\<Sum>x|P. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   896
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   897
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   898
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   899
syntax (xsymbols)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   900
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   901
syntax (HTML output)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   902
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   903
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   904
translations
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   905
  "SUM x|P. t" => "setsum (%x. t) {x. P}"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   906
  "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   907
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   908
text{* Finally we abbreviate @{term"\<Sum>x\<in>A. x"} by @{text"\<Sum>A"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   909
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   910
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   911
  "_Setsum" :: "'a set => 'a::comm_monoid_mult"  ("\<Sum>_" [1000] 999)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   912
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   913
parse_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   914
  let
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   915
    fun Setsum_tr [A] = Syntax.const "setsum" $ Abs ("", dummyT, Bound 0) $ A
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   916
  in [("_Setsum", Setsum_tr)] end;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   917
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   918
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   919
print_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   920
let
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   921
  fun setsum_tr' [Abs(_,_,Bound 0), A] = Syntax.const "_Setsum" $ A
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   922
    | setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   923
       if x<>y then raise Match
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   924
       else let val x' = Syntax.mark_bound x
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   925
                val t' = subst_bound(x',t)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   926
                val P' = subst_bound(x',P)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   927
            in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   928
in
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   929
[("setsum", setsum_tr')]
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   930
end
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   931
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   932
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   933
lemma setsum_empty [simp]: "setsum f {} = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   934
  by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   935
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   936
lemma setsum_insert [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   937
    "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   938
  by (simp add: setsum_def ACf.fold_insert [OF ACf_add])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   939
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   940
lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   941
  by (simp add: setsum_def)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   942
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   943
lemma setsum_reindex:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   944
     "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   945
by(auto simp add: setsum_def ACf.fold_reindex[OF ACf_add] dest!:finite_imageD)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   946
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   947
lemma setsum_reindex_id:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   948
     "inj_on f B ==> setsum f B = setsum id (f ` B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   949
by (auto simp add: setsum_reindex)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   950
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   951
lemma setsum_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   952
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   953
by(fastsimp simp: setsum_def intro: ACf.fold_cong[OF ACf_add])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   954
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   955
lemma setsum_reindex_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   956
     "[|inj_on f A; B = f ` A; !!a. g a = h (f a)|] 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   957
      ==> setsum h B = setsum g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   958
  by (simp add: setsum_reindex cong: setsum_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   959
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   960
lemma setsum_0: "setsum (%i. 0) A = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   961
apply (clarsimp simp: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   962
apply (erule finite_induct, auto simp:ACf.fold_insert [OF ACf_add])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   963
done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   964
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   965
lemma setsum_0': "ALL a:F. f a = 0 ==> setsum f F = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   966
  apply (subgoal_tac "setsum f F = setsum (%x. 0) F")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   967
  apply (erule ssubst, rule setsum_0)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   968
  apply (rule setsum_cong, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   969
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   970
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   971
lemma setsum_Un_Int: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   972
  setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   973
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   974
by(simp add: setsum_def ACe.fold_Un_Int[OF ACe_add,symmetric])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   975
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   976
lemma setsum_Un_disjoint: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   977
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   978
by (subst setsum_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   979
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   980
(*But we can't get rid of finite I. If infinite, although the rhs is 0, 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   981
  the lhs need not be, since UNION I A could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   982
lemma setsum_UN_disjoint:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   983
    "finite I ==> (ALL i:I. finite (A i)) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   984
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   985
      setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   986
by(simp add: setsum_def ACe.fold_UN_disjoint[OF ACe_add] cong: setsum_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   987
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   988
text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   989
directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   990
lemma setsum_Union_disjoint:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   991
  "[| (ALL A:C. finite A);
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   992
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   993
   ==> setsum f (Union C) = setsum (setsum f) C"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   994
apply (cases "finite C") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   995
 prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   996
  apply (frule setsum_UN_disjoint [of C id f])
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   997
 apply (unfold Union_def id_def, assumption+)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   998
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   999
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1000
(*But we can't get rid of finite A. If infinite, although the lhs is 0, 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1001
  the rhs need not be, since SIGMA A B could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1002
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1003
    (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1004
    (\<Sum>z\<in>(SIGMA x:A. B x). f (fst z) (snd z))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1005
by(simp add:setsum_def ACe.fold_Sigma[OF ACe_add] split_def cong:setsum_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1006
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1007
text{*Here we can eliminate the finiteness assumptions, by cases.*}
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1008
lemma setsum_cartesian_product: 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1009
   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>z\<in>A <*> B. f (fst z) (snd z))"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1010
apply (cases "finite A") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1011
 apply (cases "finite B") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1012
  apply (simp add: setsum_Sigma)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1013
 apply (cases "A={}", simp)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1014
 apply (simp add: setsum_0) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1015
apply (auto simp add: setsum_def
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1016
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1017
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1018
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1019
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1020
by(simp add:setsum_def ACe.fold_distrib[OF ACe_add])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1021
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1022
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1023
subsubsection {* Properties in more restricted classes of structures *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1024
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1025
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1026
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1027
   prefer 2 apply (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1028
  apply (erule rev_mp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1029
  apply (erule finite_induct, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1030
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1031
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1032
lemma setsum_eq_0_iff [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1033
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1034
  by (induct set: Finites) auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1035
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1036
lemma setsum_Un_nat: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1037
    (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1038
  -- {* For the natural numbers, we have subtraction. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1039
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1040
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1041
lemma setsum_Un: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1042
    (setsum f (A Un B) :: 'a :: ab_group_add) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1043
      setsum f A + setsum f B - setsum f (A Int B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1044
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1045
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1046
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1047
    (if a:A then setsum f A - f a else setsum f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1048
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1049
   prefer 2 apply (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1050
  apply (erule finite_induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1051
   apply (auto simp add: insert_Diff_if)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1052
  apply (drule_tac a = a in mk_disjoint_insert, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1053
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1054
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1055
lemma setsum_diff1: "finite A \<Longrightarrow>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1056
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1057
  (if a:A then setsum f A - f a else setsum f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1058
  by (erule finite_induct) (auto simp add: insert_Diff_if)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1059
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1060
(* By Jeremy Siek: *)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1061
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1062
lemma setsum_diff_nat: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1063
  assumes finB: "finite B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1064
  shows "B \<subseteq> A \<Longrightarrow> (setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1065
using finB
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1066
proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1067
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1068
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1069
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1070
    and xFinA: "insert x F \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1071
    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1072
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1073
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1074
    by (simp add: setsum_diff1_nat)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1075
  from xFinA have "F \<subseteq> A" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1076
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1077
  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1078
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1079
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1080
  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1081
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1082
  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1083
  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1084
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1085
  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1086
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1087
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1088
lemma setsum_diff:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1089
  assumes le: "finite A" "B \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1090
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1091
proof -
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1092
  from le have finiteB: "finite B" using finite_subset by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1093
  show ?thesis using finiteB le
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1094
    proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1095
      case empty
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1096
      thus ?case by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1097
    next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1098
      case (insert x F)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1099
      thus ?case using le finiteB 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1100
	by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1101
    qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1102
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1103
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1104
lemma setsum_mono:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1105
  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1106
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1107
proof (cases "finite K")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1108
  case True
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1109
  thus ?thesis using le
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1110
  proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1111
    case empty
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1112
    thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1113
  next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1114
    case insert
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1115
    thus ?case using add_mono 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1116
      by force
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1117
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1118
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1119
  case False
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1120
  thus ?thesis
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1121
    by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1122
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1123
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1124
lemma setsum_mono2_nat:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1125
  assumes fin: "finite B" and sub: "A \<subseteq> B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1126
shows "setsum f A \<le> (setsum f B :: nat)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1127
proof -
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1128
  have "setsum f A \<le> setsum f A + setsum f (B-A)" by arith
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1129
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1130
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1131
  also have "A \<union> (B-A) = B" using sub by blast
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1132
  finally show ?thesis .
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1133
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1134
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1135
lemma setsum_negf: "finite A ==> setsum (%x. - (f x)::'a::ab_group_add) A =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1136
  - setsum f A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1137
  by (induct set: Finites, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1138
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1139
lemma setsum_subtractf: "finite A ==> setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1140
  setsum f A - setsum g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1141
  by (simp add: diff_minus setsum_addf setsum_negf)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1142
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1143
lemma setsum_nonneg: "[| finite A;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1144
    \<forall>x \<in> A. (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) \<le> f x |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1145
    0 \<le> setsum f A";
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1146
  apply (induct set: Finites, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1147
  apply (subgoal_tac "0 + 0 \<le> f x + setsum f F", simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1148
  apply (blast intro: add_mono)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1149
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1150
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1151
lemma setsum_nonpos: "[| finite A;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1152
    \<forall>x \<in> A. f x \<le> (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1153
    setsum f A \<le> 0";
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1154
  apply (induct set: Finites, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1155
  apply (subgoal_tac "f x + setsum f F \<le> 0 + 0", simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1156
  apply (blast intro: add_mono)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1157
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1158
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1159
lemma setsum_mult: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1160
  fixes f :: "'a => ('b::semiring_0_cancel)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1161
  shows "r * setsum f A = setsum (%n. r * f n) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1162
proof (cases "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1163
  case True
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1164
  thus ?thesis
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1165
  proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1166
    case empty thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1167
  next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1168
    case (insert x A) thus ?case by (simp add: right_distrib)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1169
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1170
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1171
  case False thus ?thesis by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1172
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1173
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1174
lemma setsum_abs: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1175
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1176
  assumes fin: "finite A" 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1177
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1178
using fin 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1179
proof (induct) 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1180
  case empty thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1181
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1182
  case (insert x A)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1183
  thus ?case by (auto intro: abs_triangle_ineq order_trans)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1184
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1185
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1186
lemma setsum_abs_ge_zero: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1187
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1188
  assumes fin: "finite A" 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1189
  shows "0 \<le> setsum (%i. abs(f i)) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1190
using fin 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1191
proof (induct) 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1192
  case empty thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1193
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1194
  case (insert x A) thus ?case by (auto intro: order_trans)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1195
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1196
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1197
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1198
subsection {* Generalized product over a set *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1199
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1200
constdefs
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1201
  setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1202
  "setprod f A == if finite A then fold (op *) f 1 A else 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1203
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1204
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1205
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_:_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1206
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1207
syntax (xsymbols)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1208
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1209
syntax (HTML output)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1210
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1211
translations
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1212
  "\<Prod>i:A. b" == "setprod (%i. b) A"  -- {* Beware of argument permutation! *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1213
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1214
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1215
  "_Setprod" :: "'a set => 'a::comm_monoid_mult"  ("\<Prod>_" [1000] 999)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1216
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1217
parse_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1218
  let
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1219
    fun Setprod_tr [A] = Syntax.const "setprod" $ Abs ("", dummyT, Bound 0) $ A
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1220
  in [("_Setprod", Setprod_tr)] end;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1221
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1222
print_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1223
let fun setprod_tr' [Abs(x,Tx,t), A] =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1224
    if t = Bound 0 then Syntax.const "_Setprod" $ A else raise Match
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1225
in
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1226
[("setprod", setprod_tr')]
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1227
end
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1228
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1229
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1230
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1231
lemma setprod_empty [simp]: "setprod f {} = 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1232
  by (auto simp add: setprod_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1233
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1234
lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1235
    setprod f (insert a A) = f a * setprod f A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1236
by (simp add: setprod_def ACf.fold_insert [OF ACf_mult])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1237
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1238
lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1239
  by (simp add: setprod_def)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1240
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1241
lemma setprod_reindex:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1242
     "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1243
by(auto simp: setprod_def ACf.fold_reindex[OF ACf_mult] dest!:finite_imageD)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1244
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1245
lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1246
by (auto simp add: setprod_reindex)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1247
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1248
lemma setprod_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1249
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1250
by(fastsimp simp: setprod_def intro: ACf.fold_cong[OF ACf_mult])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1251
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1252
lemma setprod_reindex_cong: "inj_on f A ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1253
    B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1254
  by (frule setprod_reindex, simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1255
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1256
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1257
lemma setprod_1: "setprod (%i. 1) A = 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1258
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1259
  apply (erule finite_induct, auto simp add: mult_ac)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1260
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1261
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1262
lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1263
  apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1264
  apply (erule ssubst, rule setprod_1)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1265
  apply (rule setprod_cong, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1266
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1267
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1268
lemma setprod_Un_Int: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1269
    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1270
by(simp add: setprod_def ACe.fold_Un_Int[OF ACe_mult,symmetric])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1271
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1272
lemma setprod_Un_disjoint: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1273
  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1274
by (subst setprod_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1275
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1276
lemma setprod_UN_disjoint:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1277
    "finite I ==> (ALL i:I. finite (A i)) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1278
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1279
      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1280
by(simp add: setprod_def ACe.fold_UN_disjoint[OF ACe_mult] cong: setprod_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1281
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1282
lemma setprod_Union_disjoint:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1283
  "[| (ALL A:C. finite A);
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1284
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |] 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1285
   ==> setprod f (Union C) = setprod (setprod f) C"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1286
apply (cases "finite C") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1287
 prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1288
  apply (frule setprod_UN_disjoint [of C id f])
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1289
 apply (unfold Union_def id_def, assumption+)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1290
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1291
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1292
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1293
    (\<Prod>x:A. (\<Prod>y: B x. f x y)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1294
    (\<Prod>z:(SIGMA x:A. B x). f (fst z) (snd z))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1295
by(simp add:setprod_def ACe.fold_Sigma[OF ACe_mult] split_def cong:setprod_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1296
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1297
text{*Here we can eliminate the finiteness assumptions, by cases.*}
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1298
lemma setprod_cartesian_product: 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1299
     "(\<Prod>x:A. (\<Prod>y: B. f x y)) = (\<Prod>z:(A <*> B). f (fst z) (snd z))"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1300
apply (cases "finite A") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1301
 apply (cases "finite B") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1302
  apply (simp add: setprod_Sigma)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1303
 apply (cases "A={}", simp)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1304
 apply (simp add: setprod_1) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1305
apply (auto simp add: setprod_def
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1306
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1307
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1308
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1309
lemma setprod_timesf:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1310
     "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1311
by(simp add:setprod_def ACe.fold_distrib[OF ACe_mult])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1312
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1313
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1314
subsubsection {* Properties in more restricted classes of structures *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1315
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1316
lemma setprod_eq_1_iff [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1317
    "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1318
  by (induct set: Finites) auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1319
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1320
lemma setprod_zero:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1321
     "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1322
  apply (induct set: Finites, force, clarsimp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1323
  apply (erule disjE, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1324
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1325
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1326
lemma setprod_nonneg [rule_format]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1327
     "(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1328
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1329
  apply (induct set: Finites, force, clarsimp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1330
  apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1331
  apply (rule mult_mono, assumption+)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1332
  apply (auto simp add: setprod_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1333
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1334
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1335
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1336
     --> 0 < setprod f A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1337
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1338
  apply (induct set: Finites, force, clarsimp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1339
  apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1340
  apply (rule mult_strict_mono, assumption+)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1341
  apply (auto simp add: setprod_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1342
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1343
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1344
lemma setprod_nonzero [rule_format]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1345
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1346
      finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1347
  apply (erule finite_induct, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1348
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1349
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1350
lemma setprod_zero_eq:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1351
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1352
     finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1353
  apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1354
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1355
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1356
lemma setprod_nonzero_field:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1357
    "finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1358
  apply (rule setprod_nonzero, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1359
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1360
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1361
lemma setprod_zero_eq_field:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1362
    "finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1363
  apply (rule setprod_zero_eq, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1364
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1365
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1366
lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1367
    (setprod f (A Un B) :: 'a ::{field})
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1368
      = setprod f A * setprod f B / setprod f (A Int B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1369
  apply (subst setprod_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1370
  apply (subgoal_tac "finite (A Int B)")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1371
  apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1372
  apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1373
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1374