src/HOL/Finite_Set.thy
author huffman
Tue, 01 Jul 2008 20:26:48 +0200
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parent 27418 564117b58d73
child 27611 2c01c0bdb385
permissions -rw-r--r--
prove lemma finite in context of finite class
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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 Divides Transitive_Closure
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begin
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subsection {* Definition and basic properties *}
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inductive finite :: "'a set => bool"
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  where
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    emptyI [simp, intro!]: "finite {}"
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  | insertI [simp, intro!]: "finite A ==> finite (insert a A)"
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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: finite]:
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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 {}" by fact
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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 = {}"
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    thus ?thesis using `P {x}` by simp
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  next
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    assume "F \<noteq> {}"
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    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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  assumes "finite F" and "F \<subseteq> A"
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    and empty: "P {}"
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    and insert: "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
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  shows "P F"
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proof -
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  from `finite F` and `F \<subseteq> A`
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  show ?thesis
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  proof induct
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    show "P {}" by fact
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  next
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    fix x F
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    assume "finite F" and "x \<notin> F" and
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      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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      show "finite F" by fact
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      show "x \<notin> F" by fact
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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" by fact
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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: finite) simp_all
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lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
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  -- {* Every subset of a finite set is finite. *}
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proof -
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  assume "finite B"
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  thus "!!A. A \<subseteq> B ==> finite A"
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  proof induct
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    case empty
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    thus ?case by simp
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  next
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    case (insert x F A)
23389
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   161
    have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" by fact+
12396
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parents:
diff changeset
   162
    show "finite A"
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wenzelm
parents:
diff changeset
   163
    proof cases
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   164
      assume x: "x \<in> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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diff changeset
   165
      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
   166
      with r have "finite (A - {x})" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   167
      hence "finite (insert x (A - {x}))" ..
23389
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diff changeset
   168
      also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
12396
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diff changeset
   169
      finally show ?thesis .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
diff changeset
   170
    next
23389
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   171
      show "A \<subseteq> F ==> ?thesis" by fact
12396
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   172
      assume "x \<notin> A"
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wenzelm
parents:
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   173
      with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
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wenzelm
parents:
diff changeset
   174
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   175
  qed
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
18423
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nipkow
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   178
lemma finite_Collect_subset[simp]: "finite A \<Longrightarrow> finite{x \<in> A. P x}"
17761
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   179
using finite_subset[of "{x \<in> A. P x}" "A"] by blast
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nipkow
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diff changeset
   180
12396
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   181
lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
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wenzelm
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   182
  by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
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wenzelm
parents:
diff changeset
   183
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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diff changeset
   184
lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
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   185
  -- {* The converse obviously fails. *}
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wenzelm
parents:
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   186
  by (blast intro: finite_subset)
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wenzelm
parents:
diff changeset
   187
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   188
lemma finite_insert [simp]: "finite (insert a A) = finite A"
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wenzelm
parents:
diff changeset
   189
  apply (subst insert_is_Un)
14208
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paulson
parents: 13825
diff changeset
   190
  apply (simp only: finite_Un, blast)
12396
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wenzelm
parents:
diff changeset
   191
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   192
15281
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nipkow
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diff changeset
   193
lemma finite_Union[simp, intro]:
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nipkow
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diff changeset
   194
 "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
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nipkow
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diff changeset
   195
by (induct rule:finite_induct) simp_all
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nipkow
parents: 15234
diff changeset
   196
12396
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diff changeset
   197
lemma finite_empty_induct:
23389
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diff changeset
   198
  assumes "finite A"
aaca6a8e5414 tuned proofs: avoid implicit prems;
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diff changeset
   199
    and "P A"
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diff changeset
   200
    and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   201
  shows "P {}"
12396
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wenzelm
parents:
diff changeset
   202
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   203
  have "P (A - A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   204
  proof -
23389
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parents: 23277
diff changeset
   205
    {
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   206
      fix c b :: "'a set"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   207
      assume c: "finite c" and b: "finite b"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   208
	and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   209
      have "c \<subseteq> b ==> P (b - c)"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   210
	using c
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   211
      proof induct
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
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diff changeset
   212
	case empty
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   213
	from P1 show ?case by simp
aaca6a8e5414 tuned proofs: avoid implicit prems;
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parents: 23277
diff changeset
   214
      next
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   215
	case (insert x F)
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   216
	have "P (b - F - {x})"
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wenzelm
parents: 23277
diff changeset
   217
	proof (rule P2)
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   218
          from _ b show "finite (b - F)" by (rule finite_subset) blast
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   219
          from insert show "x \<in> b - F" by simp
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   220
          from insert show "P (b - F)" by simp
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   221
	qed
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   222
	also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
aaca6a8e5414 tuned proofs: avoid implicit prems;
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diff changeset
   223
	finally show ?case .
12396
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diff changeset
   224
      qed
23389
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parents: 23277
diff changeset
   225
    }
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wenzelm
parents: 23277
diff changeset
   226
    then show ?thesis by this (simp_all add: assms)
12396
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diff changeset
   227
  qed
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
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parents: 23277
diff changeset
   228
  then show ?thesis by simp
12396
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wenzelm
parents:
diff changeset
   229
qed
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 [simp]: "finite B ==> finite (B - Ba)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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diff changeset
   232
  by (rule Diff_subset [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   233
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   234
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
   235
  apply (subst Diff_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   236
  apply (case_tac "a : A - B")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   237
   apply (rule finite_insert [symmetric, THEN trans])
14208
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paulson
parents: 13825
diff changeset
   238
   apply (subst insert_Diff, simp_all)
12396
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wenzelm
parents:
diff changeset
   239
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   240
19870
ef037d1b32d1 new results
paulson
parents: 19868
diff changeset
   241
lemma finite_Diff_singleton [simp]: "finite (A - {a}) = finite A"
ef037d1b32d1 new results
paulson
parents: 19868
diff changeset
   242
  by simp
ef037d1b32d1 new results
paulson
parents: 19868
diff changeset
   243
12396
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wenzelm
parents:
diff changeset
   244
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   245
text {* Image and Inverse Image over Finite Sets *}
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   246
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   247
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
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parents: 13737
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   248
  -- {* The image of a finite set is finite. *}
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berghofe
parents: 21733
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   249
  by (induct set: finite) simp_all
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   250
14430
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paulson
parents: 14331
diff changeset
   251
lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
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paulson
parents: 14331
diff changeset
   252
  apply (frule finite_imageI)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   253
  apply (erule finite_subset, assumption)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   254
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   255
13825
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paulson
parents: 13737
diff changeset
   256
lemma finite_range_imageI:
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   257
    "finite (range g) ==> finite (range (%x. f (g x)))"
27418
564117b58d73 remove simp attribute from range_composition
huffman
parents: 27165
diff changeset
   258
  apply (drule finite_imageI, simp add: range_composition)
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   259
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   260
12396
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wenzelm
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diff changeset
   261
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
   262
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   263
  have aux: "!!A. finite (A - {}) = finite A" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   264
  fix B :: "'a set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   265
  assume "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   266
  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   267
    apply induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   268
     apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   269
    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
   270
     apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   271
     apply (simp (no_asm_use) add: inj_on_def)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   272
     apply (blast dest!: aux [THEN iffD1], atomize)
12396
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wenzelm
parents:
diff changeset
   273
    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   274
    apply (frule subsetD [OF equalityD2 insertI1], clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   275
    apply (rule_tac x = xa in bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   276
     apply (simp_all add: inj_on_image_set_diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   277
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   278
qed (rule refl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   279
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   280
13825
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paulson
parents: 13737
diff changeset
   281
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   282
  -- {* The inverse image of a singleton under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   283
         is included in a singleton. *}
14430
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paulson
parents: 14331
diff changeset
   284
  apply (auto simp add: inj_on_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   285
  apply (blast intro: the_equality [symmetric])
13825
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paulson
parents: 13737
diff changeset
   286
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   287
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   288
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   289
  -- {* The inverse image of a finite set under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   290
         is finite. *}
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   291
  apply (induct set: finite)
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
   292
   apply simp_all
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   293
  apply (subst vimage_insert)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   294
  apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   295
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   296
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   297
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   298
text {* The finite UNION of finite sets *}
12396
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wenzelm
parents:
diff changeset
   299
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   300
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   301
  by (induct set: finite) simp_all
12396
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wenzelm
parents:
diff changeset
   302
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   303
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   304
  Strengthen RHS to
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   305
  @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
12396
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wenzelm
parents:
diff changeset
   306
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   307
  We'd need to prove
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   308
  @{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
   309
  by induction. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   310
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   311
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
   312
  by (blast intro: finite_UN_I finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   313
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   314
17022
b257300c3a9c added Brian Hufmann's finite instances
nipkow
parents: 16775
diff changeset
   315
lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"
b257300c3a9c added Brian Hufmann's finite instances
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by (simp add: Plus_def)
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text {* Sigma of finite sets *}
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   320
lemma finite_SigmaI [simp]:
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   321
    "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   322
  by (unfold Sigma_def) (blast intro!: finite_UN_I)
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diff changeset
   323
15402
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   324
lemma finite_cartesian_product: "[| finite A; finite B |] ==>
97204f3b4705 REorganized Finite_Set
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   325
    finite (A <*> B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
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   326
  by (rule finite_SigmaI)
97204f3b4705 REorganized Finite_Set
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   327
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   328
lemma finite_Prod_UNIV:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
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   329
    "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:
diff changeset
   330
  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
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parents:
diff changeset
   331
   apply (erule ssubst)
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diff changeset
   332
   apply (erule finite_SigmaI, auto)
12396
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wenzelm
parents:
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   333
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
diff changeset
   334
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a063687d24eb new and stronger lemmas and improved simplification for finite sets
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   335
lemma finite_cartesian_productD1:
a063687d24eb new and stronger lemmas and improved simplification for finite sets
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   336
     "[| 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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   337
apply (auto simp add: finite_conv_nat_seg_image) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
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   338
apply (drule_tac x=n in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
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   339
apply (drule_tac x="fst o f" in spec) 
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parents: 15402
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   340
apply (auto simp add: o_def) 
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paulson
parents: 15402
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   341
 prefer 2 apply (force dest!: equalityD2) 
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paulson
parents: 15402
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   342
apply (drule equalityD1) 
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paulson
parents: 15402
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   343
apply (rename_tac y x)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
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parents: 15402
diff changeset
   344
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
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   345
 prefer 2 apply force
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paulson
parents: 15402
diff changeset
   346
apply clarify
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parents: 15402
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   347
apply (rule_tac x=k in image_eqI, auto)
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parents: 15402
diff changeset
   348
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
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parents: 15402
diff changeset
   349
a063687d24eb new and stronger lemmas and improved simplification for finite sets
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parents: 15402
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   350
lemma finite_cartesian_productD2:
a063687d24eb new and stronger lemmas and improved simplification for finite sets
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   351
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   352
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
   353
apply (drule_tac x=n in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   354
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
   355
apply (auto simp add: o_def) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   356
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   357
apply (drule equalityD1)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   358
apply (rename_tac x y)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   359
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
   360
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   361
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   362
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
   363
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   364
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   365
15392
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parents: 15376
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   366
text {* The powerset of a finite set *}
12396
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parents:
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   367
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   368
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   369
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   370
  assume "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   371
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
diff changeset
   372
  thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
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wenzelm
parents:
diff changeset
   373
next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   374
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   375
  thus "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   376
    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   377
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
diff changeset
   378
15392
290bc97038c7 First step in reorganizing Finite_Set
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parents: 15376
diff changeset
   379
290bc97038c7 First step in reorganizing Finite_Set
nipkow
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   380
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   381
by(blast intro: finite_subset[OF subset_Pow_Union])
290bc97038c7 First step in reorganizing Finite_Set
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parents: 15376
diff changeset
   382
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   383
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   384
lemma finite_converse [iff]: "finite (r^-1) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   385
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   386
   apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   387
   apply (rule iffI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   388
    apply (erule finite_imageD [unfolded inj_on_def])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   389
    apply (simp split add: split_split)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   390
   apply (erule finite_imageI)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   391
  apply (simp add: converse_def image_def, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   392
  apply (rule bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   393
   prefer 2 apply assumption
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   394
  apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   395
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   396
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   397
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   398
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   399
Ehmety) *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   400
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   401
lemma finite_Field: "finite r ==> finite (Field r)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   402
  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   403
  apply (induct set: finite)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   404
   apply (auto simp add: Field_def Domain_insert Range_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   405
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   406
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   407
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   408
  apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   409
  apply (erule trancl_induct)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   410
   apply (auto simp add: Field_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   411
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   412
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   413
lemma finite_trancl: "finite (r^+) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   414
  apply auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   415
   prefer 2
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   416
   apply (rule trancl_subset_Field2 [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   417
   apply (rule finite_SigmaI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   418
    prefer 3
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 13595
diff changeset
   419
    apply (blast intro: r_into_trancl' finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   420
   apply (auto simp add: finite_Field)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   421
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   422
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   423
26441
7914697ff104 no "attach UNIV" any more
haftmann
parents: 26146
diff changeset
   424
subsection {* Class @{text finite}  *}
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   425
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   426
setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*}
26441
7914697ff104 no "attach UNIV" any more
haftmann
parents: 26146
diff changeset
   427
class finite = itself +
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   428
  assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   429
setup {* Sign.parent_path *}
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   430
hide const finite
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   431
27430
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   432
context finite
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   433
begin
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   434
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   435
lemma finite [simp]: "finite (A \<Colon> 'a set)"
26441
7914697ff104 no "attach UNIV" any more
haftmann
parents: 26146
diff changeset
   436
  by (rule subset_UNIV finite_UNIV finite_subset)+
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   437
27430
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   438
end
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   439
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   440
lemma UNIV_unit [noatp]:
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   441
  "UNIV = {()}" by auto
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   442
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   443
instance unit :: finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   444
  by default (simp add: UNIV_unit)
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   445
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   446
lemma UNIV_bool [noatp]:
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   447
  "UNIV = {False, True}" by auto
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   448
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   449
instance bool :: finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   450
  by default (simp add: UNIV_bool)
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   451
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   452
instance * :: (finite, finite) finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   453
  by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   454
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   455
instance "+" :: (finite, finite) finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   456
  by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   457
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   458
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   459
  by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   460
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   461
instance "fun" :: (finite, finite) finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   462
proof
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   463
  show "finite (UNIV :: ('a => 'b) set)"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   464
  proof (rule finite_imageD)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   465
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
26792
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   466
    have "range ?graph \<subseteq> Pow UNIV" by simp
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   467
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   468
      by (simp only: finite_Pow_iff finite)
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   469
    ultimately show "finite (range ?graph)"
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   470
      by (rule finite_subset)
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   471
    show "inj ?graph" by (rule inj_graph)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   472
  qed
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   473
qed
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   474
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   475
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   476
subsection {* A fold functional for finite sets *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   477
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   478
text {* The intended behaviour is
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   479
@{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
   480
if @{text f} is associative-commutative. For an application of @{text fold}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   481
se the definitions of sums and products over finite sets.
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
23736
bf8d4a46452d Renamed inductive2 to inductive.
berghofe
parents: 23706
diff changeset
   484
inductive
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   485
  foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a => bool"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   486
  for f ::  "'a => 'a => 'a"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   487
  and g :: "'b => 'a"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   488
  and z :: 'a
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   489
where
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   490
  emptyI [intro]: "foldSet f g z {} z"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   491
| insertI [intro]:
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   492
     "\<lbrakk> x \<notin> A; foldSet f g z A y \<rbrakk>
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   493
      \<Longrightarrow> foldSet f g z (insert x A) (f (g x) y)"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   494
23736
bf8d4a46452d Renamed inductive2 to inductive.
berghofe
parents: 23706
diff changeset
   495
inductive_cases empty_foldSetE [elim!]: "foldSet f g z {} x"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   496
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   497
constdefs
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21626
diff changeset
   498
  fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   499
  "fold f g z A == THE x. foldSet f g z A x"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   500
15498
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   501
text{*A tempting alternative for the definiens is
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   502
@{term "if finite A then THE x. foldSet f g e A x else e"}.
15498
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   503
It allows the removal of finiteness assumptions from the theorems
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   504
@{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}.
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   505
The proofs become ugly, with @{text rule_format}. It is not worth the effort.*}
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   506
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   507
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   508
lemma Diff1_foldSet:
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   509
  "foldSet f g z (A - {x}) y ==> x: A ==> foldSet f g z A (f (g x) y)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   510
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   511
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   512
lemma foldSet_imp_finite: "foldSet f g z A x==> finite A"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   513
  by (induct set: foldSet) auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   514
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   515
lemma finite_imp_foldSet: "finite A ==> EX x. foldSet f g z A x"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   516
  by (induct set: finite) auto
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   517
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   518
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   519
subsubsection{*From @{term foldSet} to @{term fold}*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   520
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   521
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
19868
wenzelm
parents: 19793
diff changeset
   522
  by (auto simp add: less_Suc_eq) 
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   523
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   524
lemma insert_image_inj_on_eq:
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   525
     "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   526
        inj_on h {i. i < Suc m}|] 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   527
      ==> A = h ` {i. i < m}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   528
apply (auto simp add: image_less_Suc inj_on_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   529
apply (blast intro: less_trans) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   530
done
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   531
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   532
lemma insert_inj_onE:
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   533
  assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   534
      and inj_on: "inj_on h {i::nat. i<n}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   535
  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
   536
proof (cases n)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   537
  case 0 thus ?thesis using aA by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   538
next
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   539
  case (Suc m)
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   540
  have nSuc: "n = Suc m" by fact
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   541
  have mlessn: "m<n" by (simp add: nSuc)
15532
9712d41db5b8 simplified a proof
paulson
parents: 15526
diff changeset
   542
  from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
27165
e1c49eb8cee6 Hid swap
nipkow
parents: 26792
diff changeset
   543
  let ?hm = "Fun.swap k m h"
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   544
  have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   545
    by (simp add: inj_on_swap_iff inj_on)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   546
  show ?thesis
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   547
  proof (intro exI conjI)
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   548
    show "inj_on ?hm {i. i < m}" using inj_hm
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   549
      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
   550
    show "m<n" by (rule mlessn)
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   551
    show "A = ?hm ` {i. i < m}" 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   552
    proof (rule insert_image_inj_on_eq)
27165
e1c49eb8cee6 Hid swap
nipkow
parents: 26792
diff changeset
   553
      show "inj_on (Fun.swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   554
      show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   555
      show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   556
	using aA hkeq nSuc klessn
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   557
	by (auto simp add: swap_def image_less_Suc fun_upd_image 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   558
			   less_Suc_eq inj_on_image_set_diff [OF inj_on])
15479
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   559
    qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   560
  qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   561
qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   562
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   563
context ab_semigroup_mult
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   564
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   565
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   566
lemma foldSet_determ_aux:
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   567
  "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n}; 
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   568
                foldSet times g z A x; foldSet times g z A x' \<rbrakk>
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   569
   \<Longrightarrow> x' = x"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   570
proof (induct n rule: less_induct)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   571
  case (less n)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   572
    have IH: "!!m h A x x'. 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   573
               \<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m}; 
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   574
                foldSet times g z A x; foldSet times g z A x'\<rbrakk> \<Longrightarrow> x' = x" by fact
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   575
    have Afoldx: "foldSet times g z A x" and Afoldx': "foldSet times g z A x'"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   576
     and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" by fact+
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   577
    show ?case
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   578
    proof (rule foldSet.cases [OF Afoldx])
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   579
      assume "A = {}" and "x = z"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   580
      with Afoldx' show "x' = x" by blast
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   581
    next
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   582
      fix B b u
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   583
      assume AbB: "A = insert b B" and x: "x = g b * u"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   584
         and notinB: "b \<notin> B" and Bu: "foldSet times g z B u"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   585
      show "x'=x" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   586
      proof (rule foldSet.cases [OF Afoldx'])
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   587
        assume "A = {}" and "x' = z"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   588
        with AbB show "x' = x" by blast
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   589
      next
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   590
	fix C c v
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   591
	assume AcC: "A = insert c C" and x': "x' = g c * v"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   592
           and notinC: "c \<notin> C" and Cv: "foldSet times g z C v"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   593
	from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   594
        from insert_inj_onE [OF Beq notinB injh]
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   595
        obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   596
                     and Beq: "B = hB ` {i. i < mB}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   597
                     and lessB: "mB < n" by auto 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   598
	from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   599
        from insert_inj_onE [OF Ceq notinC injh]
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   600
        obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   601
                       and Ceq: "C = hC ` {i. i < mC}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   602
                       and lessC: "mC < n" by auto 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   603
	show "x'=x"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   604
	proof cases
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   605
          assume "b=c"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   606
	  then moreover have "B = C" using AbB AcC notinB notinC by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   607
	  ultimately show ?thesis  using Bu Cv x x' IH[OF lessC Ceq inj_onC]
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   608
            by auto
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   609
	next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   610
	  assume diff: "b \<noteq> c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   611
	  let ?D = "B - {c}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   612
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   613
	    using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   614
	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   615
	  with AbB have "finite ?D" by simp
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   616
	  then obtain d where Dfoldd: "foldSet times g z ?D d"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17189
diff changeset
   617
	    using finite_imp_foldSet by iprover
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   618
	  moreover have cinB: "c \<in> B" using B by auto
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   619
	  ultimately have "foldSet times g z B (g c * d)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   620
	    by(rule Diff1_foldSet)
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   621
	  then have "g c * d = u" by (rule IH [OF lessB Beq inj_onB Bu]) 
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   622
          then have "u = g c * d" ..
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   623
          moreover have "v = g b * d"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   624
	  proof (rule sym, rule IH [OF lessC Ceq inj_onC Cv])
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   625
	    show "foldSet times g z C (g b * d)" using C notinB Dfoldd
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   626
	      by fastsimp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   627
	  qed
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   628
	  ultimately show ?thesis using x x'
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   629
	    by (simp add: mult_left_commute)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   630
	qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   631
      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   632
    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   633
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   634
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   635
lemma foldSet_determ:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   636
  "foldSet times g z A x ==> foldSet times g z A y ==> y = x"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   637
apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   638
apply (blast intro: foldSet_determ_aux [rule_format])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   639
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   640
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   641
lemma fold_equality: "foldSet times g z A y ==> fold times g z A = y"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   642
  by (unfold fold_def) (blast intro: foldSet_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   643
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   644
text{* The base case for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   645
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   646
lemma (in -) fold_empty [simp]: "fold f g z {} = z"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   647
  by (unfold fold_def) blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   648
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   649
lemma fold_insert_aux: "x \<notin> A ==>
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   650
    (foldSet times g z (insert x A) v) =
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   651
    (EX y. foldSet times g z A y & v = g x * y)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   652
  apply auto
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   653
  apply (rule_tac A1 = A and f1 = times in finite_imp_foldSet [THEN exE])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   654
   apply (fastsimp dest: foldSet_imp_finite)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   655
  apply (blast intro: foldSet_determ)
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
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   658
text{* The recursion equation for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   659
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   660
lemma fold_insert [simp]:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   661
    "finite A ==> x \<notin> A ==> fold times g z (insert x A) = g x * fold times g z A"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   662
  apply (unfold fold_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   663
  apply (simp add: fold_insert_aux)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   664
  apply (rule the_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   665
  apply (auto intro: finite_imp_foldSet
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   666
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   667
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   668
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   669
lemma fold_rec:
15535
nipkow
parents: 15532
diff changeset
   670
assumes fin: "finite A" and a: "a:A"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   671
shows "fold times g z A = g a * fold times g z (A - {a})"
15535
nipkow
parents: 15532
diff changeset
   672
proof-
nipkow
parents: 15532
diff changeset
   673
  have A: "A = insert a (A - {a})" using a by blast
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   674
  hence "fold times g z A = fold times g z (insert a (A - {a}))" by simp
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   675
  also have "\<dots> = g a * fold times g z (A - {a})"
15535
nipkow
parents: 15532
diff changeset
   676
    by(rule fold_insert) (simp add:fin)+
nipkow
parents: 15532
diff changeset
   677
  finally show ?thesis .
nipkow
parents: 15532
diff changeset
   678
qed
nipkow
parents: 15532
diff changeset
   679
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   680
end
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   681
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   682
text{* A simplified version for idempotent functions: *}
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   683
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   684
context ab_semigroup_idem_mult
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   685
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   686
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   687
lemma fold_insert_idem:
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   688
assumes finA: "finite A"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   689
shows "fold times g z (insert a A) = g a * fold times g z A"
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   690
proof cases
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   691
  assume "a \<in> A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   692
  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
   693
    by(blast dest: mk_disjoint_insert)
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   694
  show ?thesis
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   695
  proof -
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   696
    from finA A have finB: "finite B" by(blast intro: finite_subset)
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   697
    have "fold times g z (insert a A) = fold times g z (insert a B)" using A by simp
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   698
    also have "\<dots> = g a * fold times g z B"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   699
      using finB disj by simp
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   700
    also have "\<dots> = g a * fold times g z A"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   701
      using A finB disj
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   702
	by (simp add: mult_idem mult_assoc [symmetric])
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   703
    finally show ?thesis .
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   704
  qed
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   705
next
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   706
  assume "a \<notin> A"
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   707
  with finA show ?thesis by simp
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   708
qed
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   709
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   710
lemma foldI_conv_id:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   711
  "finite A \<Longrightarrow> fold times g z A = fold times id z (g ` A)"
15509
c54970704285 revised fold1 proofs
paulson
parents: 15508
diff changeset
   712
by(erule finite_induct)(simp_all add: fold_insert_idem del: fold_insert)
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   713
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   714
end
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   715
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   716
subsubsection{*Lemmas about @{text fold}*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   717
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   718
context ab_semigroup_mult
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   719
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   720
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   721
lemma fold_commute:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   722
  "finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   723
  apply (induct set: finite)
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
   724
   apply simp
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   725
  apply (simp add: mult_left_commute [of x])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   726
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   727
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   728
lemma fold_nest_Un_Int:
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   729
  "finite A ==> finite B
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   730
    ==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   731
  apply (induct set: finite)
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
   732
   apply simp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   733
  apply (simp add: fold_commute Int_insert_left insert_absorb)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   734
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   735
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   736
lemma fold_nest_Un_disjoint:
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   737
  "finite A ==> finite B ==> A Int B = {}
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   738
    ==> fold times g z (A Un B) = fold times g (fold times g z B) A"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   739
  by (simp add: fold_nest_Un_Int)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   740
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   741
lemma fold_reindex:
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   742
assumes fin: "finite A"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   743
shows "inj_on h A \<Longrightarrow> fold times g z (h ` A) = fold times (g \<circ> h) z A"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   744
using fin apply induct
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   745
 apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   746
apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   747
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   748
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   749
text{*
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   750
  Fusion theorem, as described in Graham Hutton's paper,
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   751
  A Tutorial on the Universality and Expressiveness of Fold,
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   752
  JFP 9:4 (355-372), 1999.
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   753
*}
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   754
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   755
lemma fold_fusion:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   756
  includes ab_semigroup_mult g
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   757
  assumes fin: "finite A"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   758
    and hyp: "\<And>x y. h (g x y) = times x (h y)"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   759
  shows "h (fold g j w A) = fold times j (h w) A"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   760
  using fin hyp by (induct set: finite) simp_all
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   761
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   762
lemma fold_cong:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   763
  "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold times g z A = fold times h z A"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   764
  apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold times g z C = fold times h z C")
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   765
   apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   766
  apply (erule finite_induct, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   767
  apply (simp add: subset_insert_iff, clarify)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   768
  apply (subgoal_tac "finite C")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   769
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   770
  apply (subgoal_tac "C = insert x (C - {x})")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   771
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   772
  apply (erule ssubst)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   773
  apply (drule spec)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   774
  apply (erule (1) notE impE)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   775
  apply (simp add: Ball_def del: insert_Diff_single)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   776
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   777
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   778
end
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   779
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   780
context comm_monoid_mult
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   781
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   782
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   783
lemma fold_Un_Int:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   784
  "finite A ==> finite B ==>
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   785
    fold times g 1 A * fold times g 1 B =
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   786
    fold times g 1 (A Un B) * fold times g 1 (A Int B)"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   787
  by (induct set: finite) 
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   788
    (auto simp add: mult_ac insert_absorb Int_insert_left)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   789
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   790
corollary fold_Un_disjoint:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   791
  "finite A ==> finite B ==> A Int B = {} ==>
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   792
    fold times g 1 (A Un B) = fold times g 1 A * fold times g 1 B"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   793
  by (simp add: fold_Un_Int)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   794
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   795
lemma fold_UN_disjoint:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   796
  "\<lbrakk> finite I; ALL i:I. finite (A i);
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   797
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   798
   \<Longrightarrow> fold times g 1 (UNION I A) =
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   799
       fold times (%i. fold times g 1 (A i)) 1 I"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   800
  apply (induct set: finite, simp, atomize)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   801
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   802
   prefer 2 apply blast
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   803
  apply (subgoal_tac "A x Int UNION F A = {}")
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   804
   prefer 2 apply blast
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   805
  apply (simp add: fold_Un_disjoint)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   806
  done
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   807
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   808
lemma fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   809
  fold times (%x. fold times (g x) 1 (B x)) 1 A =
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   810
  fold times (split g) 1 (SIGMA x:A. B x)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   811
apply (subst Sigma_def)
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   812
apply (subst fold_UN_disjoint, assumption, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   813
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   814
apply (erule fold_cong)
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   815
apply (subst fold_UN_disjoint, simp, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   816
 apply blast
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   817
apply simp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   818
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   819
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   820
lemma fold_distrib: "finite A \<Longrightarrow>
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   821
   fold times (%x. g x * h x) 1 A = fold times g 1 A *  fold times h 1 A"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   822
  by (erule finite_induct) (simp_all add: mult_ac)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   823
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   824
end
22917
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
   825
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
   826
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   827
subsection {* Generalized summation over a set *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   828
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   829
interpretation comm_monoid_add: comm_monoid_mult ["0::'a::comm_monoid_add" "op +"]
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   830
  by unfold_locales (auto intro: add_assoc add_commute)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   831
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   832
constdefs
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   833
  setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   834
  "setsum f A == if finite A then fold (op +) f 0 A else 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   835
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   836
abbreviation
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21249
diff changeset
   837
  Setsum  ("\<Sum>_" [1000] 999) where
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   838
  "\<Sum>A == setsum (%x. x) A"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   839
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   840
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   841
written @{text"\<Sum>x\<in>A. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   842
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   843
syntax
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   844
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   845
syntax (xsymbols)
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   846
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   847
syntax (HTML output)
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   848
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   849
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   850
translations -- {* Beware of argument permutation! *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   851
  "SUM i:A. b" == "setsum (%i. b) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   852
  "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   853
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   854
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   855
 @{text"\<Sum>x|P. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   856
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   857
syntax
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   858
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   859
syntax (xsymbols)
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   860
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   861
syntax (HTML output)
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   862
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   863
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   864
translations
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   865
  "SUM x|P. t" => "setsum (%x. t) {x. P}"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   866
  "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   867
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   868
print_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   869
let
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   870
  fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   871
    if x<>y then raise Match
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   872
    else let val x' = Syntax.mark_bound x
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   873
             val t' = subst_bound(x',t)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   874
             val P' = subst_bound(x',P)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   875
         in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   876
in [("setsum", setsum_tr')] end
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   877
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   878
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   879
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   880
lemma setsum_empty [simp]: "setsum f {} = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   881
  by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   882
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   883
lemma setsum_insert [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   884
    "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
   885
  by (simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   886
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   887
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
   888
  by (simp add: setsum_def)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   889
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   890
lemma setsum_reindex:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   891
     "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   892
by(auto simp add: setsum_def comm_monoid_add.fold_reindex dest!:finite_imageD)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   893
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   894
lemma setsum_reindex_id:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   895
     "inj_on f B ==> setsum f B = setsum id (f ` B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   896
by (auto simp add: setsum_reindex)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   897
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   898
lemma setsum_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   899
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   900
by(fastsimp simp: setsum_def intro: comm_monoid_add.fold_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   901
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16632
diff changeset
   902
lemma strong_setsum_cong[cong]:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16632
diff changeset
   903
  "A = B ==> (!!x. x:B =simp=> f x = g x)
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16632
diff changeset
   904
   ==> setsum (%x. f x) A = setsum (%x. g x) B"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   905
by(fastsimp simp: simp_implies_def setsum_def intro: comm_monoid_add.fold_cong)
16632
ad2895beef79 Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents: 16550
diff changeset
   906
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
   907
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
   908
  by (rule setsum_cong[OF refl], auto);
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
   909
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   910
lemma setsum_reindex_cong:
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
   911
     "[|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
   912
      ==> setsum h B = setsum g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   913
  by (simp add: setsum_reindex cong: setsum_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   914
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   915
lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   916
apply (clarsimp simp: setsum_def)
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   917
apply (erule finite_induct, auto)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   918
done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   919
15543
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   920
lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   921
by(simp add:setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   922
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   923
lemma setsum_Un_Int: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   924
  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
   925
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   926
by(simp add: setsum_def comm_monoid_add.fold_Un_Int [symmetric])
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   927
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   928
lemma setsum_Un_disjoint: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   929
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   930
by (subst setsum_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   931
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   932
(*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
   933
  the lhs need not be, since UNION I A could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   934
lemma setsum_UN_disjoint:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   935
    "finite I ==> (ALL i:I. finite (A i)) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   936
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   937
      setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   938
by(simp add: setsum_def comm_monoid_add.fold_UN_disjoint cong: setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   939
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   940
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
   941
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
   942
lemma setsum_Union_disjoint:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   943
  "[| (ALL A:C. finite A);
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   944
      (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
   945
   ==> setsum f (Union C) = setsum (setsum f) C"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   946
apply (cases "finite C") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   947
 prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   948
  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
   949
 apply (unfold Union_def id_def, assumption+)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   950
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   951
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   952
(*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
   953
  the rhs need not be, since SIGMA A B could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   954
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   955
    (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   956
by(simp add:setsum_def comm_monoid_add.fold_Sigma split_def cong:setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   957
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   958
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
   959
lemma setsum_cartesian_product: 
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   960
   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   961
apply (cases "finite A") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   962
 apply (cases "finite B") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   963
  apply (simp add: setsum_Sigma)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   964
 apply (cases "A={}", simp)
15543
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   965
 apply (simp) 
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   966
apply (auto simp add: setsum_def
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   967
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   968
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   969
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   970
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   971
by(simp add:setsum_def comm_monoid_add.fold_distrib)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   972
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   973
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   974
subsubsection {* Properties in more restricted classes of structures *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   975
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   976
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   977
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   978
   prefer 2 apply (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   979
  apply (erule rev_mp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   980
  apply (erule finite_induct, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   981
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   982
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   983
lemma setsum_eq_0_iff [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   984
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   985
  by (induct set: finite) auto
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   986
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   987
lemma setsum_Un_nat: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   988
    (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
   989
  -- {* For the natural numbers, we have subtraction. *}
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   990
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_simps)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   991
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   992
lemma setsum_Un: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   993
    (setsum f (A Un B) :: 'a :: ab_group_add) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   994
      setsum f A + setsum f B - setsum f (A Int B)"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   995
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_simps)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   996
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   997
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   998
    (if a:A then setsum f A - f a else setsum f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   999
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1000
   prefer 2 apply (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1001
  apply (erule finite_induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1002
   apply (auto simp add: insert_Diff_if)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1003
  apply (drule_tac a = a in mk_disjoint_insert, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1004
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1005
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1006
lemma setsum_diff1: "finite A \<Longrightarrow>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1007
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1008
  (if a:A then setsum f A - f a else setsum f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1009
  by (erule finite_induct) (auto simp add: insert_Diff_if)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1010
15552
8ab8e425410b added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents: 15543
diff changeset
  1011
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
  1012
  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
  1013
  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
  1014
  done
8ab8e425410b added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents: 15543
diff changeset
  1015
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1016
(* By Jeremy Siek: *)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1017
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1018
lemma setsum_diff_nat: 
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1019
  assumes "finite B"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1020
    and "B \<subseteq> A"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1021
  shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1022
  using prems
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1023
proof induct
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1024
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1025
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1026
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1027
    and xFinA: "insert x F \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1028
    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
  1029
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1030
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1031
    by (simp add: setsum_diff1_nat)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1032
  from xFinA have "F \<subseteq> A" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1033
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1034
  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
  1035
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1036
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1037
  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
  1038
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1039
  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
  1040
  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
  1041
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1042
  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
  1043
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1044
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1045
lemma setsum_diff:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1046
  assumes le: "finite A" "B \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1047
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1048
proof -
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1049
  from le have finiteB: "finite B" using finite_subset by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1050
  show ?thesis using finiteB le
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1051
  proof induct
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1052
    case empty
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1053
    thus ?case by auto
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1054
  next
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1055
    case (insert x F)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1056
    thus ?case using le finiteB 
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1057
      by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1058
  qed
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1059
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1060
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1061
lemma setsum_mono:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1062
  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
  1063
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1064
proof (cases "finite K")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1065
  case True
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1066
  thus ?thesis using le
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1067
  proof induct
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1068
    case empty
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1069
    thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1070
  next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1071
    case insert
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1072
    thus ?case using add_mono by fastsimp
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1073
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1074
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1075
  case False
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1076
  thus ?thesis
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1077
    by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1078
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1079
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1080
lemma setsum_strict_mono:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1081
  fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1082
  assumes "finite A"  "A \<noteq> {}"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1083
    and "!!x. x:A \<Longrightarrow> f x < g x"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1084
  shows "setsum f A < setsum g A"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1085
  using prems
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1086
proof (induct rule: finite_ne_induct)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1087
  case singleton thus ?case by simp
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1088
next
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1089
  case insert thus ?case by (auto simp: add_strict_mono)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1090
qed
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1091
15535
nipkow
parents: 15532
diff changeset
  1092
lemma setsum_negf:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1093
  "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
15535
nipkow
parents: 15532
diff changeset
  1094
proof (cases "finite A")
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1095
  case True thus ?thesis by (induct set: finite) auto
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_subtractf:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1101
  "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1102
    setsum f A - setsum g A"
15535
nipkow
parents: 15532
diff changeset
  1103
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1104
  case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
nipkow
parents: 15532
diff changeset
  1105
next
nipkow
parents: 15532
diff changeset
  1106
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1107
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1108
15535
nipkow
parents: 15532
diff changeset
  1109
lemma setsum_nonneg:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1110
  assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1111
  shows "0 \<le> setsum f A"
15535
nipkow
parents: 15532
diff changeset
  1112
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1113
  case True thus ?thesis using nn
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1114
  proof induct
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1115
    case empty then show ?case by simp
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1116
  next
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1117
    case (insert x F)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1118
    then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1119
    with insert show ?case by simp
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1120
  qed
15535
nipkow
parents: 15532
diff changeset
  1121
next
nipkow
parents: 15532
diff changeset
  1122
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1123
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1124
15535
nipkow
parents: 15532
diff changeset
  1125
lemma setsum_nonpos:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1126
  assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1127
  shows "setsum f A \<le> 0"
15535
nipkow
parents: 15532
diff changeset
  1128
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1129
  case True thus ?thesis using np
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1130
  proof induct
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1131
    case empty then show ?case by simp
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1132
  next
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1133
    case (insert x F)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1134
    then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1135
    with insert show ?case by simp
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1136
  qed
15535
nipkow
parents: 15532
diff changeset
  1137
next
nipkow
parents: 15532
diff changeset
  1138
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1139
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1140
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1141
lemma setsum_mono2:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1142
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
  1143
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
  1144
shows "setsum f A \<le> setsum f B"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1145
proof -
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1146
  have "setsum f A \<le> setsum f A + setsum f (B-A)"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1147
    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1148
  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
  1149
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1150
  also have "A \<union> (B-A) = B" using sub by blast
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1151
  finally show ?thesis .
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1152
qed
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1153
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1154
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
  1155
    ALL x: B - A. 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1156
      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
  1157
        setsum f A <= setsum f B"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1158
  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
  1159
  apply (erule ssubst)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1160
  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
  1161
  apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1162
  apply (rule add_left_mono)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1163
  apply (erule setsum_nonneg)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1164
  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
  1165
  apply (erule finite_subset, assumption)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1166
  apply (rule finite_subset)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1167
  prefer 2
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1168
  apply assumption
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1169
  apply auto
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1170
  apply (rule setsum_cong)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1171
  apply auto
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1172
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16760
diff changeset
  1173
19279
48b527d0331b Renamed setsum_mult to setsum_right_distrib.
ballarin
parents: 18493
diff changeset
  1174
lemma setsum_right_distrib: 
22934
64ecb3d6790a generalize setsum lemmas from semiring_0_cancel to semiring_0
huffman
parents: 22917
diff changeset
  1175
  fixes f :: "'a => ('b::semiring_0)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1176
  shows "r * setsum f A = setsum (%n. r * f n) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1177
proof (cases "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1178
  case True
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1179
  thus ?thesis
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1180
  proof induct
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1181
    case empty thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1182
  next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1183
    case (insert x A) thus ?case by (simp add: right_distrib)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1184
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1185
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1186
  case False thus ?thesis by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1187
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1188
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1189
lemma setsum_left_distrib:
22934
64ecb3d6790a generalize setsum lemmas from semiring_0_cancel to semiring_0
huffman
parents: 22917
diff changeset
  1190
  "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1191
proof (cases "finite A")
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1192
  case True
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1193
  then show ?thesis
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1194
  proof induct
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1195
    case empty thus ?case by simp
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1196
  next
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1197
    case (insert x A) thus ?case by (simp add: left_distrib)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1198
  qed
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1199
next
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1200
  case False thus ?thesis by (simp add: setsum_def)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1201
qed
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1202
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1203
lemma setsum_divide_distrib:
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1204
  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1205
proof (cases "finite A")
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1206
  case True
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1207
  then show ?thesis
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1208
  proof induct
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1209
    case empty thus ?case by simp
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1210
  next
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1211
    case (insert x A) thus ?case by (simp add: add_divide_distrib)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1212
  qed
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1213
next
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1214
  case False thus ?thesis by (simp add: setsum_def)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1215
qed
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1216
15535
nipkow
parents: 15532
diff changeset
  1217
lemma setsum_abs[iff]: 
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25205
diff changeset
  1218
  fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1219
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
15535
nipkow
parents: 15532
diff changeset
  1220
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1221
  case True
nipkow
parents: 15532
diff changeset
  1222
  thus ?thesis
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1223
  proof induct
15535
nipkow
parents: 15532
diff changeset
  1224
    case empty thus ?case by simp
nipkow
parents: 15532
diff changeset
  1225
  next
nipkow
parents: 15532
diff changeset
  1226
    case (insert x A)
nipkow
parents: 15532
diff changeset
  1227
    thus ?case by (auto intro: abs_triangle_ineq order_trans)
nipkow
parents: 15532
diff changeset
  1228
  qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1229
next
15535
nipkow
parents: 15532
diff changeset
  1230
  case False thus ?thesis by (simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1231
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1232
15535
nipkow
parents: 15532
diff changeset
  1233
lemma setsum_abs_ge_zero[iff]: 
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25205
diff changeset
  1234
  fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1235
  shows "0 \<le> setsum (%i. abs(f i)) A"
15535
nipkow
parents: 15532
diff changeset
  1236
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1237
  case True
nipkow
parents: 15532
diff changeset
  1238
  thus ?thesis
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1239
  proof induct
15535
nipkow
parents: 15532
diff changeset
  1240
    case empty thus ?case by simp
nipkow
parents: 15532
diff changeset
  1241
  next
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21626
diff changeset
  1242
    case (insert x A) thus ?case by (auto simp: add_nonneg_nonneg)
15535
nipkow
parents: 15532
diff changeset
  1243
  qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1244
next
15535
nipkow
parents: 15532
diff changeset
  1245
  case False thus ?thesis by (simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1246
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1247
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1248
lemma abs_setsum_abs[simp]: 
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25205
diff changeset
  1249
  fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1250
  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
  1251
proof (cases "finite A")
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1252
  case True
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1253
  thus ?thesis
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1254
  proof induct
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1255
    case empty thus ?case by simp
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1256
  next
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1257
    case (insert a A)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1258
    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
  1259
    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
  1260
    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
  1261
      by (simp del: abs_of_nonneg)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1262
    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
  1263
    finally show ?case .
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1264
  qed
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1265
next
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1266
  case False thus ?thesis by (simp add: setsum_def)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1267
qed
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1268
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1269
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1270
text {* Commuting outer and inner summation *}
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1271
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1272
lemma swap_inj_on:
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1273
  "inj_on (%(i, j). (j, i)) (A \<times> B)"
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1274
  by (unfold inj_on_def) fast
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1275
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1276
lemma swap_product:
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1277
  "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1278
  by (simp add: split_def image_def) blast
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1279
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1280
lemma setsum_commute:
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1281
  "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1282
proof (simp add: setsum_cartesian_product)
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
  1283
  have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
  1284
    (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1285
    (is "?s = _")
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1286
    apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1287
    apply (simp add: split_def)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1288
    done
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
  1289
  also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1290
    (is "_ = ?t")
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1291
    apply (simp add: swap_product)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1292
    done
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1293
  finally show "?s = ?t" .
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1294
qed
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 17085
diff changeset
  1295
19279
48b527d0331b Renamed setsum_mult to setsum_right_distrib.
ballarin
parents: 18493
diff changeset
  1296
lemma setsum_product:
22934
64ecb3d6790a generalize setsum lemmas from semiring_0_cancel to semiring_0
huffman
parents: 22917
diff changeset
  1297
  fixes f :: "'a => ('b::semiring_0)"
19279
48b527d0331b Renamed setsum_mult to setsum_right_distrib.
ballarin
parents: 18493
diff changeset
  1298
  shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
48b527d0331b Renamed setsum_mult to setsum_right_distrib.
ballarin
parents: 18493
diff changeset
  1299
  by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
48b527d0331b Renamed setsum_mult to setsum_right_distrib.
ballarin
parents: 1