src/HOL/Finite_Set.thy
author nipkow
Thu, 09 Dec 2004 18:30:59 +0100
changeset 15392 290bc97038c7
parent 15376 302ef111b621
child 15402 97204f3b4705
permissions -rw-r--r--
First step in reorganizing Finite_Set
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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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FIXME: define card via fold and derive as many lemmas as possible from fold.
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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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    qed
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  qed
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qed
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   158
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   159
lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
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   160
  by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
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parents:
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   161
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   162
lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
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  -- {* The converse obviously fails. *}
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parents:
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   164
  by (blast intro: finite_subset)
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parents:
diff changeset
   165
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   166
lemma finite_insert [simp]: "finite (insert a A) = finite A"
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parents:
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   167
  apply (subst insert_is_Un)
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  apply (simp only: finite_Un, blast)
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   169
  done
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   170
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   171
lemma finite_Union[simp, intro]:
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   172
 "\<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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   174
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lemma finite_empty_induct:
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   176
  "finite A ==>
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  P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
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   178
proof -
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   179
  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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parents:
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   181
  have "P (A - A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
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   182
  proof -
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   183
    fix c b :: "'a set"
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   184
    presume c: "finite c" and b: "finite b"
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   185
      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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   186
    from c show "c \<subseteq> b ==> P (b - c)"
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parents:
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   187
    proof induct
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   188
      case empty
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parents:
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   189
      from P1 show ?case by simp
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   190
    next
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   191
      case (insert x F)
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   192
      have "P (b - F - {x})"
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parents:
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   193
      proof (rule P2)
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parents:
diff changeset
   194
        from _ b show "finite (b - F)" by (rule finite_subset) blast
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wenzelm
parents:
diff changeset
   195
        from insert show "x \<in> b - F" by simp
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parents:
diff changeset
   196
        from insert show "P (b - F)" by simp
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parents:
diff changeset
   197
      qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   198
      also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
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parents:
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   199
      finally show ?case .
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parents:
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   200
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   201
  next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   202
    show "A \<subseteq> A" ..
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
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   203
  qed
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parents:
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   204
  thus "P {}" by simp
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parents:
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   205
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
diff changeset
   206
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   207
lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
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   208
  by (rule Diff_subset [THEN finite_subset])
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wenzelm
parents:
diff changeset
   209
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   210
lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   211
  apply (subst Diff_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   212
  apply (case_tac "a : A - B")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   213
   apply (rule finite_insert [symmetric, THEN trans])
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   214
   apply (subst insert_Diff, simp_all)
12396
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parents:
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   215
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
diff changeset
   216
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   217
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   218
text {* Image and Inverse Image over Finite Sets *}
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   219
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   220
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
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   221
  -- {* The image of a finite set is finite. *}
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parents: 13737
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   222
  by (induct set: Finites) simp_all
ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   223
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   224
lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
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   225
  apply (frule finite_imageI)
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paulson
parents: 14331
diff changeset
   226
  apply (erule finite_subset, assumption)
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paulson
parents: 14331
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   227
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   228
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   229
lemma finite_range_imageI:
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   230
    "finite (range g) ==> finite (range (%x. f (g x)))"
14208
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parents: 13825
diff changeset
   231
  apply (drule finite_imageI, simp)
13825
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paulson
parents: 13737
diff changeset
   232
  done
ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   233
12396
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   234
lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
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wenzelm
parents:
diff changeset
   235
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   236
  have aux: "!!A. finite (A - {}) = finite A" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   237
  fix B :: "'a set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   238
  assume "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   239
  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   240
    apply induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   241
     apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   242
    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
   243
     apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   244
     apply (simp (no_asm_use) add: inj_on_def)
14208
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paulson
parents: 13825
diff changeset
   245
     apply (blast dest!: aux [THEN iffD1], atomize)
12396
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wenzelm
parents:
diff changeset
   246
    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
14208
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paulson
parents: 13825
diff changeset
   247
    apply (frule subsetD [OF equalityD2 insertI1], clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   248
    apply (rule_tac x = xa in bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   249
     apply (simp_all add: inj_on_image_set_diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   250
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   251
qed (rule refl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   252
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   253
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paulson
parents: 13737
diff changeset
   254
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
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parents: 13737
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   255
  -- {* The inverse image of a singleton under an injective function
ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   256
         is included in a singleton. *}
14430
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paulson
parents: 14331
diff changeset
   257
  apply (auto simp add: inj_on_def)
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paulson
parents: 14331
diff changeset
   258
  apply (blast intro: the_equality [symmetric])
13825
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paulson
parents: 13737
diff changeset
   259
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   260
ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   261
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
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paulson
parents: 13737
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   262
  -- {* The inverse image of a finite set under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   263
         is finite. *}
14430
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paulson
parents: 14331
diff changeset
   264
  apply (induct set: Finites, simp_all)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   265
  apply (subst vimage_insert)
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paulson
parents: 14331
diff changeset
   266
  apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
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parents: 13737
diff changeset
   267
  done
ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   268
ef4c41e7956a new inverse image lemmas
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diff changeset
   269
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   270
text {* The finite UNION of finite sets *}
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   271
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   272
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
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   273
  by (induct set: Finites) simp_all
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wenzelm
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diff changeset
   274
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   275
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   276
  Strengthen RHS to
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   277
  @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
12396
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wenzelm
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diff changeset
   278
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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diff changeset
   279
  We'd need to prove
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parents: 14331
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   280
  @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
12396
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wenzelm
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diff changeset
   281
  by induction. *}
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wenzelm
parents:
diff changeset
   282
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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diff changeset
   283
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
   284
  by (blast intro: finite_UN_I finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   285
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   286
15392
290bc97038c7 First step in reorganizing Finite_Set
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parents: 15376
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   287
text {* Sigma of finite sets *}
12396
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wenzelm
parents:
diff changeset
   288
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   289
lemma finite_SigmaI [simp]:
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wenzelm
parents:
diff changeset
   290
    "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   291
  by (unfold Sigma_def) (blast intro!: finite_UN_I)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   292
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   293
lemma finite_Prod_UNIV:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   294
    "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
   295
  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   296
   apply (erule ssubst)
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parents: 13825
diff changeset
   297
   apply (erule finite_SigmaI, auto)
12396
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wenzelm
parents:
diff changeset
   298
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   299
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   300
instance unit :: finite
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wenzelm
parents:
diff changeset
   301
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   302
  have "finite {()}" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   303
  also have "{()} = UNIV" by auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   304
  finally show "finite (UNIV :: unit set)" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   305
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   306
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   307
instance * :: (finite, finite) finite
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wenzelm
parents:
diff changeset
   308
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   309
  show "finite (UNIV :: ('a \<times> 'b) set)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   310
  proof (rule finite_Prod_UNIV)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   311
    show "finite (UNIV :: 'a set)" by (rule finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   312
    show "finite (UNIV :: 'b set)" by (rule finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   313
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   314
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   315
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   316
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   317
text {* The powerset of a finite set *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   318
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   319
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   320
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   321
  assume "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   322
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   323
  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
   324
next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   325
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   326
  thus "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   327
    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   328
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   329
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   330
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   331
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   332
by(blast intro: finite_subset[OF subset_Pow_Union])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   333
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   334
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   335
lemma finite_converse [iff]: "finite (r^-1) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   336
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   337
   apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   338
   apply (rule iffI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   339
    apply (erule finite_imageD [unfolded inj_on_def])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   340
    apply (simp split add: split_split)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   341
   apply (erule finite_imageI)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   342
  apply (simp add: converse_def image_def, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   343
  apply (rule bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   344
   prefer 2 apply assumption
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   345
  apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   346
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   347
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   348
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   349
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   350
Ehmety) *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   351
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   352
lemma finite_Field: "finite r ==> finite (Field r)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   353
  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   354
  apply (induct set: Finites)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   355
   apply (auto simp add: Field_def Domain_insert Range_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   356
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   357
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   358
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   359
  apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   360
  apply (erule trancl_induct)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   361
   apply (auto simp add: Field_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   362
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   363
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   364
lemma finite_trancl: "finite (r^+) = finite r"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   365
  apply auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   366
   prefer 2
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   367
   apply (rule trancl_subset_Field2 [THEN finite_subset])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   368
   apply (rule finite_SigmaI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   369
    prefer 3
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 13595
diff changeset
   370
    apply (blast intro: r_into_trancl' finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   371
   apply (auto simp add: finite_Field)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   372
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   373
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   374
lemma finite_cartesian_product: "[| finite A; finite B |] ==>
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   375
    finite (A <*> B)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   376
  by (rule finite_SigmaI)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   377
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   378
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   379
subsection {* A fold functional for finite sets *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   380
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   381
text {* The intended behaviour is
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   382
@{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
   383
if @{text f} is associative-commutative. For an application of @{text fold}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   384
se the definitions of sums and products over finite sets.
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   385
*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   386
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   387
consts
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   388
  foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   389
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   390
inductive "foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   391
intros
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   392
emptyI [intro]: "({}, e) : foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   393
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
   394
 \<Longrightarrow> (insert x A, f (g x) y) : foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   395
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   396
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   397
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   398
constdefs
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   399
  fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   400
  "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
   401
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   402
lemma Diff1_foldSet:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   403
  "(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
   404
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   405
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   406
lemma foldSet_imp_finite: "(A, x) : foldSet f g e ==> finite A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   407
  by (induct set: foldSet) auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   408
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   409
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
   410
  by (induct set: Finites) auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   411
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   412
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   413
subsubsection {* Commutative monoids *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   414
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   415
locale ACf =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   416
  fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   417
  assumes commute: "x \<cdot> y = y \<cdot> x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   418
    and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
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
locale ACe = ACf +
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   421
  fixes e :: 'a
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   422
  assumes ident [simp]: "x \<cdot> e = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   423
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   424
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
   425
proof -
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   426
  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
   427
  also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   428
  also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   429
  finally show ?thesis .
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   430
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   431
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   432
lemmas (in ACf) AC = assoc commute left_commute
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   433
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   434
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   435
proof -
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   436
  have "x \<cdot> e = x" by (rule ident)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   437
  thus ?thesis by (subst commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   438
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   439
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   440
subsubsection{*From @{term foldSet} to @{term fold}*}
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 (in ACf) foldSet_determ_aux:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   443
  "!!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
   444
   \<Longrightarrow> x' = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   445
proof (induct n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   446
  case 0 thus ?case by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   447
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   448
  case (Suc n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   449
  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
   450
           \<Longrightarrow> x' = x" and card: "A = h`{i. i<Suc n}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   451
  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
   452
  show ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   453
  proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   454
    assume "EX k<n. h n = h k"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   455
    hence card': "A = h ` {i. i < n}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   456
      using card by (auto simp:image_def less_Suc_eq)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   457
    show ?thesis by(rule IH[OF card' Afoldx Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   458
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   459
    assume new: "\<not>(EX k<n. h n = h k)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   460
    show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   461
    proof (rule foldSet.cases[OF Afoldx])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   462
      assume "(A, x) = ({}, e)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   463
      thus "x' = x" using Afoldy by (auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   464
    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   465
      fix B b y
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   466
      assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   467
	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
   468
      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
   469
      show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   470
      proof (rule foldSet.cases[OF Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   471
	assume "(A,x') = ({}, e)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   472
	thus ?thesis using A1 by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   473
      next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   474
	fix C c z
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   475
	assume eq2: "(A,x') = (insert c C, g c \<cdot> z)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   476
	  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
   477
	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
   478
	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
   479
	have finA: "finite A" by(rule foldSet_imp_finite[OF Afoldx])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   480
(* move down? *)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   481
	have less: "B = ?h`{i. i<n}" (is "_ = ?r")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   482
	proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   483
	  show "B \<subseteq> ?r"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   484
	  proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   485
	    fix u assume "u \<in> B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   486
	    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
   487
	    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
   488
	      using card by(auto simp:image_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   489
	    show "u \<in> ?r"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   490
	    proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   491
	      assume "i\<^isub>u < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   492
	      thus ?thesis using unotb by(fastsimp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   493
	    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   494
	      assume "\<not> i\<^isub>u < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   495
	      with below have [simp]: "i\<^isub>u = n" by arith
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   496
	      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
   497
		using A1 card by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   498
	      have "i\<^isub>k < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   499
	      proof (rule ccontr)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   500
		assume "\<not> i\<^isub>k < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   501
		hence "i\<^isub>k = n" using i\<^isub>k by arith
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   502
		thus False using unotb by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   503
	      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   504
	      thus ?thesis by(auto simp add:image_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   505
	    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   506
	  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   507
	next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   508
	  show "?r \<subseteq> B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   509
	  proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   510
	    fix u assume "u \<in> ?r"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   511
	    then obtain i\<^isub>u where below: "i\<^isub>u < n" and
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   512
              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
   513
	      by(auto simp:image_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   514
	    from or show "u \<in> B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   515
	    proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   516
	      assume [simp]: "b = h i\<^isub>u \<and> u = h n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   517
	      have "u \<in> A" using card by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   518
              moreover have "u \<noteq> b" using new below by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   519
	      ultimately show "u \<in> B" using A1 by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   520
	    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   521
	      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
   522
	      moreover hence "u \<in> A" using card below by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   523
	      ultimately show "u \<in> B" using A1 by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   524
	    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   525
	  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   526
	qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   527
	show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   528
	proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   529
	  assume "b = c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   530
	  then moreover have "B = C" using A1 A2 notinB notinC by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   531
	  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
   532
	next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   533
	  assume diff: "b \<noteq> c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   534
	  let ?D = "B - {c}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   535
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   536
	    using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   537
	  have "finite ?D" using finA A1 by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   538
	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   539
	    using finite_imp_foldSet by rules
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   540
	  moreover have cinB: "c \<in> B" using B by(auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   541
	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   542
	    by(rule Diff1_foldSet)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   543
	  hence "g c \<cdot> d = y" by(rule IH[OF less y])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   544
          moreover have "g b \<cdot> d = z"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   545
	  proof (rule IH[OF _ z])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   546
	    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
   547
	    show "C = ?h`{i. i<n}" (is "_ = ?r")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   548
	    proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   549
	      show "C \<subseteq> ?r"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   550
	      proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   551
		fix u assume "u \<in> C"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   552
		hence uinA: "u \<in> A" and unotc: "u \<noteq> c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   553
		  using A2 notinC by blast+
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   554
		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
   555
		  using card by(auto simp:image_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   556
		show "u \<in> ?r"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   557
		proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   558
		  assume "i\<^isub>u < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   559
		  thus ?thesis using unotc by(fastsimp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   560
		next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   561
		  assume "\<not> i\<^isub>u < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   562
		  with below have [simp]: "i\<^isub>u = n" by arith
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   563
		  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
   564
		    using A2 card by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   565
		  have "i\<^isub>k < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   566
		  proof (rule ccontr)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   567
		    assume "\<not> i\<^isub>k < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   568
		    hence "i\<^isub>k = n" using i\<^isub>k by arith
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   569
		    thus False using unotc by simp
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
		  thus ?thesis by(auto simp add:image_def)
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
	      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   574
	    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   575
	      show "?r \<subseteq> C"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   576
	      proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   577
		fix u assume "u \<in> ?r"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   578
		then obtain i\<^isub>u where below: "i\<^isub>u < n" and
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   579
		  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
   580
		  by(auto simp:image_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   581
		from or show "u \<in> C"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   582
		proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   583
		  assume [simp]: "c = h i\<^isub>u \<and> u = h n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   584
		  have "u \<in> A" using card by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   585
		  moreover have "u \<noteq> c" using new below by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   586
		  ultimately show "u \<in> C" using A2 by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   587
		next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   588
		  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
   589
		  moreover hence "u \<in> A" using card below by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   590
		  ultimately show "u \<in> C" using A2 by blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   591
		qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   592
	      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   593
	    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   594
	  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   595
	    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
   596
	      by fastsimp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   597
	  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   598
	  ultimately show ?thesis using x x' by(auto simp:AC)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   599
	qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   600
      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   601
    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   602
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   603
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   604
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   605
(* The same proof, but using card 
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   606
lemma (in ACf) foldSet_determ_aux:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   607
  "!!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
   608
   \<Longrightarrow> x' = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   609
proof (induct n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   610
  case 0 thus ?case by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   611
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   612
  case (Suc n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   613
  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
   614
           \<Longrightarrow> x' = x" and card: "card A < Suc n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   615
  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
   616
  from card have "card A < n \<or> card A = n" by arith
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   617
  thus ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   618
  proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   619
    assume less: "card A < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   620
    show ?thesis by(rule IH[OF less Afoldx Afoldy])
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
    assume cardA: "card A = n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   623
    show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   624
    proof (rule foldSet.cases[OF Afoldx])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   625
      assume "(A, x) = ({}, e)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   626
      thus "x' = x" using Afoldy by (auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   627
    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   628
      fix B b y
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   629
      assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   630
	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
   631
      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
   632
      show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   633
      proof (rule foldSet.cases[OF Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   634
	assume "(A,x') = ({}, e)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   635
	thus ?thesis using A1 by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   636
      next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   637
	fix C c z
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   638
	assume eq2: "(A,x') = (insert c C, g c \<cdot> z)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   639
	  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
   640
	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
   641
	have finA: "finite A" by(rule foldSet_imp_finite[OF Afoldx])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   642
	with cardA A1 notinB have less: "card B < n" by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   643
	show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   644
	proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   645
	  assume "b = c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   646
	  then moreover have "B = C" using A1 A2 notinB notinC by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   647
	  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
   648
	next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   649
	  assume diff: "b \<noteq> c"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   650
	  let ?D = "B - {c}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   651
	  have B: "B = insert c ?D" and C: "C = insert b ?D"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   652
	    using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   653
	  have "finite ?D" using finA A1 by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   654
	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   655
	    using finite_imp_foldSet by rules
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   656
	  moreover have cinB: "c \<in> B" using B by(auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   657
	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   658
	    by(rule Diff1_foldSet)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   659
	  hence "g c \<cdot> d = y" by(rule IH[OF less y])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   660
          moreover have "g b \<cdot> d = z"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   661
	  proof (rule IH[OF _ z])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   662
	    show "card C < n" using C cardA A1 notinB finA cinB
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   663
	      by(auto simp:card_Diff1_less)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   664
	  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   665
	    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
   666
	      by fastsimp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   667
	  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   668
	  ultimately show ?thesis using x x' by(auto simp:AC)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   669
	qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   670
      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   671
    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   672
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   673
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   674
*)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   675
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   676
lemma (in ACf) foldSet_determ:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   677
  "(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
   678
apply(frule foldSet_imp_finite)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   679
apply(simp add:finite_conv_nat_seg_image)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   680
apply(blast intro: foldSet_determ_aux [rule_format])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   681
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   682
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   683
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
   684
  by (unfold fold_def) (blast intro: foldSet_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   685
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   686
text{* The base case for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   687
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   688
lemma fold_empty [simp]: "fold f g e {} = e"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   689
  by (unfold fold_def) blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   690
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   691
lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   692
    ((insert x A, v) : foldSet f g e) =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   693
    (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
   694
  apply auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   695
  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
   696
   apply (fastsimp dest: foldSet_imp_finite)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   697
  apply (blast intro: foldSet_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   698
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   699
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   700
text{* The recursion equation for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   701
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   702
lemma (in ACf) fold_insert[simp]:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   703
    "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
   704
  apply (unfold fold_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   705
  apply (simp add: fold_insert_aux)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   706
  apply (rule the_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   707
  apply (auto intro: finite_imp_foldSet
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   708
    cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   709
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   710
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   711
text{* Its definitional form: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   712
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   713
corollary (in ACf) fold_insert_def:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   714
    "\<lbrakk> F \<equiv> fold f g e; finite A; x \<notin> A \<rbrakk> \<Longrightarrow> F (insert x A) = f (g x) (F A)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   715
by(simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   716
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   717
declare
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   718
  empty_foldSetE [rule del]  foldSet.intros [rule del]
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   719
  -- {* Delete rules to do with @{text foldSet} relation. *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   720
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   721
subsubsection{*Lemmas about @{text fold}*}
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) fold_commute:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   724
  "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
   725
  apply (induct set: Finites, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   726
  apply (simp add: left_commute)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   727
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   728
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   729
lemma (in ACf) fold_nest_Un_Int:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   730
  "finite A ==> finite B
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   731
    ==> 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
   732
  apply (induct set: Finites, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   733
  apply (simp add: fold_commute Int_insert_left insert_absorb)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   734
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   735
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   736
lemma (in ACf) fold_nest_Un_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   737
  "finite A ==> finite B ==> A Int B = {}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   738
    ==> 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
   739
  by (simp add: fold_nest_Un_Int)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   740
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   741
lemma (in ACf) fold_reindex:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   742
assumes fin: "finite B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   743
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
   744
using fin apply (induct)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   745
 apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   746
apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   747
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   748
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   749
lemma (in ACe) fold_Un_Int:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   750
  "finite A ==> finite B ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   751
    fold f g e A \<cdot> fold f g e B =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   752
    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
   753
  apply (induct set: Finites, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   754
  apply (simp add: AC insert_absorb Int_insert_left)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   755
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   756
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   757
corollary (in ACe) fold_Un_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   758
  "finite A ==> finite B ==> A Int B = {} ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   759
    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
   760
  by (simp add: fold_Un_Int)
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
lemma (in ACe) fold_UN_disjoint:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   763
  "\<lbrakk> finite I; ALL i:I. finite (A i);
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   764
     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
   765
   \<Longrightarrow> fold f g e (UNION I A) =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   766
       fold f (%i. fold f g e (A i)) e I"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   767
  apply (induct set: Finites, simp, atomize)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   768
  apply (subgoal_tac "ALL i:F. x \<noteq> i")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   769
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   770
  apply (subgoal_tac "A x Int UNION F A = {}")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   771
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   772
  apply (simp add: fold_Un_disjoint)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   773
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   774
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   775
lemma (in ACf) fold_cong:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   776
  "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
   777
  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
   778
   apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   779
  apply (erule finite_induct, simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   780
  apply (simp add: subset_insert_iff, clarify)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   781
  apply (subgoal_tac "finite C")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   782
   prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   783
  apply (subgoal_tac "C = insert x (C - {x})")
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   784
   prefer 2 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   785
  apply (erule ssubst)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   786
  apply (drule spec)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   787
  apply (erule (1) notE impE)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   788
  apply (simp add: Ball_def del: insert_Diff_single)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   789
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   790
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   791
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
   792
  fold f (%x. fold f (g x) e (B x)) e A =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   793
  fold f (split g) e (SIGMA x:A. B x)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   794
apply (subst Sigma_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   795
apply (subst fold_UN_disjoint)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   796
   apply assumption
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   797
  apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   798
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   799
apply (erule fold_cong)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   800
apply (subst fold_UN_disjoint)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   801
   apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   802
  apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   803
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   804
apply (simp)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   805
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   806
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   807
lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   808
   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
   809
apply (erule finite_induct)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   810
 apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   811
apply (simp add:AC)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   812
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   813
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   814
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   815
subsection {* Finite cardinality *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   816
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   817
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   818
  This definition, although traditional, is ugly to work with: @{text
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   819
  "card A == LEAST n. EX f. A = {f i | i. i < n}"}.  Therefore we have
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   820
  switched to an inductive one:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   821
*}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   822
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   823
consts cardR :: "('a set \<times> nat) set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   824
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   825
inductive cardR
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   826
  intros
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   827
    EmptyI: "({}, 0) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   828
    InsertI: "(A, n) : cardR ==> a \<notin> A ==> (insert a A, Suc n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   829
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   830
constdefs
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   831
  card :: "'a set => nat"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   832
  "card A == THE n. (A, n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   833
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   834
inductive_cases cardR_emptyE: "({}, n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   835
inductive_cases cardR_insertE: "(insert a A,n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   836
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   837
lemma cardR_SucD: "(A, n) : cardR ==> a : A --> (EX m. n = Suc m)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   838
  by (induct set: cardR) simp_all
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   839
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   840
lemma cardR_determ_aux1:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   841
    "(A, m): cardR ==> (!!n a. m = Suc n ==> a:A ==> (A - {a}, n) : cardR)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   842
  apply (induct set: cardR, auto)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   843
  apply (simp add: insert_Diff_if, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   844
  apply (drule cardR_SucD)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   845
  apply (blast intro!: cardR.intros)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   846
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   847
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   848
lemma cardR_determ_aux2: "(insert a A, Suc m) : cardR ==> a \<notin> A ==> (A, m) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   849
  by (drule cardR_determ_aux1) auto
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   850
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   851
lemma cardR_determ: "(A, m): cardR ==> (!!n. (A, n) : cardR ==> n = m)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   852
  apply (induct set: cardR)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   853
   apply (safe elim!: cardR_emptyE cardR_insertE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   854
  apply (rename_tac B b m)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   855
  apply (case_tac "a = b")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   856
   apply (subgoal_tac "A = B")
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   857
    prefer 2 apply (blast elim: equalityE, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   858
  apply (subgoal_tac "EX C. A = insert b C & B = insert a C")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   859
   prefer 2
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   860
   apply (rule_tac x = "A Int B" in exI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   861
   apply (blast elim: equalityE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   862
  apply (frule_tac A = B in cardR_SucD)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   863
  apply (blast intro!: cardR.intros dest!: cardR_determ_aux2)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   864
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   865
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   866
lemma cardR_imp_finite: "(A, n) : cardR ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   867
  by (induct set: cardR) simp_all
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   868
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   869
lemma finite_imp_cardR: "finite A ==> EX n. (A, n) : cardR"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   870
  by (induct set: Finites) (auto intro!: cardR.intros)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   871
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   872
lemma card_equality: "(A,n) : cardR ==> card A = n"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   873
  by (unfold card_def) (blast intro: cardR_determ)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   874
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   875
lemma card_empty [simp]: "card {} = 0"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   876
  by (unfold card_def) (blast intro!: cardR.intros elim!: cardR_emptyE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   877
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   878
lemma card_insert_disjoint [simp]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   879
  "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   880
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   881
  assume x: "x \<notin> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   882
  hence aux: "!!n. ((insert x A, n) : cardR) = (EX m. (A, m) : cardR & n = Suc m)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   883
    apply (auto intro!: cardR.intros)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   884
    apply (rule_tac A1 = A in finite_imp_cardR [THEN exE])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   885
     apply (force dest: cardR_imp_finite)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   886
    apply (blast intro!: cardR.intros intro: cardR_determ)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   887
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   888
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   889
  thus ?thesis
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   890
    apply (simp add: card_def aux)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   891
    apply (rule the_equality)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   892
     apply (auto intro: finite_imp_cardR
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   893
       cong: conj_cong simp: card_def [symmetric] card_equality)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   894
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   895
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   896
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   897
lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   898
  apply auto
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   899
  apply (drule_tac a = x in mk_disjoint_insert, clarify)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   900
  apply (rotate_tac -1, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   901
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   902
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   903
lemma card_insert_if:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   904
    "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   905
  by (simp add: insert_absorb)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   906
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   907
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   908
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   909
apply(simp del:insert_Diff_single)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   910
done
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   911
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   912
lemma card_Diff_singleton:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   913
    "finite A ==> x: A ==> card (A - {x}) = card A - 1"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   914
  by (simp add: card_Suc_Diff1 [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   915
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   916
lemma card_Diff_singleton_if:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   917
    "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
   918
  by (simp add: card_Diff_singleton)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   919
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   920
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
   921
  by (simp add: card_insert_if card_Suc_Diff1)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   922
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   923
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
   924
  by (simp add: card_insert_if)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   925
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   926
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   927
  apply (induct set: Finites, simp, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   928
  apply (subgoal_tac "finite A & A - {x} <= F")
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   929
   prefer 2 apply (blast intro: finite_subset, atomize)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   930
  apply (drule_tac x = "A - {x}" in spec)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   931
  apply (simp add: card_Diff_singleton_if split add: split_if_asm)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   932
  apply (case_tac "card A", auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   933
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   934
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   935
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
   936
  apply (simp add: psubset_def linorder_not_le [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   937
  apply (blast dest: card_seteq)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   938
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   939
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   940
lemma card_mono: "finite B ==> A <= B ==> card A <= card B"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   941
  apply (case_tac "A = B", simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   942
  apply (simp add: linorder_not_less [symmetric])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   943
  apply (blast dest: card_seteq intro: order_less_imp_le)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   944
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   945
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   946
lemma card_Un_Int: "finite A ==> finite B
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   947
    ==> card A + card B = card (A Un B) + card (A Int B)"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   948
  apply (induct set: Finites, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   949
  apply (simp add: insert_absorb Int_insert_left)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   950
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   951
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   952
lemma card_Un_disjoint: "finite A ==> finite B
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   953
    ==> A Int B = {} ==> card (A Un B) = card A + card B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   954
  by (simp add: card_Un_Int)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   955
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   956
lemma card_Diff_subset:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   957
    "finite A ==> B <= A ==> card A - card B = card (A - B)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   958
  apply (subgoal_tac "(A - B) Un B = A")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   959
   prefer 2 apply blast
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   960
  apply (rule nat_add_right_cancel [THEN iffD1])
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   961
  apply (rule card_Un_disjoint [THEN subst])
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   962
     apply (erule_tac [4] ssubst)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   963
     prefer 3 apply blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   964
    apply (simp_all add: add_commute not_less_iff_le
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   965
      add_diff_inverse card_mono finite_subset)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   966
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   967
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   968
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
   969
  apply (rule Suc_less_SucD)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   970
  apply (simp add: card_Suc_Diff1)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   971
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   972
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   973
lemma card_Diff2_less:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   974
    "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
   975
  apply (case_tac "x = y")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   976
   apply (simp add: card_Diff1_less)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   977
  apply (rule less_trans)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   978
   prefer 2 apply (auto intro!: card_Diff1_less)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   979
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   980
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   981
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   982
  apply (case_tac "x : A")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   983
   apply (simp_all add: card_Diff1_less less_imp_le)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   984
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   985
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   986
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   987
by (erule psubsetI, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   988
14889
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   989
lemma insert_partition:
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   990
     "[| x \<notin> F; \<forall>c1\<in>insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 --> c1 \<inter> c2 = {}|] 
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   991
      ==> x \<inter> \<Union> F = {}"
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   992
by auto
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   993
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   994
(* main cardinality theorem *)
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   995
lemma card_partition [rule_format]:
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   996
     "finite C ==>  
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   997
        finite (\<Union> C) -->  
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   998
        (\<forall>c\<in>C. card c = k) -->   
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
   999
        (\<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
  1000
        k * card(C) = card (\<Union> C)"
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1001
apply (erule finite_induct, simp)
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1002
apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition 
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1003
       finite_subset [of _ "\<Union> (insert x F)"])
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1004
done
d7711d6b9014 moved some cardinality results into main HOL
paulson
parents: 14748
diff changeset
  1005
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1006
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1007
subsubsection {* Cardinality of image *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1008
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1009
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
  1010
  apply (induct set: Finites, simp)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1011
  apply (simp add: le_SucI finite_imageI card_insert_if)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1012
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1013
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1014
lemma card_image: "finite A ==> inj_on f A ==> card (f ` A) = card A"
15111
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1015
by (induct set: Finites, simp_all)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1016
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1017
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
  1018
  by (simp add: card_seteq card_image)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1019
15111
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1020
lemma eq_card_imp_inj_on:
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1021
  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1022
apply(induct rule:finite_induct)
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1023
 apply simp
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1024
apply(frule card_image_le[where f = f])
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1025
apply(simp add:card_insert_if split:if_splits)
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1026
done
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1027
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1028
lemma inj_on_iff_eq_card:
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1029
  "finite A ==> inj_on f A = (card(f ` A) = card A)"
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1030
by(blast intro: card_image eq_card_imp_inj_on)
c108189645f8 added some inj_on thms
nipkow
parents: 15074
diff changeset
  1031
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1032
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1033
subsubsection {* Cardinality of the Powerset *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1034
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1035
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
  1036
  apply (induct set: Finites)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1037
   apply (simp_all add: Pow_insert)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
  1038
  apply (subst card_Un_disjoint, blast)
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
  1039
    apply (blast intro: finite_imageI, blast)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1040
  apply (subgoal_tac "inj_on (insert x) (Pow F)")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1041
   apply (simp add: card_image Pow_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1042
  apply (unfold inj_on_def)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1043
  apply (blast elim!: equalityE)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1044
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1045
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1046
text {* Relates to equivalence classes.  Based on a theorem of
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1047
F. Kammüller's.  *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1048
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1049
lemma dvd_partition:
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1050
  "finite (Union C) ==>
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1051
    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
  1052
    (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
  1053
  k dvd card (Union C)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1054
apply(frule finite_UnionD)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1055
apply(rotate_tac -1)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
  1056
  apply (induct set: Finites, simp_all, clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1057
  apply (subst card_Un_disjoint)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1058
  apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1059
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1060
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1061
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1062
subsubsection {* Theorems about @{text "choose"} *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1063
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1064
text {*
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1065
  \medskip Basic theorem about @{text "choose"}.  By Florian
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1066
  Kamm\"uller, tidied by LCP.
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1067
*}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1068
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1069
lemma card_s_0_eq_empty:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1070
    "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1071
  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
  1072
  apply (simp cong add: rev_conj_cong)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1073
  done
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1074
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1075
lemma choose_deconstruct: "finite M ==> x \<notin> M
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1076
  ==> {s. s <= insert x M & card(s) = Suc k}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1077
       = {s. s <= M & card(s) = Suc k} Un
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1078
         {s. EX t. t <= M & card(t) = k & s = insert x t}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1079
  apply safe
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1080
   apply (auto intro: finite_subset [THEN card_insert_disjoint])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1081
  apply (drule_tac x = "xa - {x}" in spec)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1082
  apply (subgoal_tac "x \<notin> xa", auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1083
  apply (erule rev_mp, subst card_Diff_singleton)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1084
  apply (auto intro: finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1085
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1086
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1087
lemma card_inj_on_le:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1088
    "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1089
apply (subgoal_tac "finite A") 
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1090
 apply (force intro: card_mono simp add: card_image [symmetric])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1091
apply (blast intro: finite_imageD dest: finite_subset) 
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1092
done
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1093
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1094
lemma card_bij_eq:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1095
    "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1096
       finite A; finite B |] ==> card A = card B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1097
  by (auto intro: le_anti_sym card_inj_on_le)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1098
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1099
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
  1100
 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
  1101
 @{term x} into each.*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1102
lemma constr_bij:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1103
   "[|finite A; x \<notin> A|] ==>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1104
    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
  1105
    card {B. B <= A & card(B) = k}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1106
  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
  1107
       apply (auto elim!: equalityE simp add: inj_on_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1108
    apply (subst Diff_insert0, auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1109
   txt {* finiteness of the two sets *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1110
   apply (rule_tac [2] B = "Pow (A)" in finite_subset)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1111
   apply (rule_tac B = "Pow (insert x A)" in finite_subset)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1112
   apply fast+
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1113
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1114
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1115
text {*
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1116
  Main theorem: combinatorial statement about number of subsets of a set.
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1117
*}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1118
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1119
lemma n_sub_lemma:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1120
  "!!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
  1121
  apply (induct k)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1122
   apply (simp add: card_s_0_eq_empty, atomize)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1123
  apply (rotate_tac -1, erule finite_induct)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1124
   apply (simp_all (no_asm_simp) cong add: conj_cong
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1125
     add: card_s_0_eq_empty choose_deconstruct)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1126
  apply (subst card_Un_disjoint)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1127
     prefer 4 apply (force simp add: constr_bij)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1128
    prefer 3 apply force
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1129
   prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1130
     finite_subset [of _ "Pow (insert x F)", standard])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1131
  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
  1132
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1133
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1134
theorem n_subsets:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1135
    "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
  1136
  by (simp add: n_sub_lemma)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1137
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1138
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1139
subsection{* A fold functional for non-empty sets *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1140
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1141
text{* Does not require start value. *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1142
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1143
consts
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1144
  foldSet1 :: "('a => 'a => 'a) => ('a set \<times> 'a) set"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1145
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1146
inductive "foldSet1 f"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1147
intros
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1148
foldSet1_singletonI [intro]: "({a}, a) : foldSet1 f"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1149
foldSet1_insertI [intro]:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1150
 "\<lbrakk> (A, x) : foldSet1 f; a \<notin> A; A \<noteq> {} \<rbrakk>
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1151
  \<Longrightarrow> (insert a A, f a x) : foldSet1 f"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1152
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1153
constdefs
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1154
  fold1 :: "('a => 'a => 'a) => 'a set => 'a"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1155
  "fold1 f A == THE x. (A, x) : foldSet1 f"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1156
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1157
lemma foldSet1_nonempty:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1158
 "(A, x) : foldSet1 f \<Longrightarrow> A \<noteq> {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1159
by(erule foldSet1.cases, simp_all) 
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1160
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1161
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1162
inductive_cases empty_foldSet1E [elim!]: "({}, x) : foldSet1 f"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1163
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1164
lemma foldSet1_sing[iff]: "(({a},b) : foldSet1 f) = (a = b)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1165
apply(rule iffI)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1166
 prefer 2 apply fast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1167
apply (erule foldSet1.cases)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1168
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1169
apply (erule foldSet1.cases)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1170
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1171
apply blast
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1172
done
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1173
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1174
lemma Diff1_foldSet1:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1175
  "(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
  1176
by (erule insert_Diff [THEN subst], rule foldSet1.intros,
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1177
    auto dest!:foldSet1_nonempty)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1178
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1179
lemma foldSet1_imp_finite: "(A, x) : foldSet1 f ==> finite A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1180
  by (induct set: foldSet1) auto
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1181
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1182
lemma finite_nonempty_imp_foldSet1:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1183
  "\<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
  1184
  by (induct set: Finites) auto
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1185
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1186
lemma (in ACf) foldSet1_determ_aux:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1187
  "!!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
  1188
proof (induct n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1189
  case 0 thus ?case by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1190
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1191
  case (Suc n)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1192
  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
  1193
           \<Longrightarrow> y = x" and card: "card A < Suc n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1194
  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
  1195
  from card have "card A < n \<or> card A = n" by arith
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1196
  thus ?case
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1197
  proof
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1198
    assume less: "card A < n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1199
    show ?thesis by(rule IH[OF less Afoldx Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1200
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1201
    assume cardA: "card A = n"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1202
    show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1203
    proof (rule foldSet1.cases[OF Afoldx])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1204
      fix a assume "(A, x) = ({a}, a)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1205
      thus "y = x" using Afoldy by (simp add:foldSet1_sing)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1206
    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1207
      fix Ax ax x'
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1208
      assume eq1: "(A, x) = (insert ax Ax, ax \<cdot> x')"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1209
	and x': "(Ax, x') \<in> foldSet1 f" and notinx: "ax \<notin> Ax"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1210
	and Axnon: "Ax \<noteq> {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1211
      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
  1212
      show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1213
      proof (rule foldSet1.cases[OF Afoldy])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1214
	fix ay assume "(A, y) = ({ay}, ay)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1215
	thus ?thesis using eq1 x' Axnon notinx
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1216
	  by (fastsimp simp:foldSet1_sing)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1217
      next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1218
	fix Ay ay y'
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1219
	assume eq2: "(A, y) = (insert ay Ay, ay \<cdot> y')"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1220
	  and y': "(Ay, y') \<in> foldSet1 f" and notiny: "ay \<notin> Ay"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1221
	  and Aynon: "Ay \<noteq> {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1222
	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
  1223
	have finA: "finite A" by(rule foldSet1_imp_finite[OF Afoldx])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1224
	with cardA A1 notinx have less: "card Ax < n" by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1225
	show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1226
	proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1227
	  assume "ax = ay"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1228
	  then moreover have "Ax = Ay" using A1 A2 notinx notiny by auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1229
	  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
  1230
	next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1231
	  assume diff: "ax \<noteq> ay"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1232
	  let ?B = "Ax - {ay}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1233
	  have Ax: "Ax = insert ay ?B" and Ay: "Ay = insert ax ?B"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1234
	    using A1 A2 notinx notiny diff by(blast elim!:equalityE)+
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1235
	  show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1236
	  proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1237
	    assume "?B = {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1238
	    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
  1239
	  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1240
	    assume Bnon: "?B \<noteq> {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1241
	    moreover have "finite ?B" using finA A1 by simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1242
	    ultimately obtain b where Bfoldb: "(?B,b) \<in> foldSet1 f"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1243
	      using finite_nonempty_imp_foldSet1 by(blast)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1244
	    moreover have ayinAx: "ay \<in> Ax" using Ax by(auto)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1245
	    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
  1246
	    hence "ay\<cdot>b = x'" by(rule IH[OF less x'])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1247
            moreover have "ax\<cdot>b = y'"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1248
	    proof (rule IH[OF _ y'])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1249
	      show "card Ay < n" using Ay cardA A1 notinx finA ayinAx
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1250
		by(auto simp:card_Diff1_less)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1251
	    next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1252
	      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
  1253
		by fastsimp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1254
	    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1255
	    ultimately show ?thesis using x y by(auto simp:AC)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1256
	  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1257
	qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1258
      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1259
    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1260
  qed
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1261
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1262
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1263
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1264
lemma (in ACf) foldSet1_determ:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1265
  "(A, x) : foldSet1 f ==> (A, y) : foldSet1 f ==> y = x"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1266
by (blast intro: foldSet1_determ_aux [rule_format])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1267
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1268
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
  1269
  by (unfold fold1_def) (blast intro: foldSet1_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1270
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1271
lemma fold1_singleton: "fold1 f {a} = a"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1272
  by (unfold fold1_def) blast
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1273
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1274
lemma (in ACf) foldSet1_insert_aux: "x \<notin> A ==> A \<noteq> {} \<Longrightarrow> 
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1275
    ((insert x A, v) : foldSet1 f) =
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1276
    (EX y. (A, y) : foldSet1 f & v = f x y)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1277
apply auto
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1278
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
  1279
  apply (fastsimp dest: foldSet1_imp_finite)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1280
 apply blast
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1281
apply (blast intro: foldSet1_determ)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1282
done
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1283
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1284
lemma (in ACf) fold1_insert:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1285
  "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
  1286
apply (unfold fold1_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1287
apply (simp add: foldSet1_insert_aux)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1288
apply (rule the_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1289
apply (auto intro: finite_nonempty_imp_foldSet1
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1290
    cong add: conj_cong simp add: fold1_def [symmetric] foldSet1_equality)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1291
done
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1292
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1293
locale ACIf = ACf +
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1294
  assumes idem: "x \<cdot> x = x"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1295
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1296
lemma (in ACIf) fold1_insert2:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1297
assumes finA: "finite A" and nonA: "A \<noteq> {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1298
shows "fold1 f (insert a A) = f a (fold1 f A)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1299
proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1300
  assume "a \<in> A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1301
  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
  1302
    by(blast dest: mk_disjoint_insert)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1303
  show ?thesis
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1304
  proof cases
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1305
    assume "B = {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1306
    thus ?thesis using A by(simp add:idem fold1_singleton)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1307
  next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1308
    assume nonB: "B \<noteq> {}"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1309
    from finA A have finB: "finite B" by(blast intro: finite_subset)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1310
    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
  1311
    also have "\<dots> = f a (fold1 f B)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1312
      using finB nonB disj by(simp add: fold1_insert)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1313
    also have "\<dots> = f a (fold1 f A)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1314
      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
  1315
    finally show ?thesis .
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1316
  qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1317
next
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1318
  assume "a \<notin> A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1319
  with finA nonA show ?thesis by(simp add:fold1_insert)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1320
qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1321
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1322
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1323
text{* Now the recursion rules for definitions: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1324
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1325
lemma fold1_singleton_def: "g \<equiv> fold1 f \<Longrightarrow> g {a} = a"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1326
by(simp add:fold1_singleton)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1327
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1328
lemma (in ACf) fold1_insert_def:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1329
  "\<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
  1330
by(simp add:fold1_insert)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1331
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1332
lemma (in ACIf) fold1_insert2_def:
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1333
  "\<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
  1334
by(simp add:fold1_insert2)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1335
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1336
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1337
subsection{*Min and Max*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1338
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1339
text{* As an application of @{text fold1} we define the minimal and
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1340
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
  1341
show that @{text min} and @{text max} are ACI: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1342
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1343
lemma ACf_min: "ACf(min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1344
apply(rule ACf.intro)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1345
apply(auto simp:min_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1346
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1347
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1348
lemma ACIf_min: "ACIf(min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1349
apply(rule ACIf.intro[OF ACf_min])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1350
apply(rule ACIf_axioms.intro)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1351
apply(auto simp:min_def)
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1352
done
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1353
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1354
lemma ACf_max: "ACf(max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1355
apply(rule ACf.intro)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1356
apply(auto simp:max_def)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1357
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1358
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1359
lemma ACIf_max: "ACIf(max:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1360
apply(rule ACIf.intro[OF ACf_max])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1361
apply(rule ACIf_axioms.intro)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1362
apply(auto simp:max_def)
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1363
done
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1364
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1365
text{* Now we make the definitions: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1366
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1367
constdefs
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1368
  Min :: "('a::linorder)set => 'a"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1369
  "Min  ==  fold1 min"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1370
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1371
  Max :: "('a::linorder)set => 'a"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1372
  "Max  ==  fold1 max"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1373
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1374
text{* Now we instantiate the recursiuon equations and declare them
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1375
simplification rules: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1376
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1377
declare
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1378
  fold1_singleton_def[OF Min_def, simp]
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff chan