src/HOL/Finite_Set.thy
author paulson
Fri, 21 Jan 2005 13:52:09 +0100
changeset 15447 177ffdbabf80
parent 15409 a063687d24eb
child 15479 fbc473ea9d3c
permissions -rw-r--r--
new theorem image_eq_fold
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title:      HOL/Finite_Set.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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                Additions by Jeremy Avigad in Feb 2004
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Get rid of a couple of superfluous finiteness assumptions in lemmas
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about setsum and card - see FIXME.
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NB: the analogous lemmas for setprod should also be simplified!
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*)
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header {* Finite sets *}
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theory Finite_Set
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imports Divides Power Inductive
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begin
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subsection {* Definition and basic properties *}
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consts Finites :: "'a set set"
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syntax
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  finite :: "'a set => bool"
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translations
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  "finite A" == "A : Finites"
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inductive Finites
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  intros
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    emptyI [simp, intro!]: "{} : Finites"
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    insertI [simp, intro!]: "A : Finites ==> insert a A : Finites"
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axclass finite \<subseteq> type
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  finite: "finite UNIV"
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lemma ex_new_if_finite: -- "does not depend on def of finite at all"
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  assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
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  shows "\<exists>a::'a. a \<notin> A"
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proof -
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  from prems have "A \<noteq> UNIV" by blast
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  thus ?thesis by blast
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qed
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lemma finite_induct [case_names empty insert, induct set: Finites]:
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  "finite F ==>
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    P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
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  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
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proof -
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  assume "P {}" and
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    insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
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  assume "finite F"
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  thus "P F"
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  proof induct
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    show "P {}" .
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    fix x F assume F: "finite F" and P: "P F"
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    show "P (insert x F)"
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    proof cases
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      assume "x \<in> F"
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      hence "insert x F = F" by (rule insert_absorb)
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      with P show ?thesis by (simp only:)
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    next
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      assume "x \<notin> F"
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      from F this P show ?thesis by (rule insert)
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    qed
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  qed
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qed
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lemma finite_subset_induct [consumes 2, case_names empty insert]:
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  "finite F ==> F \<subseteq> A ==>
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    P {} ==> (!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==>
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    P F"
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proof -
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  assume "P {}" and insert:
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    "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
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  assume "finite F"
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  thus "F \<subseteq> A ==> P F"
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  proof induct
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    show "P {}" .
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    fix x F assume "finite F" and "x \<notin> F"
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      and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
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    show "P (insert x F)"
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    proof (rule insert)
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      from i show "x \<in> A" by blast
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      from i have "F \<subseteq> A" by blast
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      with P show "P F" .
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    qed
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  qed
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qed
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text{* Finite sets are the images of initial segments of natural numbers: *}
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lemma finite_imp_nat_seg_image:
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assumes fin: "finite A" shows "\<exists> (n::nat) f. A = f ` {i::nat. i<n}"
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using fin
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proof induct
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  case empty
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  show ?case
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  proof show "\<exists>f. {} = f ` {i::nat. i < 0}" by(simp add:image_def) qed
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next
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  case (insert a A)
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  from insert.hyps obtain n f where "A = f ` {i::nat. i < n}" by blast
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  hence "insert a A = (%i. if i<n then f i else a) ` {i. i < n+1}"
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    by (auto simp add:image_def Ball_def)
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  thus ?case by blast
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qed
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lemma nat_seg_image_imp_finite:
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  "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
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proof (induct n)
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  case 0 thus ?case by simp
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next
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  case (Suc n)
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  let ?B = "f ` {i. i < n}"
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  have finB: "finite ?B" by(rule Suc.hyps[OF refl])
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  show ?case
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  proof cases
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    assume "\<exists>k<n. f n = f k"
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    hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  next
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    assume "\<not>(\<exists> k<n. f n = f k)"
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    hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  qed
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qed
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lemma finite_conv_nat_seg_image:
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  "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
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by(blast intro: finite_imp_nat_seg_image nat_seg_image_imp_finite)
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subsubsection{* Finiteness and set theoretic constructions *}
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lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
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  -- {* The union of two finite sets is finite. *}
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  by (induct set: Finites) simp_all
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lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
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  -- {* Every subset of a finite set is finite. *}
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proof -
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  assume "finite B"
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  thus "!!A. A \<subseteq> B ==> finite A"
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  proof induct
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    case empty
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    thus ?case by simp
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  next
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    case (insert x F A)
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    have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" .
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    show "finite A"
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    proof cases
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      assume x: "x \<in> A"
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      with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
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      with r have "finite (A - {x})" .
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      hence "finite (insert x (A - {x}))" ..
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      also have "insert x (A - {x}) = A" by (rule insert_Diff)
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      finally show ?thesis .
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    next
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      show "A \<subseteq> F ==> ?thesis" .
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      assume "x \<notin> A"
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      with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
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   157
    qed
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  qed
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parents:
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   159
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
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   160
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   161
lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
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   162
  by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
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2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   164
lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
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   165
  -- {* The converse obviously fails. *}
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   166
  by (blast intro: finite_subset)
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parents:
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   167
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   168
lemma finite_insert [simp]: "finite (insert a A) = finite A"
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   169
  apply (subst insert_is_Un)
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  apply (simp only: finite_Un, blast)
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   171
  done
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   172
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   173
lemma finite_Union[simp, intro]:
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 "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
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by (induct rule:finite_induct) simp_all
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   176
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lemma finite_empty_induct:
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  "finite A ==>
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  P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
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   180
proof -
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  assume "finite A"
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    and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
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  have "P (A - A)"
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   184
  proof -
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   185
    fix c b :: "'a set"
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    presume c: "finite c" and b: "finite b"
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   187
      and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
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parents:
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   188
    from c show "c \<subseteq> b ==> P (b - c)"
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   189
    proof induct
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   190
      case empty
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   191
      from P1 show ?case by simp
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   192
    next
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   193
      case (insert x F)
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   194
      have "P (b - F - {x})"
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   195
      proof (rule P2)
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diff changeset
   196
        from _ b show "finite (b - F)" by (rule finite_subset) blast
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parents:
diff changeset
   197
        from insert show "x \<in> b - F" by simp
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parents:
diff changeset
   198
        from insert show "P (b - F)" by simp
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parents:
diff changeset
   199
      qed
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   200
      also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
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   201
      finally show ?case .
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parents:
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   202
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   203
  next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   204
    show "A \<subseteq> A" ..
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parents:
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   205
  qed
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parents:
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   206
  thus "P {}" by simp
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parents:
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   207
qed
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diff changeset
   208
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   209
lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
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   210
  by (rule Diff_subset [THEN finite_subset])
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parents:
diff changeset
   211
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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   212
lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   213
  apply (subst Diff_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
diff changeset
   214
  apply (case_tac "a : A - B")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   215
   apply (rule finite_insert [symmetric, THEN trans])
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   216
   apply (subst insert_Diff, simp_all)
12396
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   217
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
diff changeset
   218
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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diff changeset
   219
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   220
text {* Image and Inverse Image over Finite Sets *}
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   221
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   222
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
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   223
  -- {* The image of a finite set is finite. *}
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   224
  by (induct set: Finites) simp_all
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diff changeset
   225
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   226
lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
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   227
  apply (frule finite_imageI)
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   228
  apply (erule finite_subset, assumption)
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   229
  done
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   230
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   231
lemma finite_range_imageI:
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   232
    "finite (range g) ==> finite (range (%x. f (g x)))"
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   233
  apply (drule finite_imageI, simp)
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diff changeset
   234
  done
ef4c41e7956a new inverse image lemmas
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diff changeset
   235
12396
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   236
lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
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parents:
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   237
proof -
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wenzelm
parents:
diff changeset
   238
  have aux: "!!A. finite (A - {}) = finite A" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   239
  fix B :: "'a set"
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wenzelm
parents:
diff changeset
   240
  assume "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   241
  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   242
    apply induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   243
     apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   244
    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
   245
     apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   246
     apply (simp (no_asm_use) add: inj_on_def)
14208
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paulson
parents: 13825
diff changeset
   247
     apply (blast dest!: aux [THEN iffD1], atomize)
12396
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parents:
diff changeset
   248
    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
14208
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paulson
parents: 13825
diff changeset
   249
    apply (frule subsetD [OF equalityD2 insertI1], clarify)
12396
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wenzelm
parents:
diff changeset
   250
    apply (rule_tac x = xa in bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   251
     apply (simp_all add: inj_on_image_set_diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   252
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   253
qed (rule refl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   254
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   255
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parents: 13737
diff changeset
   256
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
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   257
  -- {* The inverse image of a singleton under an injective function
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   258
         is included in a singleton. *}
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   259
  apply (auto simp add: inj_on_def)
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paulson
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diff changeset
   260
  apply (blast intro: the_equality [symmetric])
13825
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paulson
parents: 13737
diff changeset
   261
  done
ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   262
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   263
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
ef4c41e7956a new inverse image lemmas
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parents: 13737
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   264
  -- {* The inverse image of a finite set under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   265
         is finite. *}
14430
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paulson
parents: 14331
diff changeset
   266
  apply (induct set: Finites, simp_all)
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paulson
parents: 14331
diff changeset
   267
  apply (subst vimage_insert)
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paulson
parents: 14331
diff changeset
   268
  apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
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parents: 13737
diff changeset
   269
  done
ef4c41e7956a new inverse image lemmas
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diff changeset
   270
ef4c41e7956a new inverse image lemmas
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diff changeset
   271
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   272
text {* The finite UNION of finite sets *}
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   273
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   274
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
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   275
  by (induct set: Finites) simp_all
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   276
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   277
text {*
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   278
  Strengthen RHS to
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   279
  @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
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diff changeset
   280
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   281
  We'd need to prove
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   282
  @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
12396
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diff changeset
   283
  by induction. *}
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parents:
diff changeset
   284
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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diff changeset
   285
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
   286
  by (blast intro: finite_UN_I finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   287
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   288
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   289
text {* Sigma of finite sets *}
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wenzelm
parents:
diff changeset
   290
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   291
lemma finite_SigmaI [simp]:
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diff changeset
   292
    "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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parents:
diff changeset
   293
  by (unfold Sigma_def) (blast intro!: finite_UN_I)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   294
15402
97204f3b4705 REorganized Finite_Set
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   295
lemma finite_cartesian_product: "[| finite A; finite B |] ==>
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   296
    finite (A <*> B)"
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diff changeset
   297
  by (rule finite_SigmaI)
97204f3b4705 REorganized Finite_Set
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parents: 15392
diff changeset
   298
12396
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diff changeset
   299
lemma finite_Prod_UNIV:
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diff changeset
   300
    "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   301
  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
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wenzelm
parents:
diff changeset
   302
   apply (erule ssubst)
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parents: 13825
diff changeset
   303
   apply (erule finite_SigmaI, auto)
12396
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wenzelm
parents:
diff changeset
   304
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   305
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
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parents: 15402
diff changeset
   306
lemma finite_cartesian_productD1:
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parents: 15402
diff changeset
   307
     "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
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paulson
parents: 15402
diff changeset
   308
apply (auto simp add: finite_conv_nat_seg_image) 
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paulson
parents: 15402
diff changeset
   309
apply (drule_tac x=n in spec) 
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paulson
parents: 15402
diff changeset
   310
apply (drule_tac x="fst o f" in spec) 
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paulson
parents: 15402
diff changeset
   311
apply (auto simp add: o_def) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   312
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   313
apply (drule equalityD1) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   314
apply (rename_tac y x)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   315
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
   316
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   317
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   318
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
   319
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   320
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   321
lemma finite_cartesian_productD2:
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   322
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   323
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
   324
apply (drule_tac x=n in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   325
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
   326
apply (auto simp add: o_def) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   327
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   328
apply (drule equalityD1)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   329
apply (rename_tac x y)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   330
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
   331
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   332
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   333
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
   334
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   335
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   336
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   337
instance unit :: finite
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   338
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   339
  have "finite {()}" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   340
  also have "{()} = UNIV" by auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   341
  finally show "finite (UNIV :: unit set)" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   342
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   343
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   344
instance * :: (finite, finite) finite
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   345
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   346
  show "finite (UNIV :: ('a \<times> 'b) set)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   347
  proof (rule finite_Prod_UNIV)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   348
    show "finite (UNIV :: 'a set)" by (rule finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   349
    show "finite (UNIV :: 'b set)" by (rule finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   350
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   351
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   352
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   353
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   354
text {* The powerset of a finite set *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   355
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   356
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   357
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   358
  assume "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   359
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   360
  thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   361
next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   362
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   363
  thus "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   364
    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   365
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   366
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   367
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   368
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   369
by(blast intro: finite_subset[OF subset_Pow_Union])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   370
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   371
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   372
lemma finite_converse [iff]: "finite (r^-1) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   373
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   374
   apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   375
   apply (rule iffI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   376
    apply (erule finite_imageD [unfolded inj_on_def])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   377
    apply (simp split add: split_split)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   378
   apply (erule finite_imageI)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   379
  apply (simp add: converse_def image_def, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   380
  apply (rule bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   381
   prefer 2 apply assumption
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   382
  apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   383
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   384
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   385
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   386
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   387
Ehmety) *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   388
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   389
lemma finite_Field: "finite r ==> finite (Field r)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   390
  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   391
  apply (induct set: Finites)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   392
   apply (auto simp add: Field_def Domain_insert Range_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   393
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   394
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   395
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   396
  apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   397
  apply (erule trancl_induct)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   398
   apply (auto simp add: Field_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   399
  done
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_trancl: "finite (r^+) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   402
  apply auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   403
   prefer 2
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   404
   apply (rule trancl_subset_Field2 [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   405
   apply (rule finite_SigmaI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   406
    prefer 3
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 13595
diff changeset
   407
    apply (blast intro: r_into_trancl' finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   408
   apply (auto simp add: finite_Field)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   409
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   410
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   411
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   412
subsection {* A fold functional for finite sets *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   413
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   414
text {* The intended behaviour is
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   415
@{text "fold f g e {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) e)\<dots>)"}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   416
if @{text f} is associative-commutative. For an application of @{text fold}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   417
se the definitions of sums and products over finite sets.
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   418
*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   419
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   420
consts
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   421
  foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   422
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   423
inductive "foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   424
intros
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   425
emptyI [intro]: "({}, e) : foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   426
insertI [intro]: "\<lbrakk> x \<notin> A; (A, y) : foldSet f g e \<rbrakk>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   427
 \<Longrightarrow> (insert x A, f (g x) y) : foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   428
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   429
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   430
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   431
constdefs
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   432
  fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   433
  "fold f g e A == THE x. (A, x) : foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   434
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   435
lemma Diff1_foldSet:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   436
  "(A - {x}, y) : foldSet f g e ==> x: A ==> (A, f (g x) y) : foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   437
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   438
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   439
lemma foldSet_imp_finite: "(A, x) : foldSet f g e ==> finite A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   440
  by (induct set: foldSet) auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   441
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   442
lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   443
  by (induct set: Finites) auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   444
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   445
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   446
subsubsection {* Commutative monoids *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   447
locale ACf =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   448
  fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   449
  assumes commute: "x \<cdot> y = y \<cdot> x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   450
    and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   451
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   452
locale ACe = ACf +
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   453
  fixes e :: 'a
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   454
  assumes ident [simp]: "x \<cdot> e = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   455
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   456
lemma (in ACf) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   457
proof -
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   458
  have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   459
  also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   460
  also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   461
  finally show ?thesis .
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   462
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   463
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   464
lemmas (in ACf) AC = assoc commute left_commute
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   465
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   466
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   467
proof -
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   468
  have "x \<cdot> e = x" by (rule ident)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   469
  thus ?thesis by (subst commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   470
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   471
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   472
text{* Instantiation of locales: *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   473
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   474
lemma ACf_add: "ACf (op + :: 'a::comm_monoid_add \<Rightarrow> 'a \<Rightarrow> 'a)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   475
by(fastsimp intro: ACf.intro add_assoc add_commute)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   476
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   477
lemma ACe_add: "ACe (op +) (0::'a::comm_monoid_add)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   478
by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_add)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   479
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   480
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   481
lemma ACf_mult: "ACf (op * :: 'a::comm_monoid_mult \<Rightarrow> 'a \<Rightarrow> 'a)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   482
by(fast intro: ACf.intro mult_assoc ab_semigroup_mult.mult_commute)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   483
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   484
lemma ACe_mult: "ACe (op *) (1::'a::comm_monoid_mult)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   485
by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_mult)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   486
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   487
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   488
subsubsection{*From @{term foldSet} to @{term fold}*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   489
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   490
lemma (in ACf) foldSet_determ_aux:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   491
  "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; (A,x) : foldSet f g e; (A,x') : foldSet f g e \<rbrakk>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   492
   \<Longrightarrow> x' = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   493
proof (induct n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   494
  case 0 thus ?case by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   495
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   496
  case (Suc n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   497
  have IH: "!!A x x' h. \<lbrakk>A = h`{i::nat. i<n}; (A,x) \<in> foldSet f g e; (A,x') \<in> foldSet f g e\<rbrakk>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   498
           \<Longrightarrow> x' = x" and card: "A = h`{i. i<Suc n}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   499
  and Afoldx: "(A, x) \<in> foldSet f g e" and Afoldy: "(A,x') \<in> foldSet f g e" .
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   500
  show ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   501
  proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   502
    assume "EX k<n. h n = h k"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   503
    hence card': "A = h ` {i. i < n}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   504
      using card by (auto simp:image_def less_Suc_eq)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   505
    show ?thesis by(rule IH[OF card' Afoldx Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   506
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   507
    assume new: "\<not>(EX k<n. h n = h k)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   508
    show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   509
    proof (rule foldSet.cases[OF Afoldx])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   510
      assume "(A, x) = ({}, e)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   511
      thus "x' = x" using Afoldy by (auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   512
    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   513
      fix B b y
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   514
      assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   515
	and y: "(B,y) \<in> foldSet f g e" and notinB: "b \<notin> B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   516
      hence A1: "A = insert b B" and x: "x = g b \<cdot> y" by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   517
      show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   518
      proof (rule foldSet.cases[OF Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   519
	assume "(A,x') = ({}, e)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   520
	thus ?thesis using A1 by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   521
      next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   522
	fix C c z
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   523
	assume eq2: "(A,x') = (insert c C, g c \<cdot> z)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   524
	  and z: "(C,z) \<in> foldSet f g e" and notinC: "c \<notin> C"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   525
	hence A2: "A = insert c C" and x': "x' = g c \<cdot> z" by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   526
	let ?h = "%i. if h i = b then h n else h i"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   527
	have less: "B = ?h`{i. i<n}" (is "_ = ?r")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   528
	proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   529
	  show "B \<subseteq> ?r"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   530
	  proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   531
	    fix u assume "u \<in> B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   532
	    hence uinA: "u \<in> A" and unotb: "u \<noteq> b" using A1 notinB by blast+
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   533
	    then obtain i\<^isub>u where below: "i\<^isub>u < Suc n" and [simp]: "u = h i\<^isub>u"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   534
	      using card by(auto simp:image_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   535
	    show "u \<in> ?r"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   536
	    proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   537
	      assume "i\<^isub>u < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   538
	      thus ?thesis using unotb by(fastsimp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   539
	    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   540
	      assume "\<not> i\<^isub>u < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   541
	      with below have [simp]: "i\<^isub>u = n" by arith
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   542
	      obtain i\<^isub>k where i\<^isub>k: "i\<^isub>k < Suc n" and [simp]: "b = h i\<^isub>k"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   543
		using A1 card by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   544
	      have "i\<^isub>k < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   545
	      proof (rule ccontr)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   546
		assume "\<not> i\<^isub>k < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   547
		hence "i\<^isub>k = n" using i\<^isub>k by arith
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   548
		thus False using unotb by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   549
	      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   550
	      thus ?thesis by(auto simp add:image_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   551
	    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   552
	  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   553
	next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   554
	  show "?r \<subseteq> B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   555
	  proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   556
	    fix u assume "u \<in> ?r"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   557
	    then obtain i\<^isub>u where below: "i\<^isub>u < n" and
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   558
              or: "b = h i\<^isub>u \<and> u = h n \<or> h i\<^isub>u \<noteq> b \<and> h i\<^isub>u = u"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   559
	      by(auto simp:image_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   560
	    from or show "u \<in> B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   561
	    proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   562
	      assume [simp]: "b = h i\<^isub>u \<and> u = h n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   563
	      have "u \<in> A" using card by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   564
              moreover have "u \<noteq> b" using new below by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   565
	      ultimately show "u \<in> B" using A1 by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   566
	    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   567
	      assume "h i\<^isub>u \<noteq> b \<and> h i\<^isub>u = u"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   568
	      moreover hence "u \<in> A" using card below by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   569
	      ultimately show "u \<in> B" using A1 by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   570
	    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   571
	  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   572
	qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   573
	show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   574
	proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   575
	  assume "b = c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   576
	  then moreover have "B = C" using A1 A2 notinB notinC by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   577
	  ultimately show ?thesis using IH[OF less] y z x x' by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   578
	next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   579
	  assume diff: "b \<noteq> c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   580
	  let ?D = "B - {c}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   581
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   582
	    using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   583
	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   584
	  with A1 have "finite ?D" by simp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   585
	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   586
	    using finite_imp_foldSet by rules
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   587
	  moreover have cinB: "c \<in> B" using B by(auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   588
	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   589
	    by(rule Diff1_foldSet)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   590
	  hence "g c \<cdot> d = y" by(rule IH[OF less y])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   591
          moreover have "g b \<cdot> d = z"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   592
	  proof (rule IH[OF _ z])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   593
	    let ?h = "%i. if h i = c then h n else h i"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   594
	    show "C = ?h`{i. i<n}" (is "_ = ?r")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   595
	    proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   596
	      show "C \<subseteq> ?r"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   597
	      proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   598
		fix u assume "u \<in> C"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   599
		hence uinA: "u \<in> A" and unotc: "u \<noteq> c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   600
		  using A2 notinC by blast+
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   601
		then obtain i\<^isub>u where below: "i\<^isub>u < Suc n" and [simp]: "u = h i\<^isub>u"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   602
		  using card by(auto simp:image_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   603
		show "u \<in> ?r"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   604
		proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   605
		  assume "i\<^isub>u < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   606
		  thus ?thesis using unotc by(fastsimp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   607
		next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   608
		  assume "\<not> i\<^isub>u < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   609
		  with below have [simp]: "i\<^isub>u = n" by arith
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   610
		  obtain i\<^isub>k where i\<^isub>k: "i\<^isub>k < Suc n" and [simp]: "c = h i\<^isub>k"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   611
		    using A2 card by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   612
		  have "i\<^isub>k < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   613
		  proof (rule ccontr)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   614
		    assume "\<not> i\<^isub>k < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   615
		    hence "i\<^isub>k = n" using i\<^isub>k by arith
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   616
		    thus False using unotc by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   617
		  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   618
		  thus ?thesis by(auto simp add:image_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   619
		qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   620
	      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   621
	    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   622
	      show "?r \<subseteq> C"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   623
	      proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   624
		fix u assume "u \<in> ?r"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   625
		then obtain i\<^isub>u where below: "i\<^isub>u < n" and
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   626
		  or: "c = h i\<^isub>u \<and> u = h n \<or> h i\<^isub>u \<noteq> c \<and> h i\<^isub>u = u"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   627
		  by(auto simp:image_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   628
		from or show "u \<in> C"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   629
		proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   630
		  assume [simp]: "c = h i\<^isub>u \<and> u = h n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   631
		  have "u \<in> A" using card by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   632
		  moreover have "u \<noteq> c" using new below by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   633
		  ultimately show "u \<in> C" using A2 by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   634
		next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   635
		  assume "h i\<^isub>u \<noteq> c \<and> h i\<^isub>u = u"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   636
		  moreover hence "u \<in> A" using card below by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   637
		  ultimately show "u \<in> C" using A2 by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   638
		qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   639
	      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   640
	    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   641
	  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   642
	    show "(C,g b \<cdot> d) \<in> foldSet f g e" using C notinB Dfoldd
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   643
	      by fastsimp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   644
	  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   645
	  ultimately show ?thesis using x x' by(auto simp:AC)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   646
	qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   647
      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   648
    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   649
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   650
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   651
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   652
(* The same proof, but using card 
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   653
lemma (in ACf) foldSet_determ_aux:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   654
  "!!A x x'. \<lbrakk> card A < n; (A,x) : foldSet f g e; (A,x') : foldSet f g e \<rbrakk>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   655
   \<Longrightarrow> x' = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   656
proof (induct n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   657
  case 0 thus ?case by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   658
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   659
  case (Suc n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   660
  have IH: "!!A x x'. \<lbrakk>card A < n; (A,x) \<in> foldSet f g e; (A,x') \<in> foldSet f g e\<rbrakk>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   661
           \<Longrightarrow> x' = x" and card: "card A < Suc n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   662
  and Afoldx: "(A, x) \<in> foldSet f g e" and Afoldy: "(A,x') \<in> foldSet f g e" .
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   663
  from card have "card A < n \<or> card A = n" by arith
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   664
  thus ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   665
  proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   666
    assume less: "card A < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   667
    show ?thesis by(rule IH[OF less Afoldx Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   668
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   669
    assume cardA: "card A = n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   670
    show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   671
    proof (rule foldSet.cases[OF Afoldx])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   672
      assume "(A, x) = ({}, e)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   673
      thus "x' = x" using Afoldy by (auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   674
    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   675
      fix B b y
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   676
      assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   677
	and y: "(B,y) \<in> foldSet f g e" and notinB: "b \<notin> B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   678
      hence A1: "A = insert b B" and x: "x = g b \<cdot> y" by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   679
      show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   680
      proof (rule foldSet.cases[OF Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   681
	assume "(A,x') = ({}, e)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   682
	thus ?thesis using A1 by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   683
      next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   684
	fix C c z
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   685
	assume eq2: "(A,x') = (insert c C, g c \<cdot> z)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   686
	  and z: "(C,z) \<in> foldSet f g e" and notinC: "c \<notin> C"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   687
	hence A2: "A = insert c C" and x': "x' = g c \<cdot> z" by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   688
	have finA: "finite A" by(rule foldSet_imp_finite[OF Afoldx])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   689
	with cardA A1 notinB have less: "card B < n" by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   690
	show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   691
	proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   692
	  assume "b = c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   693
	  then moreover have "B = C" using A1 A2 notinB notinC by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   694
	  ultimately show ?thesis using IH[OF less] y z x x' by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   695
	next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   696
	  assume diff: "b \<noteq> c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   697
	  let ?D = "B - {c}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   698
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   699
	    using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   700
	  have "finite ?D" using finA A1 by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   701
	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   702
	    using finite_imp_foldSet by rules
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   703
	  moreover have cinB: "c \<in> B" using B by(auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   704
	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   705
	    by(rule Diff1_foldSet)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   706
	  hence "g c \<cdot> d = y" by(rule IH[OF less y])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   707
          moreover have "g b \<cdot> d = z"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   708
	  proof (rule IH[OF _ z])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   709
	    show "card C < n" using C cardA A1 notinB finA cinB
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   710
	      by(auto simp:card_Diff1_less)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   711
	  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   712
	    show "(C,g b \<cdot> d) \<in> foldSet f g e" using C notinB Dfoldd
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   713
	      by fastsimp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   714
	  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   715
	  ultimately show ?thesis using x x' by(auto simp:AC)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   716
	qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   717
      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   718
    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   719
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   720
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   721
*)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   722
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   723
lemma (in ACf) foldSet_determ:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   724
  "(A, x) : foldSet f g e ==> (A, y) : foldSet f g e ==> y = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   725
apply(frule foldSet_imp_finite)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   726
apply(simp add:finite_conv_nat_seg_image)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   727
apply(blast intro: foldSet_determ_aux [rule_format])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   728
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   729
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   730
lemma (in ACf) fold_equality: "(A, y) : foldSet f g e ==> fold f g e A = y"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   731
  by (unfold fold_def) (blast intro: foldSet_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   732
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   733
text{* The base case for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   734
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   735
lemma fold_empty [simp]: "fold f g e {} = e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   736
  by (unfold fold_def) blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   737
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   738
lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   739
    ((insert x A, v) : foldSet f g e) =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   740
    (EX y. (A, y) : foldSet f g e & v = f (g x) y)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   741
  apply auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   742
  apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   743
   apply (fastsimp dest: foldSet_imp_finite)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   744
  apply (blast intro: foldSet_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   745
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   746
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   747
text{* The recursion equation for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   748
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   749
lemma (in ACf) fold_insert[simp]:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   750
    "finite A ==> x \<notin> A ==> fold f g e (insert x A) = f (g x) (fold f g e A)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   751
  apply (unfold fold_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   752
  apply (simp add: fold_insert_aux)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   753
  apply (rule the_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   754
  apply (auto intro: finite_imp_foldSet
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   755
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   756
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   757
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   758
declare
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   759
  empty_foldSetE [rule del]  foldSet.intros [rule del]
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   760
  -- {* Delete rules to do with @{text foldSet} relation. *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   761
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   762
subsubsection{*Lemmas about @{text fold}*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   763
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   764
lemma (in ACf) fold_commute:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   765
  "finite A ==> (!!e. f (g x) (fold f g e A) = fold f g (f (g x) e) A)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   766
  apply (induct set: Finites, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   767
  apply (simp add: left_commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   768
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   769
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   770
lemma (in ACf) fold_nest_Un_Int:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   771
  "finite A ==> finite B
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   772
    ==> fold f g (fold f g e B) A = fold f g (fold f g e (A Int B)) (A Un B)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   773
  apply (induct set: Finites, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   774
  apply (simp add: fold_commute Int_insert_left insert_absorb)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   775
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   776
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   777
lemma (in ACf) fold_nest_Un_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   778
  "finite A ==> finite B ==> A Int B = {}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   779
    ==> fold f g e (A Un B) = fold f g (fold f g e B) A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   780
  by (simp add: fold_nest_Un_Int)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   781
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   782
lemma (in ACf) fold_reindex:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   783
assumes fin: "finite B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   784
shows "inj_on h B \<Longrightarrow> fold f g e (h ` B) = fold f (g \<circ> h) e B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   785
using fin apply (induct)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   786
 apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   787
apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   788
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   789
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   790
lemma (in ACe) fold_Un_Int:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   791
  "finite A ==> finite B ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   792
    fold f g e A \<cdot> fold f g e B =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   793
    fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   794
  apply (induct set: Finites, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   795
  apply (simp add: AC insert_absorb Int_insert_left)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   796
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   797
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   798
corollary (in ACe) fold_Un_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   799
  "finite A ==> finite B ==> A Int B = {} ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   800
    fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   801
  by (simp add: fold_Un_Int)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   802
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   803
lemma (in ACe) fold_UN_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   804
  "\<lbrakk> finite I; ALL i:I. finite (A i);
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   805
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   806
   \<Longrightarrow> fold f g e (UNION I A) =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   807
       fold f (%i. fold f g e (A i)) e I"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   808
  apply (induct set: Finites, simp, atomize)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   809
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   810
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   811
  apply (subgoal_tac "A x Int UNION F A = {}")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   812
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   813
  apply (simp add: fold_Un_disjoint)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   814
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   815
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   816
lemma (in ACf) fold_cong:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   817
  "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g a A = fold f h a A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   818
  apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g a C = fold f h a C")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   819
   apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   820
  apply (erule finite_induct, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   821
  apply (simp add: subset_insert_iff, clarify)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   822
  apply (subgoal_tac "finite C")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   823
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   824
  apply (subgoal_tac "C = insert x (C - {x})")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   825
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   826
  apply (erule ssubst)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   827
  apply (drule spec)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   828
  apply (erule (1) notE impE)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   829
  apply (simp add: Ball_def del: insert_Diff_single)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   830
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   831
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   832
lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   833
  fold f (%x. fold f (g x) e (B x)) e A =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   834
  fold f (split g) e (SIGMA x:A. B x)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   835
apply (subst Sigma_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   836
apply (subst fold_UN_disjoint)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   837
   apply assumption
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   838
  apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   839
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   840
apply (erule fold_cong)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   841
apply (subst fold_UN_disjoint)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   842
   apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   843
  apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   844
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   845
apply (simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   846
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   847
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   848
lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   849
   fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   850
apply (erule finite_induct)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   851
 apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   852
apply (simp add:AC)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   853
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   854
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   855
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   856
subsection {* Generalized summation over a set *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   857
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   858
constdefs
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   859
  setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   860
  "setsum f A == if finite A then fold (op +) f 0 A else 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   861
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   862
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   863
written @{text"\<Sum>x\<in>A. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   864
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   865
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   866
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   867
syntax (xsymbols)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   868
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   869
syntax (HTML output)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   870
  "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   871
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   872
translations -- {* Beware of argument permutation! *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   873
  "SUM i:A. b" == "setsum (%i. b) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   874
  "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   875
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   876
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   877
 @{text"\<Sum>x|P. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   878
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   879
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   880
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   881
syntax (xsymbols)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   882
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   883
syntax (HTML output)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   884
  "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   885
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   886
translations
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   887
  "SUM x|P. t" => "setsum (%x. t) {x. P}"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   888
  "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   889
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   890
text{* Finally we abbreviate @{term"\<Sum>x\<in>A. x"} by @{text"\<Sum>A"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   891
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   892
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   893
  "_Setsum" :: "'a set => 'a::comm_monoid_mult"  ("\<Sum>_" [1000] 999)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   894
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   895
parse_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   896
  let
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   897
    fun Setsum_tr [A] = Syntax.const "setsum" $ Abs ("", dummyT, Bound 0) $ A
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   898
  in [("_Setsum", Setsum_tr)] end;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   899
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   900
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   901
print_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   902
let
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   903
  fun setsum_tr' [Abs(_,_,Bound 0), A] = Syntax.const "_Setsum" $ A
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   904
    | setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   905
       if x<>y then raise Match
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   906
       else let val x' = Syntax.mark_bound x
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   907
                val t' = subst_bound(x',t)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   908
                val P' = subst_bound(x',P)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   909
            in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   910
in
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   911
[("setsum", setsum_tr')]
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   912
end
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   913
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   914
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   915
lemma setsum_empty [simp]: "setsum f {} = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   916
  by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   917
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   918
lemma setsum_insert [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   919
    "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   920
  by (simp add: setsum_def ACf.fold_insert [OF ACf_add])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   921
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   922
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
   923
  by (simp add: setsum_def)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   924
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   925
lemma setsum_reindex:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   926
     "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   927
by(auto simp add: setsum_def ACf.fold_reindex[OF ACf_add] dest!:finite_imageD)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   928
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   929
lemma setsum_reindex_id:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   930
     "inj_on f B ==> setsum f B = setsum id (f ` B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   931
by (auto simp add: setsum_reindex)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   932
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   933
lemma setsum_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   934
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   935
by(fastsimp simp: setsum_def intro: ACf.fold_cong[OF ACf_add])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   936
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   937
lemma setsum_reindex_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   938
     "[|inj_on f A; B = f ` A; !!a. g a = h (f a)|] 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   939
      ==> setsum h B = setsum g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   940
  by (simp add: setsum_reindex cong: setsum_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   941
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   942
lemma setsum_0: "setsum (%i. 0) A = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   943
apply (clarsimp simp: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   944
apply (erule finite_induct, auto simp:ACf.fold_insert [OF ACf_add])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   945
done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   946
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   947
lemma setsum_0': "ALL a:F. f a = 0 ==> setsum f F = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   948
  apply (subgoal_tac "setsum f F = setsum (%x. 0) F")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   949
  apply (erule ssubst, rule setsum_0)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   950
  apply (rule setsum_cong, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   951
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   952
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   953
lemma setsum_Un_Int: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   954
  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
   955
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   956
by(simp add: setsum_def ACe.fold_Un_Int[OF ACe_add,symmetric])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   957
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   958
lemma setsum_Un_disjoint: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   959
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   960
by (subst setsum_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   961
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   962
(*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
   963
  the lhs need not be, since UNION I A could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   964
lemma setsum_UN_disjoint:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   965
    "finite I ==> (ALL i:I. finite (A i)) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   966
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   967
      setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   968
by(simp add: setsum_def ACe.fold_UN_disjoint[OF ACe_add] cong: setsum_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   969
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   970
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
   971
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
   972
lemma setsum_Union_disjoint:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   973
  "[| (ALL A:C. finite A);
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   974
      (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
   975
   ==> setsum f (Union C) = setsum (setsum f) C"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   976
apply (cases "finite C") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   977
 prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   978
  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
   979
 apply (unfold Union_def id_def, assumption+)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   980
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   981
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   982
(*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
   983
  the rhs need not be, since SIGMA A B could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   984
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   985
    (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   986
    (\<Sum>z\<in>(SIGMA x:A. B x). f (fst z) (snd z))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   987
by(simp add:setsum_def ACe.fold_Sigma[OF ACe_add] split_def cong:setsum_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   988
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   989
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
   990
lemma setsum_cartesian_product: 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   991
   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>z\<in>A <*> B. f (fst z) (snd z))"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   992
apply (cases "finite A") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   993
 apply (cases "finite B") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   994
  apply (simp add: setsum_Sigma)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   995
 apply (cases "A={}", simp)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   996
 apply (simp add: setsum_0) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   997
apply (auto simp add: setsum_def
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   998
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   999
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1000
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1001
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1002
by(simp add:setsum_def ACe.fold_distrib[OF ACe_add])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1003
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1004
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1005
subsubsection {* Properties in more restricted classes of structures *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1006
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1007
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1008
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1009
   prefer 2 apply (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1010
  apply (erule rev_mp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1011
  apply (erule finite_induct, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1012
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1013
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1014
lemma setsum_eq_0_iff [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1015
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1016
  by (induct set: Finites) auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1017
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1018
lemma setsum_Un_nat: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1019
    (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
  1020
  -- {* For the natural numbers, we have subtraction. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1021
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1022
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1023
lemma setsum_Un: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1024
    (setsum f (A Un B) :: 'a :: ab_group_add) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1025
      setsum f A + setsum f B - setsum f (A Int B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1026
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1027
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1028
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1029
    (if a:A then setsum f A - f a else setsum f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1030
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1031
   prefer 2 apply (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1032
  apply (erule finite_induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1033
   apply (auto simp add: insert_Diff_if)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1034
  apply (drule_tac a = a in mk_disjoint_insert, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1035
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1036
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1037
lemma setsum_diff1: "finite A \<Longrightarrow>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1038
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1039
  (if a:A then setsum f A - f a else setsum f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1040
  by (erule finite_induct) (auto simp add: insert_Diff_if)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1041
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1042
(* By Jeremy Siek: *)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1043
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1044
lemma setsum_diff_nat: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1045
  assumes finB: "finite B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1046
  shows "B \<subseteq> A \<Longrightarrow> (setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1047
using finB
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1048
proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1049
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1050
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1051
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1052
    and xFinA: "insert x F \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1053
    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
  1054
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1055
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1056
    by (simp add: setsum_diff1_nat)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1057
  from xFinA have "F \<subseteq> A" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1058
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1059
  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
  1060
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1061
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1062
  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
  1063
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1064
  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
  1065
  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
  1066
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1067
  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
  1068
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1069
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1070
lemma setsum_diff:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1071
  assumes le: "finite A" "B \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1072
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1073
proof -
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1074
  from le have finiteB: "finite B" using finite_subset by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1075
  show ?thesis using finiteB le
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1076
    proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1077
      case empty
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1078
      thus ?case by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1079
    next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1080
      case (insert x F)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1081
      thus ?case using le finiteB 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1082
	by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1083
    qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1084
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1085
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1086
lemma setsum_mono:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1087
  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
  1088
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1089
proof (cases "finite K")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1090
  case True
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1091
  thus ?thesis using le
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1092
  proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1093
    case empty
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1094
    thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1095
  next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1096
    case insert
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1097
    thus ?case using add_mono 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1098
      by force
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1099
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1100
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1101
  case False
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1102
  thus ?thesis
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1103
    by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1104
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1105
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1106
lemma setsum_mono2_nat:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1107
  assumes fin: "finite B" and sub: "A \<subseteq> B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1108
shows "setsum f A \<le> (setsum f B :: nat)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1109
proof -
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1110
  have "setsum f A \<le> setsum f A + setsum f (B-A)" by arith
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1111
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1112
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1113
  also have "A \<union> (B-A) = B" using sub by blast
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1114
  finally show ?thesis .
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1115
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1116
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1117
lemma setsum_negf: "finite A ==> setsum (%x. - (f x)::'a::ab_group_add) A =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1118
  - setsum f A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1119
  by (induct set: Finites, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1120
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1121
lemma setsum_subtractf: "finite A ==> setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1122
  setsum f A - setsum g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1123
  by (simp add: diff_minus setsum_addf setsum_negf)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1124
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1125
lemma setsum_nonneg: "[| finite A;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1126
    \<forall>x \<in> A. (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) \<le> f x |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1127
    0 \<le> setsum f A";
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1128
  apply (induct set: Finites, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1129
  apply (subgoal_tac "0 + 0 \<le> f x + setsum f F", simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1130
  apply (blast intro: add_mono)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1131
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1132
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1133
lemma setsum_nonpos: "[| finite A;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1134
    \<forall>x \<in> A. f x \<le> (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1135
    setsum f A \<le> 0";
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1136
  apply (induct set: Finites, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1137
  apply (subgoal_tac "f x + setsum f F \<le> 0 + 0", simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1138
  apply (blast intro: add_mono)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1139
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1140
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1141
lemma setsum_mult: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1142
  fixes f :: "'a => ('b::semiring_0_cancel)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1143
  shows "r * setsum f A = setsum (%n. r * f n) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1144
proof (cases "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1145
  case True
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1146
  thus ?thesis
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1147
  proof (induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1148
    case empty thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1149
  next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1150
    case (insert x A) thus ?case by (simp add: right_distrib)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1151
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1152
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1153
  case False thus ?thesis by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1154
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1155
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1156
lemma setsum_abs: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1157
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1158
  assumes fin: "finite A" 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1159
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1160
using fin 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1161
proof (induct) 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1162
  case empty thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1163
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1164
  case (insert x A)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1165
  thus ?case by (auto intro: abs_triangle_ineq order_trans)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1166
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1167
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1168
lemma setsum_abs_ge_zero: 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1169
  fixes f :: "'a => ('b::lordered_ab_group_abs)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1170
  assumes fin: "finite A" 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1171
  shows "0 \<le> setsum (%i. abs(f i)) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1172
using fin 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1173
proof (induct) 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1174
  case empty thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1175
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1176
  case (insert x A) thus ?case by (auto intro: order_trans)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1177
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1178
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1179
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1180
subsection {* Generalized product over a set *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1181
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1182
constdefs
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1183
  setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1184
  "setprod f A == if finite A then fold (op *) f 1 A else 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1185
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1186
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1187
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_:_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1188
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1189
syntax (xsymbols)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1190
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1191
syntax (HTML output)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1192
  "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1193
translations
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1194
  "\<Prod>i:A. b" == "setprod (%i. b) A"  -- {* Beware of argument permutation! *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1195
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1196
syntax
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1197
  "_Setprod" :: "'a set => 'a::comm_monoid_mult"  ("\<Prod>_" [1000] 999)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1198
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1199
parse_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1200
  let
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1201
    fun Setprod_tr [A] = Syntax.const "setprod" $ Abs ("", dummyT, Bound 0) $ A
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1202
  in [("_Setprod", Setprod_tr)] end;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1203
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1204
print_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1205
let fun setprod_tr' [Abs(x,Tx,t), A] =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1206
    if t = Bound 0 then Syntax.const "_Setprod" $ A else raise Match
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1207
in
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1208
[("setprod", setprod_tr')]
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1209
end
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1210
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1211
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1212
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1213
lemma setprod_empty [simp]: "setprod f {} = 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1214
  by (auto simp add: setprod_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1215
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1216
lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1217
    setprod f (insert a A) = f a * setprod f A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1218
by (simp add: setprod_def ACf.fold_insert [OF ACf_mult])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1219
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1220
lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1221
  by (simp add: setprod_def)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1222
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1223
lemma setprod_reindex:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1224
     "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1225
by(auto simp: setprod_def ACf.fold_reindex[OF ACf_mult] dest!:finite_imageD)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1226
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1227
lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1228
by (auto simp add: setprod_reindex)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1229
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1230
lemma setprod_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1231
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1232
by(fastsimp simp: setprod_def intro: ACf.fold_cong[OF ACf_mult])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1233
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1234
lemma setprod_reindex_cong: "inj_on f A ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1235
    B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1236
  by (frule setprod_reindex, simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1237
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1238
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1239
lemma setprod_1: "setprod (%i. 1) A = 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1240
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1241
  apply (erule finite_induct, auto simp add: mult_ac)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1242
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1243
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1244
lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1245
  apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1246
  apply (erule ssubst, rule setprod_1)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1247
  apply (rule setprod_cong, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1248
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1249
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1250
lemma setprod_Un_Int: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1251
    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1252
by(simp add: setprod_def ACe.fold_Un_Int[OF ACe_mult,symmetric])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1253
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1254
lemma setprod_Un_disjoint: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1255
  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1256
by (subst setprod_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1257
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1258
lemma setprod_UN_disjoint:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1259
    "finite I ==> (ALL i:I. finite (A i)) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1260
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1261
      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1262
by(simp add: setprod_def ACe.fold_UN_disjoint[OF ACe_mult] cong: setprod_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1263
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1264
lemma setprod_Union_disjoint:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1265
  "[| (ALL A:C. finite A);
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1266
      (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
  1267
   ==> setprod f (Union C) = setprod (setprod f) C"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1268
apply (cases "finite C") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1269
 prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1270
  apply (frule setprod_UN_disjoint [of C id f])
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1271
 apply (unfold Union_def id_def, assumption+)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1272
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1273
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1274
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1275
    (\<Prod>x:A. (\<Prod>y: B x. f x y)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1276
    (\<Prod>z:(SIGMA x:A. B x). f (fst z) (snd z))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1277
by(simp add:setprod_def ACe.fold_Sigma[OF ACe_mult] split_def cong:setprod_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1278
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1279
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
  1280
lemma setprod_cartesian_product: 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1281
     "(\<Prod>x:A. (\<Prod>y: B. f x y)) = (\<Prod>z:(A <*> B). f (fst z) (snd z))"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1282
apply (cases "finite A") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1283
 apply (cases "finite B") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1284
  apply (simp add: setprod_Sigma)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1285
 apply (cases "A={}", simp)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1286
 apply (simp add: setprod_1) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1287
apply (auto simp add: setprod_def
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1288
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1289
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1290
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1291
lemma setprod_timesf:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1292
     "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1293
by(simp add:setprod_def ACe.fold_distrib[OF ACe_mult])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1294
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1295
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1296
subsubsection {* Properties in more restricted classes of structures *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1297
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1298
lemma setprod_eq_1_iff [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1299
    "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1300
  by (induct set: Finites) auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1301
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1302
lemma setprod_zero:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1303
     "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1304
  apply (induct set: Finites, force, clarsimp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1305
  apply (erule disjE, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1306
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1307
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1308
lemma setprod_nonneg [rule_format]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1309
     "(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1310
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1311
  apply (induct set: Finites, force, clarsimp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1312
  apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1313
  apply (rule mult_mono, assumption+)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1314
  apply (auto simp add: setprod_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1315
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1316
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1317
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1318
     --> 0 < setprod f A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1319
  apply (case_tac "finite A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1320
  apply (induct set: Finites, force, clarsimp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1321
  apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1322
  apply (rule mult_strict_mono, assumption+)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1323
  apply (auto simp add: setprod_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1324
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1325
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1326
lemma setprod_nonzero [rule_format]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1327
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1328
      finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1329
  apply (erule finite_induct, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1330
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1331
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1332
lemma setprod_zero_eq:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1333
    "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1334
     finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1335
  apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1336
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1337
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1338
lemma setprod_nonzero_field:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1339
    "finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1340
  apply (rule setprod_nonzero, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1341
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1342
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1343
lemma setprod_zero_eq_field:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1344
    "finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1345
  apply (rule setprod_zero_eq, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1346
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1347
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1348
lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1349
    (setprod f (A Un B) :: 'a ::{field})
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1350
      = setprod f A * setprod f B / setprod f (A Int B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1351
  apply (subst setprod_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1352
  apply (subgoal_tac "finite (A Int B)")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1353
  apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1354
  apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1355
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1356
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1357
lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1358
    (setprod f (A - {a}) :: 'a :: {field}) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1359
      (if a:A then setprod f A / f a else setprod f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1360
  apply (erule finite_induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1361
   apply (auto simp add: insert_Diff_if)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1362
  apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1363
  apply (erule ssubst)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1364
  apply (subst times_divide_eq_right [THEN sym])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1365
  apply (auto simp add: mult_ac times_divide_eq_right divide_self)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1366
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1367
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1368
lemma setprod_inversef: "finite A ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1369
    ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1370
      setprod (inverse \<circ> f) A = inverse (setprod f A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1371
  apply (erule finite_induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1372
  apply (simp, simp)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1373
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1374
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1375
lemma setprod_dividef:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1376
     "[|finite A;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1377
        \<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1378
      ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1379
  apply (subgoal_tac
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1380
         "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1381
  apply (erule ssubst)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1382
  apply (subst divide_inverse)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1383
  apply (subst setprod_timesf)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1384
  apply (subst setprod_inversef, assumption+, rule refl)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1385
  apply (rule setprod_cong, rule refl)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1386
  apply (subst divide_inverse, auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1387
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1388
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1389
subsection {* Finite cardinality *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1390
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1391
text {* This definition, although traditional, is ugly to work with:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1392
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1393
But now that we have @{text setsum} things are easy:
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1394
*}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1395
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1396
constdefs
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1397
  card :: "'a set => nat"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1398
  "card A == setsum (%x. 1::nat) A"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1399
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1400
lemma card_empty [simp]: "card {} = 0"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1401
  by (simp add: card_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1402
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1403
lemma card_infinite [simp]: "~ finite A ==> card A = 0"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1404
  by (simp add: card_def)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1405
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1406
lemma card_eq_setsum: "card A = setsum (%x. 1) A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1407
by (simp add: card_def)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1408
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1409
lemma card_insert_disjoint [simp]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1410
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1411
by(simp add: card_def ACf.fold_insert[OF ACf_add])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1412
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1413
lemma card_insert_if:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1414
    "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1415
  by (simp add: insert_absorb)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1416
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1417
lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1418
  apply auto
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
  1419
  apply (drule_tac a = x in mk_disjoint_insert, clarify)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1420
  apply (auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1421
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1422
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1423
lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1424
by auto
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1425
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1426
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1427
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1428
apply(simp del:insert_Diff_single)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1429
done
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1430
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1431
lemma card_Diff_singleton:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1432
    "finite A ==> x: A ==> card (A - {x}) = card A - 1"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1433
  by (simp add: card_Suc_Diff1 [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1434
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1435
lemma card_Diff_singleton_if:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1436
    "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1437
  by (simp add: card_Diff_singleton)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1438
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1439
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1440
  by (simp add: card_insert_if card_Suc_Diff1)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1441
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1442
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1443
  by (simp add: card_insert_if)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1444
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1445
lemma card_mono: "\<lbrakk> finite B; A \<subseteq> B \<rbrakk> \<Longrightarrow> card A \<le> card B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1446
by (simp add: card_def setsum_mono2_nat)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1447
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1448
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
  1449
  apply (induct set: Finites, simp, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1450
  apply (subgoal_tac "finite A & A - {x} <= F")
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
  1451
   prefer 2 apply (blast intro: finite_subset, atomize)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1452
  apply (drule_tac x = "A - {x}" in spec)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1453
  apply (simp add: card_Diff_singleton_if split add: split_if_asm)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
  1454
  apply (case_tac "card A", auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1455
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1456
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1457
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1458
  apply (simp add: psubset_def linorder_not_le [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1459
  apply (blast dest: card_seteq)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1460
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1461
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1462
lemma card_Un_Int: "finite A ==> finite B
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1463
    ==> card A + card B = card (A Un B) + card (A Int B)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1464
by(simp add:card_def setsum_Un_Int)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1465
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1466
lemma card_Un_disjoint: "finite A ==> finite B
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1467
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1468
  by (simp add: card_Un_Int)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1469
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1470
lemma card_Diff_subset:
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1471
  "finite B ==> B <= A ==> card (A - B) = card A - card B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1472
by(simp add:card_def setsum_diff_nat)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1473
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1474
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1475
  apply (rule Suc_less_SucD)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1476
  apply (simp add: card_Suc_Diff1)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1477
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1478
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1479
lemma card_Diff2_less:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1480
    "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1481
  apply (case_tac "x = y")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1482
   apply (simp add: card_Diff1_less)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1483
  apply (rule less_trans)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1484
   prefer 2 apply (auto intro!: card_Diff1_less)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1485
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1486
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1487
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1488
  apply (case_tac "x : A")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1489
   apply (simp_all add: card_Diff1_less less_imp_le)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1490
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1491
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1492
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
  1493
by (erule psubsetI, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1494
14889
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1495
lemma insert_partition:
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1496
  "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1497
  \<Longrightarrow> x \<inter> \<Union> F = {}"
14889
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1498
by auto
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1499
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1500
(* main cardinality theorem *)
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1501
lemma card_partition [rule_format]:
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1502
     "finite C ==>  
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1503
        finite (\<Union> C) -->  
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1504
        (\<forall>c\<in>C. card c = k) -->   
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1505
        (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->  
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1506
        k * card(C) = card (\<Union> C)"
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1507
apply (erule finite_induct, simp)
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1508
apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition 
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1509
       finite_subset [of _ "\<Union> (insert x F)"])
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1510
done
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1511
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1512
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1513
lemma setsum_constant_nat: "(\<Sum>x\<in>A. y) = (card A) * y"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1514
  -- {* Generalized to any @{text comm_semiring_1_cancel} in
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1515
        @{text IntDef} as @{text setsum_constant}. *}
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1516
apply (cases "finite A") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1517
apply (erule finite_induct, auto)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1518
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1519
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1520
lemma setprod_constant: "finite A ==> (\<Prod>x: A. (y::'a::recpower)) = y^(card A)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1521
  apply (erule finite_induct)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1522
  apply (auto simp add: power_Suc)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1523
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1524
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1525
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1526
subsubsection {* Cardinality of unions *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1527
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1528
lemma card_UN_disjoint:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1529
    "finite I ==> (ALL i:I. finite (A i)) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1530
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1531
      card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1532
  apply (simp add: card_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1533
  apply (subgoal_tac
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1534
           "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1535
  apply (simp add: setsum_UN_disjoint)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1536
  apply (simp add: setsum_constant_nat cong: setsum_cong)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1537
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1538
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1539
lemma card_Union_disjoint:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1540
  "finite C ==> (ALL A:C. finite A) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1541
        (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1542
      card (Union C) = setsum card C"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1543
  apply (frule card_UN_disjoint [of C id])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1544
  apply (unfold Union_def id_def, assumption+)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1545
  done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1546
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1547
subsubsection {* Cardinality of image *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1548
15447
177ffdbabf80 new theorem image_eq_fold
paulson
parents: 15409
diff changeset
  1549
text{*The image of a finite set can be expressed using @{term fold}.*}
177ffdbabf80 new theorem image_eq_fold
paulson
parents: 15409
diff changeset
  1550
lemma image_eq_fold: "finite A ==> f ` A = fold (op Un) (%x. {f x}) {} A"
177ffdbabf80 new theorem image_eq_fold
paulson
parents: 15409
diff changeset
  1551
  apply (erule finite_induct, simp)
177ffdbabf80 new theorem image_eq_fold
paulson
parents: 15409
diff changeset
  1552
  apply (subst ACf.fold_insert) 
177ffdbabf80 new theorem image_eq_fold
paulson
parents: 15409
diff changeset
  1553
  apply (auto simp add: ACf_def) 
177ffdbabf80 new theorem image_eq_fold
paulson
parents: 15409
diff changeset
  1554
  done
177ffdbabf80 new theorem image_eq_fold
paulson
parents: 15409
diff changeset
  1555
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1556
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
  1557
  apply (induct set: Finites, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1558
  apply (simp add: le_SucI finite_imageI card_insert_if)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1559
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1560
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1561
lemma card_image: "inj_on f A ==> card (f ` A) = card A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1562
by(simp add:card_def setsum_reindex o_def)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1563
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1564
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1565
  by (simp add: card_seteq card_image)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1566
15111
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1567
lemma eq_card_imp_inj_on:
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1568
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1569
apply(induct rule:finite_induct)
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1570
 apply simp
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1571
apply(frule card_image_le[where f = f])
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1572
apply(simp add:card_insert_if split:if_splits)
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1573
done
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1574
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1575
lemma inj_on_iff_eq_card:
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1576
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1577
by(blast intro: card_image eq_card_imp_inj_on)
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1578
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1579
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1580
lemma card_inj_on_le:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1581
    "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1582
apply (subgoal_tac "finite A") 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1583
 apply (force intro: card_mono simp add: card_image [symmetric])
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1584
apply (blast intro: finite_imageD dest: finite_subset) 
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1585
done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1586
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1587
lemma card_bij_eq:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1588
    "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1589
       finite A; finite B |] ==> card A = card B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1590
  by (auto intro: le_anti_sym card_inj_on_le)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1591
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1592
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1593
subsubsection {* Cardinality of products *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1594
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1595
(*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1596
lemma SigmaI_insert: "y \<notin> A ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1597
  (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1598
  by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1599
*)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1600
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1601
lemma card_SigmaI [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1602
  "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1603
  \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1604
by(simp add:card_def setsum_Sigma)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1605
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1606
lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1607
apply (cases "finite A") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1608
apply (cases "finite B") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1609
  apply (simp add: setsum_constant_nat) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1610
apply (auto simp add: card_eq_0_iff
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1611
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1612
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1613
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1614
lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1615
by (simp add: card_cartesian_product) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1616
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1617
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1618
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1619
subsubsection {* Cardinality of the Powerset *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1620
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1621
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1622
  apply (induct set: Finites)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1623
   apply (simp_all add: Pow_insert)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
  1624
  apply (subst card_Un_disjoint, blast)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
  1625
    apply (blast intro: finite_imageI, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1626
  apply (subgoal_tac "inj_on (insert x) (Pow F)")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1627
   apply (simp add: card_image Pow_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1628
  apply (unfold inj_on_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1629
  apply (blast elim!: equalityE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1630
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1631
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1632
text {* Relates to equivalence classes.  Based on a theorem of
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1633
F. Kammüller's.  *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1634
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1635
lemma dvd_partition:
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1636
  "finite (Union C) ==>
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1637
    ALL c : C. k dvd card c ==>
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
  1638
    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1639
  k dvd card (Union C)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1640
apply(frule finite_UnionD)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1641
apply(rotate_tac -1)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
  1642
  apply (induct set: Finites, simp_all, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1643
  apply (subst card_Un_disjoint)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1644
  apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1645
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1646
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1647
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1648
subsubsection {* Theorems about @{text "choose"} *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1649
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1650
text {*
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1651
  \medskip Basic theorem about @{text "choose"}.  By Florian
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1652
  Kamm\"uller, tidied by LCP.
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1653
*}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1654
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1655
lemma card_s_0_eq_empty:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1656
    "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1657
  apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1658
  apply (simp cong add: rev_conj_cong)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1659
  done
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1660
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1661
lemma choose_deconstruct: "finite M ==> x \<notin> M
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1662
  ==> {s. s <= insert x M & card(s) = Suc k}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1663
       = {s. s <= M & card(s) = Suc k} Un
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1664
         {s. EX t. t <= M & card(t) = k & s = insert x t}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1665
  apply safe
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1666
   apply (auto intro: finite_subset [THEN card_insert_disjoint])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1667
  apply (drule_tac x = "xa - {x}" in spec)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1668
  apply (subgoal_tac "x \<notin> xa", auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1669
  apply (erule rev_mp, subst card_Diff_singleton)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1670
  apply (auto intro: finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1671
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1672
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1673
text{*There are as many subsets of @{term A} having cardinality @{term k}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1674
 as there are sets obtained from the former by inserting a fixed element
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1675
 @{term x} into each.*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1676
lemma constr_bij:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1677
   "[|finite A; x \<notin> A|] ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1678
    card {B. EX C. C <= A & card(C) = k & B = insert x C} =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1679
    card {B. B <= A & card(B) = k}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1680
  apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1681
       apply (auto elim!: equalityE simp add: inj_on_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1682
    apply (subst Diff_insert0, auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1683
   txt {* finiteness of the two sets *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1684
   apply (rule_tac [2] B = "Pow (A)" in finite_subset)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1685
   apply (rule_tac B = "Pow (insert x A)" in finite_subset)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1686
   apply fast+
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1687
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1688
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1689
text {*
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1690
  Main theorem: combinatorial statement about number of subsets of a set.
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1691
*}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1692
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1693
lemma n_sub_lemma:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1694
  "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1695
  apply (induct k)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1696
   apply (simp add: card_s_0_eq_empty, atomize)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1697
  apply (rotate_tac -1, erule finite_induct)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1698
   apply (simp_all (no_asm_simp) cong add: conj_cong
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1699
     add: card_s_0_eq_empty choose_deconstruct)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1700
  apply (subst card_Un_disjoint)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1701
     prefer 4 apply (force simp add: constr_bij)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1702
    prefer 3 apply force
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1703
   prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1704
     finite_subset [of _ "Pow (insert x F)", standard])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1705
  apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1706
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1707
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1708
theorem n_subsets:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1709
    "finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1710
  by (simp add: n_sub_lemma)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1711
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1712
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1713
subsection{* A fold functional for non-empty sets *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1714
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1715
text{* Does not require start value. *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1716
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1717
consts
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1718
  foldSet1 :: "('a => 'a => 'a) => ('a set \<times> 'a) set"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1719
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1720
inductive "foldSet1 f"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1721
intros
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1722
foldSet1_singletonI [intro]: "({a}, a) : foldSet1 f"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1723
foldSet1_insertI [intro]:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1724
 "\<lbrakk> (A, x) : foldSet1 f; a \<notin> A; A \<noteq> {} \<rbrakk>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1725
  \<Longrightarrow> (insert a A, f a x) : foldSet1 f"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1726
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1727
constdefs
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1728
  fold1 :: "('a => 'a => 'a) => 'a set => 'a"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1729
  "fold1 f A == THE x. (A, x) : foldSet1 f"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1730
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1731
lemma foldSet1_nonempty:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1732
 "(A, x) : foldSet1 f \<Longrightarrow> A \<noteq> {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1733
by(erule foldSet1.cases, simp_all) 
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1734
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1735
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1736
inductive_cases empty_foldSet1E [elim!]: "({}, x) : foldSet1 f"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1737
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1738
lemma foldSet1_sing[iff]: "(({a},b) : foldSet1 f) = (a = b)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1739
apply(rule iffI)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1740
 prefer 2 apply fast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1741
apply (erule foldSet1.cases)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1742
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1743
apply (erule foldSet1.cases)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1744
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1745
apply blast
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1746
done
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1747
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1748
lemma Diff1_foldSet1:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1749
  "(A - {x}, y) : foldSet1 f ==> x: A ==> (A, f x y) : foldSet1 f"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1750
by (erule insert_Diff [THEN subst], rule foldSet1.intros,
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1751
    auto dest!:foldSet1_nonempty)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1752
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1753
lemma foldSet1_imp_finite: "(A, x) : foldSet1 f ==> finite A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1754
  by (induct set: foldSet1) auto
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1755
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1756
lemma finite_nonempty_imp_foldSet1:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1757
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. (A, x) : foldSet1 f"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1758
  by (induct set: Finites) auto
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1759
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1760
lemma (in ACf) foldSet1_determ_aux:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1761
  "!!A x y. \<lbrakk> card A < n; (A, x) : foldSet1 f; (A, y) : foldSet1 f \<rbrakk> \<Longrightarrow> y = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1762
proof (induct n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1763
  case 0 thus ?case by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1764
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1765
  case (Suc n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1766
  have IH: "!!A x y. \<lbrakk>card A < n; (A, x) \<in> foldSet1 f; (A, y) \<in> foldSet1 f\<rbrakk>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1767
           \<Longrightarrow> y = x" and card: "card A < Suc n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1768
  and Afoldx: "(A, x) \<in> foldSet1 f" and Afoldy: "(A, y) \<in> foldSet1 f" .
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1769
  from card have "card A < n \<or> card A = n" by arith
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1770
  thus ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1771
  proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1772
    assume less: "card A < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1773
    show ?thesis by(rule IH[OF less Afoldx Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1774
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1775
    assume cardA: "card A = n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1776
    show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1777
    proof (rule foldSet1.cases[OF Afoldx])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1778
      fix a assume "(A, x) = ({a}, a)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1779
      thus "y = x" using Afoldy by (simp add:foldSet1_sing)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1780
    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1781
      fix Ax ax x'
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1782
      assume eq1: "(A, x) = (insert ax Ax, ax \<cdot> x')"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1783
	and x': "(Ax, x') \<in> foldSet1 f" and notinx: "ax \<notin> Ax"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1784
	and Axnon: "Ax \<noteq> {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1785
      hence A1: "A = insert ax Ax" and x: "x = ax \<cdot> x'" by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1786
      show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1787
      proof (rule foldSet1.cases[OF Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1788
	fix ay assume "(A, y) = ({ay}, ay)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1789
	thus ?thesis using eq1 x' Axnon notinx
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1790
	  by (fastsimp simp:foldSet1_sing)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1791
      next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1792
	fix Ay ay y'
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1793
	assume eq2: "(A, y) = (insert ay Ay, ay \<cdot> y')"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1794
	  and y': "(Ay, y') \<in> foldSet1 f" and notiny: "ay \<notin> Ay"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1795
	  and Aynon: "Ay \<noteq> {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1796
	hence A2: "A = insert ay Ay" and y: "y = ay \<cdot> y'" by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1797
	have finA: "finite A" by(rule foldSet1_imp_finite[OF Afoldx])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1798
	with cardA A1 notinx have less: "card Ax < n" by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1799
	show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1800
	proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1801
	  assume "ax = ay"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1802
	  then moreover have "Ax = Ay" using A1 A2 notinx notiny by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1803
	  ultimately show ?thesis using IH[OF less x'] y' eq1 eq2 by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1804
	next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1805
	  assume diff: "ax \<noteq> ay"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1806
	  let ?B = "Ax - {ay}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1807
	  have Ax: "Ax = insert ay ?B" and Ay: "Ay = insert ax ?B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1808
	    using A1 A2 notinx notiny diff by(blast elim!:equalityE)+
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1809
	  show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1810
	  proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1811
	    assume "?B = {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1812
	    with Ax Ay show ?thesis using x' y' x y by(simp add:commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1813
	  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1814
	    assume Bnon: "?B \<noteq> {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1815
	    moreover have "finite ?B" using finA A1 by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1816
	    ultimately obtain b where Bfoldb: "(?B,b) \<in> foldSet1 f"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1817
	      using finite_nonempty_imp_foldSet1 by(blast)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1818
	    moreover have ayinAx: "ay \<in> Ax" using Ax by(auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1819
	    ultimately have "(Ax,ay\<cdot>b) \<in> foldSet1 f" by(rule Diff1_foldSet1)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1820
	    hence "ay\<cdot>b = x'" by(rule IH[OF less x'])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1821
            moreover have "ax\<cdot>b = y'"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1822
	    proof (rule IH[OF _ y'])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1823
	      show "card Ay < n" using Ay cardA A1 notinx finA ayinAx
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1824
		by(auto simp:card_Diff1_less)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1825
	    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1826
	      show "(Ay,ax\<cdot>b) \<in> foldSet1 f" using Ay notinx Bfoldb Bnon
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1827
		by fastsimp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1828
	    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1829
	    ultimately show ?thesis using x y by(auto simp:AC)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1830
	  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1831
	qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1832
      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1833
    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1834
  qed
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1835
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1836
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1837
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1838
lemma (in ACf) foldSet1_determ:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1839
  "(A, x) : foldSet1 f ==> (A, y) : foldSet1 f ==> y = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1840
by (blast intro: foldSet1_determ_aux [rule_format])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1841
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1842
lemma (in ACf) foldSet1_equality: "(A, y) : foldSet1 f ==> fold1 f A = y"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1843
  by (unfold fold1_def) (blast intro: foldSet1_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1844
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1845
lemma fold1_singleton: "fold1 f {a} = a"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1846
  by (unfold fold1_def) blast
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1847
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1848
lemma (in ACf) foldSet1_insert_aux: "x \<notin> A ==> A \<noteq> {} \<Longrightarrow> 
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1849
    ((insert x A, v) : foldSet1 f) =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1850
    (EX y. (A, y) : foldSet1 f & v = f x y)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1851
apply auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1852
apply (rule_tac A1 = A and f1 = f in finite_nonempty_imp_foldSet1 [THEN exE])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1853
  apply (fastsimp dest: foldSet1_imp_finite)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1854
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1855
apply (blast intro: foldSet1_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1856
done
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1857
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1858
lemma (in ACf) fold1_insert:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1859
  "finite A ==> x \<notin> A ==> A \<noteq> {} \<Longrightarrow> fold1 f (insert x A) = f x (fold1 f A)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1860
apply (unfold fold1_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1861
apply (simp add: foldSet1_insert_aux)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1862
apply (rule the_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1863
apply (auto intro: finite_nonempty_imp_foldSet1
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1864
    cong add: conj_cong simp add: fold1_def [symmetric] foldSet1_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1865
done
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1866
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1867
locale ACIf = ACf +
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1868
  assumes idem: "x \<cdot> x = x"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1869
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1870
lemma (in ACIf) fold1_insert2:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1871
assumes finA: "finite A" and nonA: "A \<noteq> {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1872
shows "fold1 f (insert a A) = f a (fold1 f A)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1873
proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1874
  assume "a \<in> A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1875
  then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1876
    by(blast dest: mk_disjoint_insert)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1877
  show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1878
  proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1879
    assume "B = {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1880
    thus ?thesis using A by(simp add:idem fold1_singleton)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1881
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1882
    assume nonB: "B \<noteq> {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1883
    from finA A have finB: "finite B" by(blast intro: finite_subset)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1884
    have "fold1 f (insert a A) = fold1 f (insert a B)" using A by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1885
    also have "\<dots> = f a (fold1 f B)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1886
      using finB nonB disj by(simp add: fold1_insert)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1887
    also have "\<dots> = f a (fold1 f A)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1888
      using A finB nonB disj by(simp add:idem fold1_insert assoc[symmetric])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1889
    finally show ?thesis .
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1890
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1891
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1892
  assume "a \<notin> A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1893
  with finA nonA show ?thesis by(simp add:fold1_insert)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1894
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1895
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1896
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1897
text{* Now the recursion rules for definitions: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1898
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1899
lemma fold1_singleton_def: "g \<equiv> fold1 f \<Longrightarrow> g {a} = a"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1900
by(simp add:fold1_singleton)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1901
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1902
lemma (in ACf) fold1_insert_def:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1903
  "\<lbrakk> g \<equiv> fold1 f; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1904
by(simp add:fold1_insert)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1905
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1906
lemma (in ACIf) fold1_insert2_def:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1907
  "\<lbrakk> g \<equiv> fold1 f; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1908
by(simp add:fold1_insert2)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1909
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1910
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1911
subsection{*Min and Max*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1912
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1913
text{* As an application of @{text fold1} we define the minimal and
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1914
maximal element of a (non-empty) set over a linear order. First we
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1915
show that @{text min} and @{text max} are ACI: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1916
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1917
lemma ACf_min: "ACf(min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1918
apply(rule ACf.intro)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1919
apply(auto simp:min_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1920
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1921
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1922
lemma ACIf_min: "ACIf(min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1923
apply(rule ACIf.intro[OF ACf_min])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1924
apply(rule ACIf_axioms.intro)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1925
apply(auto simp:min_def)
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1926
done
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1927
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1928
lemma ACf_max: "ACf(max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1929
apply(rule ACf.intro)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1930
apply(auto simp:max_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1931
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1932
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1933
lemma ACIf_max: "ACIf(max:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1934
apply(rule ACIf.intro[OF ACf_max])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1935
apply(rule ACIf_axioms.intro)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1936
apply(auto simp:max_def)
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1937
done
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1938
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1939
text{* Now we make the definitions: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1940
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1941
constdefs
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1942
  Min :: "('a::linorder)set => 'a"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1943
  "Min  ==  fold1 min"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1944
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1945
  Max :: "('a::linorder)set => 'a"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1946
  "Max  ==  fold1 max"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1947
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1948
text{* Now we instantiate the recursion equations and declare them
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1949
simplification rules: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1950
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1951
declare
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1952
  fold1_singleton_def[OF Min_def, simp]
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1953
  ACIf.fold1_insert2_def[OF ACIf_min Min_def, simp]
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1954
  fold1_singleton_def[OF Max_def, simp]
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1955
  ACIf.fold1_insert2_def[OF ACIf_max Max_def, simp]
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1956
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1957
text{* Now we prove some properties by induction: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1958
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1959
lemma Min_in [simp]:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1960
  assumes a: "finite S"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1961
  shows "S \<noteq> {} \<Longrightarrow> Min S \<in> S"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1962
using a
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1963
proof induct
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1964
  case empty thus ?case by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1965
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1966
  case (insert x S)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1967
  show ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1968
  proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1969
    assume "S = {}" thus ?thesis by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1970
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1971
    assume "S \<noteq> {}" thus ?thesis using insert by (simp add:min_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1972
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1973
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1974
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1975
lemma Min_le [simp]:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1976
  assumes a: "finite S"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1977
  shows "\<lbrakk> S \<noteq> {}; x \<in> S \<rbrakk> \<Longrightarrow> Min S \<le> x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1978
using a
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1979
proof induct
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1980
  case empty thus ?case by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1981
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1982
  case (insert y S)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1983
  show ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1984
  proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1985
    assume "S = {}" thus ?thesis using insert by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1986
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1987
    assume "S \<noteq> {}" thus ?thesis using insert by (auto simp add:min_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1988
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1989
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1990
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1991
lemma Max_in [simp]:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1992
  assumes a: "finite S"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1993
  shows "S \<noteq> {} \<Longrightarrow> Max S \<in> S"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1994
using a
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1995
proof induct
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1996
  case empty thus ?case by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1997
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1998
  case (insert x S)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1999
  show ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2000
  proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2001
    assume "S = {}" thus ?thesis by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2002
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2003
    assume "S \<noteq> {}" thus ?thesis using insert by (simp add:max_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2004
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2005
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2006
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2007
lemma Max_le [simp]:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2008
  assumes a: "finite S"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2009
  shows "\<lbrakk> S \<noteq> {}; x \<in> S \<rbrakk> \<Longrightarrow> x \<le> Max S"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2010
using a
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2011
proof induct
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2012
  case empty thus ?case by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2013
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2014
  case (insert y S)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2015
  show ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2016
  proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2017
    assume "S = {}" thus ?thesis using insert by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2018
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2019
    assume "S \<noteq> {}" thus ?thesis using insert by (auto simp add:max_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2020
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2021
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  2022
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  2023
15042
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15004
diff changeset
  2024
end