src/HOL/Finite_Set.thy
author nipkow
Fri, 05 Aug 2005 12:20:30 +0200
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child 17085 5b57f995a179
permissions -rw-r--r--
added Brian Hufmann's finite instances
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(*  Title:      HOL/Finite_Set.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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                with contributions by Jeremy Avigad
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*)
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header {* Finite sets *}
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theory Finite_Set
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imports Power Inductive Lattice_Locales
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begin
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subsection {* Definition and basic properties *}
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consts Finites :: "'a set set"
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syntax
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  finite :: "'a set => bool"
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translations
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  "finite A" == "A : Finites"
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inductive Finites
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  intros
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    emptyI [simp, intro!]: "{} : Finites"
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    insertI [simp, intro!]: "A : Finites ==> insert a A : Finites"
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axclass finite \<subseteq> type
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  finite: "finite UNIV"
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lemma ex_new_if_finite: -- "does not depend on def of finite at all"
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  assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
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  shows "\<exists>a::'a. a \<notin> A"
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proof -
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  from prems have "A \<noteq> UNIV" by blast
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  thus ?thesis by blast
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qed
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lemma finite_induct [case_names empty insert, induct set: Finites]:
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  "finite F ==>
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    P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
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  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
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proof -
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  assume "P {}" and
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    insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
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  assume "finite F"
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  thus "P F"
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  proof induct
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    show "P {}" .
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    fix x F assume F: "finite F" and P: "P F"
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    show "P (insert x F)"
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    proof cases
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      assume "x \<in> F"
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      hence "insert x F = F" by (rule insert_absorb)
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      with P show ?thesis by (simp only:)
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    next
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      assume "x \<notin> F"
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      from F this P show ?thesis by (rule insert)
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    qed
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  qed
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qed
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lemma finite_ne_induct[case_names singleton insert, consumes 2]:
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assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>
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 \<lbrakk> \<And>x. P{x};
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   \<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk>
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 \<Longrightarrow> P F"
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using fin
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proof induct
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  case empty thus ?case by simp
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next
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  case (insert x F)
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  show ?case
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  proof cases
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    assume "F = {}" thus ?thesis using insert(4) by simp
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  next
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    assume "F \<noteq> {}" thus ?thesis using insert by blast
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  qed
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qed
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lemma finite_subset_induct [consumes 2, case_names empty insert]:
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  "finite F ==> F \<subseteq> A ==>
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    P {} ==> (!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==>
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    P F"
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proof -
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  assume "P {}" and insert:
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    "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
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  assume "finite F"
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  thus "F \<subseteq> A ==> P F"
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  proof induct
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    show "P {}" .
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    fix x F assume "finite F" and "x \<notin> F"
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      and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
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    show "P (insert x F)"
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    proof (rule insert)
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      from i show "x \<in> A" by blast
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      from i have "F \<subseteq> A" by blast
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      with P show "P F" .
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    qed
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  qed
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qed
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text{* Finite sets are the images of initial segments of natural numbers: *}
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lemma finite_imp_nat_seg_image_inj_on:
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  assumes fin: "finite A" 
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  shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}"
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using fin
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proof induct
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  case empty
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  show ?case  
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  proof show "\<exists>f. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp 
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  qed
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next
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  case (insert a A)
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  have notinA: "a \<notin> A" .
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  from insert.hyps obtain n f
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    where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
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  hence "insert a A = f(n:=a) ` {i. i < Suc n}"
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        "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
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    by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
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  thus ?case by blast
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qed
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lemma nat_seg_image_imp_finite:
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  "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
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proof (induct n)
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  case 0 thus ?case by simp
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next
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  case (Suc n)
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  let ?B = "f ` {i. i < n}"
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  have finB: "finite ?B" by(rule Suc.hyps[OF refl])
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  show ?case
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  proof cases
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    assume "\<exists>k<n. f n = f k"
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    hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  next
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    assume "\<not>(\<exists> k<n. f n = f k)"
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    hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  qed
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qed
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lemma finite_conv_nat_seg_image:
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  "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
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by(blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
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subsubsection{* Finiteness and set theoretic constructions *}
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lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
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  -- {* The union of two finite sets is finite. *}
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  by (induct set: Finites) simp_all
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lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
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  -- {* Every subset of a finite set is finite. *}
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proof -
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  assume "finite B"
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  thus "!!A. A \<subseteq> B ==> finite A"
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  proof induct
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    case empty
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    thus ?case by simp
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  next
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   162
    case (insert x F A)
12396
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   163
    have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   164
    show "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   165
    proof cases
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   166
      assume x: "x \<in> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   167
      with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   168
      with r have "finite (A - {x})" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   169
      hence "finite (insert x (A - {x}))" ..
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   170
      also have "insert x (A - {x}) = A" by (rule insert_Diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   171
      finally show ?thesis .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   172
    next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   173
      show "A \<subseteq> F ==> ?thesis" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   174
      assume "x \<notin> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   175
      with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   176
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   177
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   178
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   179
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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   180
lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   181
  by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   182
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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   183
lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
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   184
  -- {* The converse obviously fails. *}
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wenzelm
parents:
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   185
  by (blast intro: finite_subset)
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wenzelm
parents:
diff changeset
   186
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   187
lemma finite_insert [simp]: "finite (insert a A) = finite A"
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parents:
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   188
  apply (subst insert_is_Un)
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   189
  apply (simp only: finite_Un, blast)
12396
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wenzelm
parents:
diff changeset
   190
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   191
15281
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   192
lemma finite_Union[simp, intro]:
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   193
 "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
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   194
by (induct rule:finite_induct) simp_all
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diff changeset
   195
12396
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   196
lemma finite_empty_induct:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   197
  "finite A ==>
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wenzelm
parents:
diff changeset
   198
  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
   199
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   200
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   201
    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
   202
  have "P (A - A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   203
  proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   204
    fix c b :: "'a set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   205
    presume c: "finite c" and b: "finite b"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   206
      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
   207
    from c show "c \<subseteq> b ==> P (b - c)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   208
    proof induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   209
      case empty
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   210
      from P1 show ?case by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   211
    next
15327
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nipkow
parents: 15318
diff changeset
   212
      case (insert x F)
12396
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wenzelm
parents:
diff changeset
   213
      have "P (b - F - {x})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   214
      proof (rule P2)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   215
        from _ b show "finite (b - F)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   216
        from insert show "x \<in> b - F" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   217
        from insert show "P (b - F)" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   218
      qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   219
      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
   220
      finally show ?case .
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
  next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   223
    show "A \<subseteq> A" ..
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   224
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   225
  thus "P {}" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   226
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   227
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   228
lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   229
  by (rule Diff_subset [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   230
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   231
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
   232
  apply (subst Diff_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   233
  apply (case_tac "a : A - B")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   234
   apply (rule finite_insert [symmetric, THEN trans])
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   235
   apply (subst insert_Diff, simp_all)
12396
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wenzelm
parents:
diff changeset
   236
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   237
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   238
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   239
text {* Image and Inverse Image over Finite Sets *}
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ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   240
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   241
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
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parents: 13737
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   242
  -- {* The image of a finite set is finite. *}
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   243
  by (induct set: Finites) simp_all
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   244
14430
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diff changeset
   245
lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
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paulson
parents: 14331
diff changeset
   246
  apply (frule finite_imageI)
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paulson
parents: 14331
diff changeset
   247
  apply (erule finite_subset, assumption)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   248
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   249
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   250
lemma finite_range_imageI:
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   251
    "finite (range g) ==> finite (range (%x. f (g x)))"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   252
  apply (drule finite_imageI, simp)
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   253
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   254
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   255
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
   256
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   257
  have aux: "!!A. finite (A - {}) = finite A" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   258
  fix B :: "'a set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   259
  assume "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   260
  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   261
    apply induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   262
     apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   263
    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
   264
     apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   265
     apply (simp (no_asm_use) add: inj_on_def)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   266
     apply (blast dest!: aux [THEN iffD1], atomize)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   267
    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   268
    apply (frule subsetD [OF equalityD2 insertI1], clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   269
    apply (rule_tac x = xa in bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   270
     apply (simp_all add: inj_on_image_set_diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   271
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   272
qed (rule refl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   273
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   274
13825
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paulson
parents: 13737
diff changeset
   275
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   276
  -- {* The inverse image of a singleton under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   277
         is included in a singleton. *}
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   278
  apply (auto simp add: inj_on_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   279
  apply (blast intro: the_equality [symmetric])
13825
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paulson
parents: 13737
diff changeset
   280
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   281
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   282
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   283
  -- {* The inverse image of a finite set under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   284
         is finite. *}
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   285
  apply (induct set: Finites, simp_all)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   286
  apply (subst vimage_insert)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   287
  apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   288
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   289
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   290
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   291
text {* The finite UNION of finite sets *}
12396
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wenzelm
parents:
diff changeset
   292
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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diff changeset
   293
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
   294
  by (induct set: Finites) simp_all
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   295
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   296
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   297
  Strengthen RHS to
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   298
  @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
12396
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wenzelm
parents:
diff changeset
   299
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   300
  We'd need to prove
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   301
  @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   302
  by induction. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   303
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   304
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
   305
  by (blast intro: finite_UN_I finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   306
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   307
17022
b257300c3a9c added Brian Hufmann's finite instances
nipkow
parents: 16775
diff changeset
   308
lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"
b257300c3a9c added Brian Hufmann's finite instances
nipkow
parents: 16775
diff changeset
   309
by (simp add: Plus_def)
b257300c3a9c added Brian Hufmann's finite instances
nipkow
parents: 16775
diff changeset
   310
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   311
text {* Sigma of finite sets *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   312
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   313
lemma finite_SigmaI [simp]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   314
    "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
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   315
  by (unfold Sigma_def) (blast intro!: finite_UN_I)
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lemma finite_cartesian_product: "[| finite A; finite B |] ==>
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   318
    finite (A <*> B)"
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   319
  by (rule finite_SigmaI)
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lemma finite_Prod_UNIV:
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   322
    "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
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   323
  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   324
   apply (erule ssubst)
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parents: 13825
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   325
   apply (erule finite_SigmaI, auto)
12396
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parents:
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   326
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
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   327
15409
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   328
lemma finite_cartesian_productD1:
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parents: 15402
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   329
     "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
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parents: 15402
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   330
apply (auto simp add: finite_conv_nat_seg_image) 
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parents: 15402
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   331
apply (drule_tac x=n in spec) 
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parents: 15402
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   332
apply (drule_tac x="fst o f" in spec) 
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parents: 15402
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   333
apply (auto simp add: o_def) 
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paulson
parents: 15402
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   334
 prefer 2 apply (force dest!: equalityD2) 
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paulson
parents: 15402
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   335
apply (drule equalityD1) 
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paulson
parents: 15402
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   336
apply (rename_tac y x)
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parents: 15402
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   337
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
   338
 prefer 2 apply force
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paulson
parents: 15402
diff changeset
   339
apply clarify
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paulson
parents: 15402
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   340
apply (rule_tac x=k in image_eqI, auto)
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paulson
parents: 15402
diff changeset
   341
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   342
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   343
lemma finite_cartesian_productD2:
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parents: 15402
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   344
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   345
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
   346
apply (drule_tac x=n in spec) 
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paulson
parents: 15402
diff changeset
   347
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
   348
apply (auto simp add: o_def) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   349
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   350
apply (drule equalityD1)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   351
apply (rename_tac x y)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   352
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
   353
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   354
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   355
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
   356
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   357
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   358
15392
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parents: 15376
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   359
text {* The powerset of a finite set *}
12396
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wenzelm
parents:
diff changeset
   360
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   361
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   362
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   363
  assume "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   364
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   365
  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
   366
next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   367
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   368
  thus "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   369
    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   370
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   371
15392
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   372
290bc97038c7 First step in reorganizing Finite_Set
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   373
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
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diff changeset
   374
by(blast intro: finite_subset[OF subset_Pow_Union])
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diff changeset
   375
290bc97038c7 First step in reorganizing Finite_Set
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diff changeset
   376
12396
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wenzelm
parents:
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   377
lemma finite_converse [iff]: "finite (r^-1) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   378
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   379
   apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   380
   apply (rule iffI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   381
    apply (erule finite_imageD [unfolded inj_on_def])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   382
    apply (simp split add: split_split)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   383
   apply (erule finite_imageI)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   384
  apply (simp add: converse_def image_def, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   385
  apply (rule bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   386
   prefer 2 apply assumption
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   387
  apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   388
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   389
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   390
15392
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nipkow
parents: 15376
diff changeset
   391
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   392
Ehmety) *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   393
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   394
lemma finite_Field: "finite r ==> finite (Field r)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   395
  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   396
  apply (induct set: Finites)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   397
   apply (auto simp add: Field_def Domain_insert Range_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   398
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   399
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   400
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   401
  apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   402
  apply (erule trancl_induct)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   403
   apply (auto simp add: Field_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   404
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   405
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   406
lemma finite_trancl: "finite (r^+) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   407
  apply auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   408
   prefer 2
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   409
   apply (rule trancl_subset_Field2 [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   410
   apply (rule finite_SigmaI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   411
    prefer 3
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 13595
diff changeset
   412
    apply (blast intro: r_into_trancl' finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   413
   apply (auto simp add: finite_Field)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   414
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   415
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   416
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   417
subsection {* A fold functional for finite sets *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   418
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   419
text {* The intended behaviour is
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   420
@{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<dots>)"}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   421
if @{text f} is associative-commutative. For an application of @{text fold}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   422
se the definitions of sums and products over finite sets.
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   423
*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   424
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   425
consts
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   426
  foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   427
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   428
inductive "foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   429
intros
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   430
emptyI [intro]: "({}, z) : foldSet f g z"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   431
insertI [intro]:
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   432
     "\<lbrakk> x \<notin> A; (A, y) : foldSet f g z \<rbrakk>
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   433
      \<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
   434
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   435
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   436
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   437
constdefs
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   438
  fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   439
  "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
   440
15498
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   441
text{*A tempting alternative for the definiens is
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   442
@{term "if finite A then THE x. (A, x) : foldSet f g e else e"}.
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   443
It allows the removal of finiteness assumptions from the theorems
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   444
@{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}.
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   445
The proofs become ugly, with @{text rule_format}. It is not worth the effort.*}
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   446
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   447
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   448
lemma Diff1_foldSet:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   449
  "(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
   450
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   451
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   452
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
   453
  by (induct set: foldSet) auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   454
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   455
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
   456
  by (induct set: Finites) auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   457
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
subsubsection {* Commutative monoids *}
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   460
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   461
locale ACf =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   462
  fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   463
  assumes commute: "x \<cdot> y = y \<cdot> x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   464
    and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   465
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   466
locale ACe = ACf +
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   467
  fixes e :: 'a
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   468
  assumes ident [simp]: "x \<cdot> e = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   469
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   470
locale ACIf = ACf +
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   471
  assumes idem: "x \<cdot> x = x"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   472
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   473
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
   474
proof -
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   475
  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
   476
  also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   477
  also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   478
  finally show ?thesis .
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   479
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   480
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   481
lemmas (in ACf) AC = assoc commute left_commute
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   482
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   483
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   484
proof -
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   485
  have "x \<cdot> e = x" by (rule ident)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   486
  thus ?thesis by (subst commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   487
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   488
15497
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   489
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
   490
proof -
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   491
  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
   492
  also have "\<dots> = x \<cdot> y" by(simp add:idem)
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   493
  finally show ?thesis .
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   494
qed
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   495
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   496
lemmas (in ACIf) ACI = AC idem idem2
53bca254719a Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents: 15487
diff changeset
   497
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   498
text{* Interpretation of locales: *}
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   499
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   500
interpretation AC_add: ACe ["op +" "0::'a::comm_monoid_add"]
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   501
by(auto intro: ACf.intro ACe_axioms.intro add_assoc add_commute)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   502
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   503
interpretation AC_mult: ACe ["op *" "1::'a::comm_monoid_mult"]
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   504
  apply -
15780
6744bba5561d Used locale interpretations everywhere.
nipkow
parents: 15770
diff changeset
   505
   apply (fast intro: ACf.intro mult_assoc mult_commute)
6744bba5561d Used locale interpretations everywhere.
nipkow
parents: 15770
diff changeset
   506
  apply (fastsimp intro: ACe_axioms.intro mult_assoc mult_commute)
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   507
  done
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   508
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   509
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   510
subsubsection{*From @{term foldSet} to @{term fold}*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   511
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   512
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   513
by (auto simp add: less_Suc_eq) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   514
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   515
lemma insert_image_inj_on_eq:
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   516
     "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   517
        inj_on h {i. i < Suc m}|] 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   518
      ==> A = h ` {i. i < m}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   519
apply (auto simp add: image_less_Suc inj_on_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   520
apply (blast intro: less_trans) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   521
done
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   522
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   523
lemma insert_inj_onE:
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   524
  assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   525
      and inj_on: "inj_on h {i::nat. i<n}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   526
  shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   527
proof (cases n)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   528
  case 0 thus ?thesis using aA by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   529
next
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   530
  case (Suc m)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   531
  have nSuc: "n = Suc m" . 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   532
  have mlessn: "m<n" by (simp add: nSuc)
15532
9712d41db5b8 simplified a proof
paulson
parents: 15526
diff changeset
   533
  from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   534
  let ?hm = "swap k m h"
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   535
  have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   536
    by (simp add: inj_on_swap_iff inj_on)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   537
  show ?thesis
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   538
  proof (intro exI conjI)
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   539
    show "inj_on ?hm {i. i < m}" using inj_hm
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   540
      by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   541
    show "m<n" by (rule mlessn)
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   542
    show "A = ?hm ` {i. i < m}" 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   543
    proof (rule insert_image_inj_on_eq)
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   544
      show "inj_on (swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   545
      show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   546
      show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   547
	using aA hkeq nSuc klessn
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   548
	by (auto simp add: swap_def image_less_Suc fun_upd_image 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   549
			   less_Suc_eq inj_on_image_set_diff [OF inj_on])
15479
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   550
    qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   551
  qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   552
qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   553
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   554
lemma (in ACf) foldSet_determ_aux:
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   555
  "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n}; 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   556
                (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
   557
   \<Longrightarrow> x' = x"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   558
proof (induct n rule: less_induct)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   559
  case (less n)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   560
    have IH: "!!m h A x x'. 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   561
               \<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m}; 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   562
                (A,x) \<in> foldSet f g z; (A, x') \<in> foldSet f g z\<rbrakk> \<Longrightarrow> x' = x" .
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   563
    have Afoldx: "(A,x) \<in> foldSet f g z" and Afoldx': "(A,x') \<in> foldSet f g z"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   564
     and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" .
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   565
    show ?case
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   566
    proof (rule foldSet.cases [OF Afoldx])
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   567
      assume "(A, x) = ({}, z)"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   568
      with Afoldx' show "x' = x" by blast
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   569
    next
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   570
      fix B b u
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   571
      assume "(A,x) = (insert b B, g b \<cdot> u)" and notinB: "b \<notin> B"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   572
         and Bu: "(B,u) \<in> foldSet f g z"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   573
      hence AbB: "A = insert b B" and x: "x = g b \<cdot> u" by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   574
      show "x'=x" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   575
      proof (rule foldSet.cases [OF Afoldx'])
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   576
        assume "(A, x') = ({}, z)"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   577
        with AbB show "x' = x" by blast
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   578
      next
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   579
	fix C c v
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   580
	assume "(A,x') = (insert c C, g c \<cdot> v)" and notinC: "c \<notin> C"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   581
	   and Cv: "(C,v) \<in> foldSet f g z"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   582
	hence AcC: "A = insert c C" and x': "x' = g c \<cdot> v" by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   583
	from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   584
        from insert_inj_onE [OF Beq notinB injh]
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   585
        obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   586
                     and Beq: "B = hB ` {i. i < mB}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   587
                     and lessB: "mB < n" by auto 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   588
	from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   589
        from insert_inj_onE [OF Ceq notinC injh]
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   590
        obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   591
                       and Ceq: "C = hC ` {i. i < mC}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   592
                       and lessC: "mC < n" by auto 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   593
	show "x'=x"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   594
	proof cases
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   595
          assume "b=c"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   596
	  then moreover have "B = C" using AbB AcC notinB notinC by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   597
	  ultimately show ?thesis  using Bu Cv x x' IH[OF lessC Ceq inj_onC]
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   598
            by auto
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   599
	next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   600
	  assume diff: "b \<noteq> c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   601
	  let ?D = "B - {c}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   602
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   603
	    using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   604
	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   605
	  with AbB have "finite ?D" by simp
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   606
	  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
   607
	    using finite_imp_foldSet by rules
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   608
	  moreover have cinB: "c \<in> B" using B by auto
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   609
	  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
   610
	    by(rule Diff1_foldSet)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   611
	  hence "g c \<cdot> d = u" by (rule IH [OF lessB Beq inj_onB Bu]) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   612
          moreover have "g b \<cdot> d = v"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   613
	  proof (rule IH[OF lessC Ceq inj_onC Cv])
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   614
	    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
   615
	      by fastsimp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   616
	  qed
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   617
	  ultimately show ?thesis using x x' by (auto simp: AC)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   618
	qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   619
      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   620
    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   621
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   622
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   623
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   624
lemma (in ACf) foldSet_determ:
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   625
  "(A,x) : foldSet f g z ==> (A,y) : foldSet f g z ==> y = x"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   626
apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   627
apply (blast intro: foldSet_determ_aux [rule_format])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   628
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   629
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   630
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
   631
  by (unfold fold_def) (blast intro: foldSet_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   632
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   633
text{* The base case for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   634
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   635
lemma fold_empty [simp]: "fold f g z {} = z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   636
  by (unfold fold_def) blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   637
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   638
lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   639
    ((insert x A, v) : foldSet f g z) =
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   640
    (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
   641
  apply auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   642
  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
   643
   apply (fastsimp dest: foldSet_imp_finite)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   644
  apply (blast intro: foldSet_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   645
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   646
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   647
text{* The recursion equation for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   648
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   649
lemma (in ACf) fold_insert[simp]:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   650
    "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
   651
  apply (unfold fold_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   652
  apply (simp add: fold_insert_aux)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   653
  apply (rule the_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   654
  apply (auto intro: finite_imp_foldSet
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   655
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   656
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   657
15535
nipkow
parents: 15532
diff changeset
   658
lemma (in ACf) fold_rec:
nipkow
parents: 15532
diff changeset
   659
assumes fin: "finite A" and a: "a:A"
nipkow
parents: 15532
diff changeset
   660
shows "fold f g z A = f (g a) (fold f g z (A - {a}))"
nipkow
parents: 15532
diff changeset
   661
proof-
nipkow
parents: 15532
diff changeset
   662
  have A: "A = insert a (A - {a})" using a by blast
nipkow
parents: 15532
diff changeset
   663
  hence "fold f g z A = fold f g z (insert a (A - {a}))" by simp
nipkow
parents: 15532
diff changeset
   664
  also have "\<dots> = f (g a) (fold f g z (A - {a}))"
nipkow
parents: 15532
diff changeset
   665
    by(rule fold_insert) (simp add:fin)+
nipkow
parents: 15532
diff changeset
   666
  finally show ?thesis .
nipkow
parents: 15532
diff changeset
   667
qed
nipkow
parents: 15532
diff changeset
   668
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   669
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   670
text{* A simplified version for idempotent functions: *}
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   671
15509
c54970704285 revised fold1 proofs
paulson
parents: 15508
diff changeset
   672
lemma (in ACIf) fold_insert_idem:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   673
assumes finA: "finite A"
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
   674
shows "fold f g z (insert a A) = g a \<cdot> fold f g z A"
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   675
proof cases
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   676
  assume "a \<in> A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   677
  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
   678
    by(blast dest: mk_disjoint_insert)
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   679
  show ?thesis
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   680
  proof -
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   681
    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
   682
    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
   683
    also have "\<dots> = (g a) \<cdot> (fold f g z B)"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   684
      using finB disj by simp
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   685
    also have "\<dots> = g a \<cdot> fold f g z A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   686
      using A finB disj by(simp add:idem assoc[symmetric])
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   687
    finally show ?thesis .
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   688
  qed
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   689
next
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   690
  assume "a \<notin> A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   691
  with finA show ?thesis by simp
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   692
qed
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   693
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   694
lemma (in ACIf) foldI_conv_id:
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   695
  "finite A \<Longrightarrow> fold f g z A = fold f id z (g ` A)"
15509
c54970704285 revised fold1 proofs
paulson
parents: 15508
diff changeset
   696
by(erule finite_induct)(simp_all add: fold_insert_idem del: fold_insert)
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   697
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   698
subsubsection{*Lemmas about @{text fold}*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   699
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   700
lemma (in ACf) fold_commute:
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   701
  "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
   702
  apply (induct set: Finites, simp)
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   703
  apply (simp add: left_commute [of x])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   704
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   705
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   706
lemma (in ACf) fold_nest_Un_Int:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   707
  "finite A ==> finite B
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   708
    ==> 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
   709
  apply (induct set: Finites, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   710
  apply (simp add: fold_commute Int_insert_left insert_absorb)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   711
  done
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) fold_nest_Un_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   714
  "finite A ==> finite B ==> A Int B = {}
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   715
    ==> 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
   716
  by (simp add: fold_nest_Un_Int)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   717
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   718
lemma (in ACf) fold_reindex:
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   719
assumes fin: "finite A"
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   720
shows "inj_on h A \<Longrightarrow> fold f g z (h ` A) = fold f (g \<circ> h) z A"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   721
using fin apply induct
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   722
 apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   723
apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   724
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   725
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   726
lemma (in ACe) fold_Un_Int:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   727
  "finite A ==> finite B ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   728
    fold f g e A \<cdot> fold f g e B =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   729
    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
   730
  apply (induct set: Finites, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   731
  apply (simp add: AC insert_absorb Int_insert_left)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   732
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   733
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   734
corollary (in ACe) fold_Un_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   735
  "finite A ==> finite B ==> A Int B = {} ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   736
    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
   737
  by (simp add: fold_Un_Int)
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 ACe) fold_UN_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   740
  "\<lbrakk> finite I; ALL i:I. finite (A i);
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   741
     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
   742
   \<Longrightarrow> fold f g e (UNION I A) =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   743
       fold f (%i. fold f g e (A i)) e I"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   744
  apply (induct set: Finites, simp, atomize)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   745
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   746
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   747
  apply (subgoal_tac "A x Int UNION F A = {}")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   748
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   749
  apply (simp add: fold_Un_disjoint)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   750
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   751
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   752
text{*Fusion theorem, as described in
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   753
Graham Hutton's paper,
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   754
A Tutorial on the Universality and Expressiveness of Fold,
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   755
JFP 9:4 (355-372), 1999.*}
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   756
lemma (in ACf) fold_fusion:
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   757
      includes ACf g
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   758
      shows
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   759
	"finite A ==> 
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   760
	 (!!x y. h (g x y) = f x (h y)) ==>
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   761
         h (fold g j w A) = fold f j (h w) A"
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   762
  by (induct set: Finites, simp_all)
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   763
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   764
lemma (in ACf) fold_cong:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   765
  "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
   766
  apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g z C = fold f h z C")
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   767
   apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   768
  apply (erule finite_induct, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   769
  apply (simp add: subset_insert_iff, clarify)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   770
  apply (subgoal_tac "finite C")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   771
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   772
  apply (subgoal_tac "C = insert x (C - {x})")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   773
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   774
  apply (erule ssubst)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   775
  apply (drule spec)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   776
  apply (erule (1) notE impE)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   777
  apply (simp add: Ball_def del: insert_Diff_single)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   778
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   779
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   780
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
   781
  fold f (%x. fold f (g x) e (B x)) e A =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   782
  fold f (split g) e (SIGMA x:A. B x)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   783
apply (subst Sigma_def)
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   784
apply (subst fold_UN_disjoint, assumption, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   785
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   786
apply (erule fold_cong)
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   787
apply (subst fold_UN_disjoint, simp, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   788
 apply blast
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   789
apply simp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   790
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   791
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   792
lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   793
   fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   794
apply (erule finite_induct, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   795
apply (simp add:AC)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   796
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   797
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   798
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   799
subsection {* Generalized summation over a set *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   800
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   801
constdefs
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   802
  setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   803
  "setsum f A == if finite A then fold (op +) f 0 A else 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   804
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   805
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   806
written @{text"\<Sum>x\<in>A. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   807
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   808
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   809
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   810
syntax (xsymbols)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   811
  "_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
   812
syntax (HTML output)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   813
  "_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
   814
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   815
translations -- {* Beware of argument permutation! *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   816
  "SUM i:A. b" == "setsum (%i. b) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   817
  "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   818
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   819
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   820
 @{text"\<Sum>x|P. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   821
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   822
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   823
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   824
syntax (xsymbols)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   825
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   826
syntax (HTML output)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   827
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   828
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   829
translations
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   830
  "SUM x|P. t" => "setsum (%x. t) {x. P}"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   831
  "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   832
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   833
text{* Finally we abbreviate @{term"\<Sum>x\<in>A. x"} by @{text"\<Sum>A"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   834
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   835
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   836
  "_Setsum" :: "'a set => 'a::comm_monoid_mult"  ("\<Sum>_" [1000] 999)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   837
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   838
parse_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   839
  let
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   840
    fun Setsum_tr [A] = Syntax.const "setsum" $ Abs ("", dummyT, Bound 0) $ A
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   841
  in [("_Setsum", Setsum_tr)] end;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   842
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   843
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   844
print_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   845
let
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   846
  fun setsum_tr' [Abs(_,_,Bound 0), A] = Syntax.const "_Setsum" $ A
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   847
    | setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   848
       if x<>y then raise Match
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   849
       else let val x' = Syntax.mark_bound x
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   850
                val t' = subst_bound(x',t)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   851
                val P' = subst_bound(x',P)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   852
            in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   853
in
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   854
[("setsum", setsum_tr')]
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   855
end
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   856
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   857
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   858
lemma setsum_empty [simp]: "setsum f {} = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   859
  by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   860
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   861
lemma setsum_insert [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   862
    "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   863
  by (simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   864
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   865
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
   866
  by (simp add: setsum_def)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   867
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   868
lemma setsum_reindex:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   869
     "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   870
by(auto simp add: setsum_def AC_add.fold_reindex dest!:finite_imageD)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   871
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   872
lemma setsum_reindex_id:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   873
     "inj_on f B ==> setsum f B = setsum id (f ` B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   874
by (auto simp add: setsum_reindex)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   875
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   876
lemma setsum_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   877
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   878
by(fastsimp simp: setsum_def intro: AC_add.fold_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   879
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16632
diff changeset
   880
lemma strong_setsum_cong[cong]:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16632
diff changeset
   881
  "A = B ==> (!!x. x:B =simp=> f x = g x)
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16632
diff changeset
   882
   ==> setsum (%x. f x) A = setsum (%x. g x) B"
16632
ad2895beef79 Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents: 16550
diff changeset
   883
by(fastsimp simp: simp_implies_def setsum_def intro: AC_add.fold_cong)
ad2895beef79 Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents: 16550
diff changeset
   884
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
   885
lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A";
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
   886
  by (rule setsum_cong[OF refl], auto);
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
   887
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   888
lemma setsum_reindex_cong:
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
   889
     "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] 
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   890
      ==> setsum h B = setsum g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   891
  by (simp add: setsum_reindex cong: setsum_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   892
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   893
lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   894
apply (clarsimp simp: setsum_def)
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   895
apply (erule finite_induct, auto)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   896
done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   897
15543
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   898
lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   899
by(simp add:setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   900
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   901
lemma setsum_Un_Int: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   902
  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
   903
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   904
by(simp add: setsum_def AC_add.fold_Un_Int [symmetric])
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   905
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   906
lemma setsum_Un_disjoint: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   907
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   908
by (subst setsum_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   909
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   910
(*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
   911
  the lhs need not be, since UNION I A could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   912
lemma setsum_UN_disjoint:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   913
    "finite I ==> (ALL i:I. finite (A i)) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   914
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   915
      setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   916
by(simp add: setsum_def AC_add.fold_UN_disjoint cong: setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   917
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   918
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
   919
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
   920
lemma setsum_Union_disjoint:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   921
  "[| (ALL A:C. finite A);
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   922
      (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
   923
   ==> setsum f (Union C) = setsum (setsum f) C"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   924
apply (cases "finite C") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   925
 prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   926
  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
   927
 apply (unfold Union_def id_def, assumption+)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   928
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   929
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   930
(*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
   931
  the rhs need not be, since SIGMA A B could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   932
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   933
    (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   934
    (\<Sum>z\<in>(SIGMA x:A. B x). f (fst z) (snd z))"
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   935
by(simp add:setsum_def AC_add.fold_Sigma split_def cong:setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   936
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   937
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
   938
lemma setsum_cartesian_product: 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   939
   "(\<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
   940
apply (cases "finite A") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   941
 apply (cases "finite B") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   942
  apply (simp add: setsum_Sigma)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   943
 apply (cases "A={}", simp)
15543
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   944
 apply (simp) 
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   945
apply (auto simp add: setsum_def
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   946
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   947
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   948
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   949
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   950
by(simp add:setsum_def AC_add.fold_distrib)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   951
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   952
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   953
subsubsection {* Properties in more restricted classes of structures *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   954
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   955
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   956
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   957
   prefer 2 apply (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   958
  apply (erule rev_mp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   959
  apply (erule finite_induct, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   960
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   961
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   962
lemma setsum_eq_0_iff [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   963
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   964
  by (induct set: Finites) auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   965
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   966
lemma setsum_Un_nat: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   967
    (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
   968
  -- {* For the natural numbers, we have subtraction. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   969
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   970
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   971
lemma setsum_Un: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   972
    (setsum f (A Un B) :: 'a :: ab_group_add) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   973
      setsum f A + setsum f B - setsum f (A Int B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   974
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   975
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   976
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   977
    (if a:A then setsum f A - f a else setsum f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   978
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   979
   prefer 2 apply (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   980
  apply (erule finite_induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   981
   apply (auto simp add: insert_Diff_if)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   982
  apply (drule_tac a = a in mk_disjoint_insert, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   983
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   984
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   985
lemma setsum_diff1: "finite A \<Longrightarrow>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   986
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   987
  (if a:A then setsum f A - f a else setsum f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   988
  by (erule finite_induct) (auto simp add: insert_Diff_if)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   989
15552
8ab8e425410b added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents: 15543
diff changeset
   990
lemma setsum_diff1'[rule_format]: "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
8ab8e425410b added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents: 15543
diff changeset
   991
  apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
8ab8e425410b added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents: 15543
diff changeset
   992
  apply (auto simp add: insert_Diff_if add_ac)
8ab8e425410b added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents: 15543
diff changeset
   993
  done
8ab8e425410b added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents: 15543
diff changeset
   994
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   995
(* By Jeremy Siek: *)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   996
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   997
lemma setsum_diff_nat: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   998
  assumes finB: "finite B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   999
  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
  1000
using finB
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1001
proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1002
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1003
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1004
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1005
    and xFinA: "insert x F \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1006
    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
  1007
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1008
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1009
    by (simp add: setsum_diff1_nat)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1010
  from xFinA have "F \<subseteq> A" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1011
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1012
  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
  1013
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1014
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1015
  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
  1016
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1017
  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
  1018
  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
  1019
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1020
  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
  1021
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1022
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1023
lemma setsum_diff:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1024
  assumes le: "finite A" "B \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1025
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1026
proof -
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1027
  from le have finiteB: "finite B" using finite_subset by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1028
  show ?thesis using finiteB le
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1029
    proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1030
      case empty
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1031
      thus ?case by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1032
    next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1033
      case (insert x F)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1034
      thus ?case using le finiteB 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1035
	by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1036
    qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1037
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1038
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1039
lemma setsum_mono:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1040
  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
  1041
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1042
proof (cases "finite K")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1043
  case True
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1044
  thus ?thesis using le
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1045
  proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1046
    case empty
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1047
    thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1048
  next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1049
    case insert
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1050
    thus ?case using add_mono 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1051
      by force
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1052
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1053
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1054
  case False
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1055
  thus ?thesis
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1056
    by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1057
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1058
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1059
lemma setsum_strict_mono:
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1060
fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1061
assumes fin_ne: "finite A"  "A \<noteq> {}"
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1062
shows "(!!x. x:A \<Longrightarrow> f x < g x) \<Longrightarrow> setsum f A < setsum g A"
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1063
using fin_ne
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1064
proof (induct rule: finite_ne_induct)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1065
  case singleton thus ?case by simp
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1066
next
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1067
  case insert thus ?case by (auto simp: add_strict_mono)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1068
qed
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1069
15535
nipkow
parents: 15532
diff changeset
  1070
lemma setsum_negf:
nipkow
parents: 15532
diff changeset
  1071
 "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
nipkow
parents: 15532
diff changeset
  1072
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1073
  case True thus ?thesis by (induct set: Finites, auto)
nipkow
parents: 15532
diff changeset
  1074
next
nipkow
parents: 15532
diff changeset
  1075
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1076
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1077
15535
nipkow
parents: 15532
diff changeset
  1078
lemma setsum_subtractf:
nipkow
parents: 15532
diff changeset
  1079
 "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1080
  setsum f A - setsum g A"
15535
nipkow
parents: 15532
diff changeset
  1081
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1082
  case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
nipkow
parents: 15532
diff changeset
  1083
next
nipkow
parents: 15532
diff changeset
  1084
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1085
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1086
15535
nipkow
parents: 15532
diff changeset
  1087
lemma setsum_nonneg:
nipkow
parents: 15532
diff changeset
  1088
assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
nipkow
parents: 15532
diff changeset
  1089
shows "0 \<le> setsum f A"
nipkow
parents: 15532
diff changeset
  1090
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1091
  case True thus ?thesis using nn
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1092
  apply (induct set: Finites, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1093
  apply (subgoal_tac "0 + 0 \<le> f x + setsum f F", simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1094
  apply (blast intro: add_mono)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1095
  done
15535
nipkow
parents: 15532
diff changeset
  1096
next
nipkow
parents: 15532
diff changeset
  1097
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1098
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1099
15535
nipkow
parents: 15532
diff changeset
  1100
lemma setsum_nonpos:
nipkow
parents: 15532
diff changeset
  1101
assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
nipkow
parents: 15532
diff changeset
  1102
shows "setsum f A \<le> 0"
nipkow
parents: 15532
diff changeset
  1103
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1104
  case True thus ?thesis using np
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1105
  apply (induct set: Finites, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1106
  apply (subgoal_tac "f x + setsum f F \<le> 0 + 0", simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1107
  apply (blast intro: add_mono)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1108
  done
15535
nipkow
parents: 15532
diff changeset
  1109
next
nipkow
parents: 15532
diff changeset
  1110
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1111
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1112
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1113
lemma setsum_mono2:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1114
fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1115
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1116
shows "setsum f A \<le> setsum f B"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1117
proof -
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1118
  have "setsum f A \<le> setsum f A + setsum f (B-A)"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1119
    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1120
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1121
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1122
  also have "A \<union> (B-A) = B" using sub by blast
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1123
  finally show ?thesis .
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1124
qed
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1125
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1126
lemma setsum_mono3: "finite B ==> A <= B ==> 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1127
    ALL x: B - A. 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1128
      0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==>
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1129
        setsum f A <= setsum f B"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1130
  apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1131
  apply (erule ssubst)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1132
  apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1133
  apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1134
  apply (rule add_left_mono)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1135
  apply (erule setsum_nonneg)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1136
  apply (subst setsum_Un_disjoint [THEN sym])
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1137
  apply (erule finite_subset, assumption)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1138
  apply (rule finite_subset)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1139
  prefer 2
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1140
  apply assumption
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1141
  apply auto
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1142
  apply (rule setsum_cong)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1143
  apply auto
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1144
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1145
15837
7a567dcd4cda Subsumption of locale interpretations.
ballarin
parents: 15791
diff changeset
  1146
(* FIXME: this is distributitivty, name as such! *)
7a567dcd4cda Subsumption of locale interpretations.
ballarin
parents: 15791
diff changeset
  1147
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1148
lemma setsum_mult: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1149
  fixes f :: "'a => ('b::semiring_0_cancel)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1150
  shows "r * setsum f A = setsum (%n. r * f n) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1151
proof (cases "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1152
  case True
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1153
  thus ?thesis
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1154
  proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1155
    case empty thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1156
  next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1157
    case (insert x A) thus ?case by (simp add: right_distrib)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1158
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1159
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1160
  case False thus ?thesis by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1161
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1162
15535
nipkow
parents: 15532
diff changeset
  1163
lemma setsum_abs[iff]: 
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1164
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1165
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
15535
nipkow
parents: 15532
diff changeset
  1166
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1167
  case True
nipkow
parents: 15532
diff changeset
  1168
  thus ?thesis
nipkow
parents: 15532
diff changeset
  1169
  proof (induct)
nipkow
parents: 15532
diff changeset
  1170
    case empty thus ?case by simp
nipkow
parents: 15532
diff changeset
  1171
  next
nipkow
parents: 15532
diff changeset
  1172
    case (insert x A)
nipkow
parents: 15532
diff changeset
  1173
    thus ?case by (auto intro: abs_triangle_ineq order_trans)
nipkow
parents: 15532
diff changeset
  1174
  qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1175
next
15535
nipkow
parents: 15532
diff changeset
  1176
  case False thus ?thesis by (simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1177
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1178
15535
nipkow
parents: 15532
diff changeset
  1179
lemma setsum_abs_ge_zero[iff]: 
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1180
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1181
  shows "0 \<le> setsum (%i. abs(f i)) A"
15535
nipkow
parents: 15532
diff changeset
  1182
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1183
  case True
nipkow
parents: 15532
diff changeset
  1184
  thus ?thesis
nipkow
parents: 15532
diff changeset
  1185
  proof (induct)
nipkow
parents: 15532
diff changeset
  1186
    case empty thus ?case by simp
nipkow
parents: 15532
diff changeset
  1187
  next
nipkow
parents: 15532
diff changeset
  1188
    case (insert x A) thus ?case by (auto intro: order_trans)
nipkow
parents: 15532
diff changeset
  1189
  qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1190
next
15535
nipkow
parents: 15532
diff changeset
  1191
  case False thus ?thesis by (simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1192
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1193
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1194
lemma abs_setsum_abs[simp]: 
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1195
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1196
  shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1197
proof (cases "finite A")
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1198
  case True
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1199
  thus ?thesis
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1200
  proof (induct)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1201
    case empty thus ?case by simp
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1202
  next
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1203
    case (insert a A)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1204
    hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1205
    also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1206
    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1207
      by (simp del: abs_of_nonneg)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1208
    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1209
    finally show ?case .
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1210
  qed
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1211
next
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1212
  case False thus ?thesis by (simp add: setsum_def)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1213
qed
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1214
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1215
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1216
subsection {* Generalized product over a set *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1217
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1218
constdefs
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1219
  setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1220
  "setprod f A == if finite A then fold (op *) f 1 A else 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1221
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1222
syntax
16550
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1223
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1224
syntax (xsymbols)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1225
  "_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
  1226
syntax (HTML output)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1227
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
16550
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1228
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1229
translations -- {* Beware of argument permutation! *}
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1230
  "PROD i:A. b" == "setprod (%i. b) A" 
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1231
  "\<Prod>i\<in>A. b" == "setprod (%i. b) A" 
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1232
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1233
text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1234
 @{text"\<Prod>x|P. e"}. *}
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1235
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1236
syntax
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1237
  "_qsetprod" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1238
syntax (xsymbols)
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1239
  "_qsetprod" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1240
syntax (HTML output)
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1241
  "_qsetprod" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1242
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1243
translations
16550
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1244
  "PROD x|P. t" => "setprod (%x. t) {x. P}"
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1245
  "\<Prod>x|P. t" => "setprod (%x. t) {x. P}"
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1246
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1247
text{* Finally we abbreviate @{term"\<Prod>x\<in>A. x"} by @{text"\<Prod>A"}. *}
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1248
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1249
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1250
  "_Setprod" :: "'a set => 'a::comm_monoid_mult"  ("\<Prod>_" [1000] 999)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1251
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1252
parse_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1253
  let
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1254
    fun Setprod_tr [A] = Syntax.const "setprod" $ Abs ("", dummyT, Bound 0) $ A
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1255
  in [("_Setprod", Setprod_tr)] end;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1256
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1257
print_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1258
let fun setprod_tr' [Abs(x,Tx,t), A] =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1259
    if t = Bound 0 then Syntax.const "_Setprod" $ A else raise Match
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1260
in
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1261
[("setprod", setprod_tr')]
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1262
end
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1263
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1264
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1265
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1266
lemma setprod_empty [simp]: "setprod f {} = 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1267
  by (auto simp add: setprod_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1268
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1269
lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1270
    setprod f (insert a A) = f a * setprod f A"
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
  1271
by (simp add: setprod_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1272
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1273
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
  1274
  by (simp add: setprod_def)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1275
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1276
lemma setprod_reindex:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1277
     "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
  1278
by(auto simp: setprod_def AC_mult.fold_reindex dest!:finite_imageD)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1279
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1280
lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1281
by (auto simp add: setprod_reindex)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1282
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1283
lemma setprod_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1284
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
  1285
by(fastsimp simp: setprod_def intro: AC_mult.fold_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1286
16632
ad2895beef79 Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents: 16550
diff changeset
  1287
lemma strong_setprod_cong:
ad2895beef79 Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents: 16550
diff changeset
  1288
  "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
ad2895beef79 Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents: 16550
diff changeset
  1289
by(fastsimp simp: simp_implies_def setprod_def intro: AC_mult.fold_cong)
ad2895beef79 Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents: 16550
diff changeset
  1290
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1291
lemma setprod_reindex_cong: "inj_on f A ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1292
    B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1293
  by (frule setprod_reindex, simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1294
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1295
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1296
lemma setprod_1: "setprod (%i. 1) A = 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1297
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1298
  apply (erule finite_induct, auto simp add: mult_ac)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1299
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1300
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1301
lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1302
  apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1303
  apply (erule ssubst, rule setprod_1)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1304
  apply (rule setprod_cong, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1305
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1306
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1307
lemma setprod_Un_Int: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1308
    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
  1309
by(simp add: setprod_def AC_mult.fold_Un_Int[symmetric])
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1310
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1311
lemma setprod_Un_disjoint: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1312
  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1313
by (subst setprod_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1314
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1315
lemma setprod_UN_disjoint:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1316
    "finite I ==> (ALL i:I. finite (A i)) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1317
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1318
      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
  1319
by(simp add: setprod_def AC_mult.fold_UN_disjoint cong: setprod_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1320
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1321
lemma setprod_Union_disjoint:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1322
  "[| (ALL A:C. finite A);
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1323
      (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
  1324
   ==> setprod f (Union C) = setprod (setprod f) C"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1325
apply (cases "finite C") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1326
 prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1327
  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
  1328
 apply (unfold Union_def id_def, assumption+)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1329
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1330
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1331
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
16550
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1332
    (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1333
    (\<Prod>z\<in>(SIGMA x:A. B x). f (fst z) (snd z))"
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
  1334
by(simp add:setprod_def AC_mult.fold_Sigma split_def cong:setprod_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1335
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1336
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
  1337
lemma setprod_cartesian_product: 
16550
e14b89d6ef13 fixed \<Prod> syntax
nipkow
parents: 15837
diff changeset
  1338
     "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>z\<in>(A <*> B). f (fst z) (snd z))"
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1339
apply (cases "finite A") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1340
 apply (cases "finite B") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1341
  apply (simp add: setprod_Sigma)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1342
 apply (cases "A={}", simp)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1343
 apply (simp add: setprod_1) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1344
apply (auto simp add: setprod_def
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1345
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1346
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1347
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1348
lemma setprod_timesf:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1349
     "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
  1350
by(simp add:setprod_def AC_mult.fold_distrib)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1351
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1352
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1353
subsubsection {* Properties in more restricted classes of structures *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1354
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1355
lemma setprod_eq_1_iff [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1356
    "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
</