src/HOL/Finite_Set.thy
author kuncar
Tue, 09 Oct 2012 16:57:58 +0200
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rename Set.project to Set.filter - more appropriate name
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(*  Title:      HOL/Finite_Set.thy
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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                with contributions by Jeremy Avigad
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*)
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header {* Finite sets *}
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theory Finite_Set
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imports Option Power
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begin
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subsection {* Predicate for finite sets *}
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inductive finite :: "'a set \<Rightarrow> bool"
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  where
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    emptyI [simp, intro!]: "finite {}"
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  | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
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(* FIXME: move to Set theory *)
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ML_file "Tools/set_comprehension_pointfree.ML"
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Rafal Kolanski <rafal.kolanski@nicta.com.au>
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simproc_setup finite_Collect ("finite (Collect P)") =
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  {* K Set_Comprehension_Pointfree.simproc *}
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(* FIXME: move to Set theory*)
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setup {*
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  Code_Preproc.map_pre (fn ss => ss addsimprocs
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    [Raw_Simplifier.make_simproc {name = "set comprehension", lhss = [@{cpat "Collect ?P"}],
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    proc = K Set_Comprehension_Pointfree.code_simproc, identifier = []}])
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*}
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lemma finite_induct [case_names empty insert, induct set: finite]:
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  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
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  assumes "finite F"
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  assumes "P {}"
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    and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
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  shows "P F"
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using `finite F`
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proof induct
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  show "P {}" by fact
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  fix x F assume F: "finite F" and P: "P F"
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  show "P (insert x F)"
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  proof cases
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    assume "x \<in> F"
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    hence "insert x F = F" by (rule insert_absorb)
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    with P show ?thesis by (simp only:)
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  next
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    assume "x \<notin> F"
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    from F this P show ?thesis by (rule insert)
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  qed
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qed
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subsubsection {* Choice principles *}
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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 assms have "A \<noteq> UNIV" by blast
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  then show ?thesis by blast
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qed
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text {* A finite choice principle. Does not need the SOME choice operator. *}
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lemma finite_set_choice:
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  "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
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proof (induct rule: finite_induct)
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  case empty then show ?case by simp
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next
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  case (insert a A)
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  then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
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  show ?case (is "EX f. ?P f")
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  proof
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    show "?P(%x. if x = a then b else f x)" using f ab by auto
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  qed
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qed
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subsubsection {* Finite sets are the images of initial segments of natural numbers *}
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lemma finite_imp_nat_seg_image_inj_on:
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  assumes "finite A" 
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  shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
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using assms
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proof induct
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  case empty
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  show ?case
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  proof
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    show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp 
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  qed
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next
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  case (insert a A)
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  have notinA: "a \<notin> A" by fact
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  from insert.hyps obtain n f
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    where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
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  hence "insert a A = f(n:=a) ` {i. i < Suc n}"
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        "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
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    by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
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  thus ?case by blast
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qed
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lemma nat_seg_image_imp_finite:
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  "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
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proof (induct n arbitrary: A)
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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 \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
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  by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
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lemma finite_imp_inj_to_nat_seg:
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  assumes "finite A"
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  shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
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proof -
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  from finite_imp_nat_seg_image_inj_on[OF `finite A`]
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  obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
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    by (auto simp:bij_betw_def)
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  let ?f = "the_inv_into {i. i<n} f"
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  have "inj_on ?f A & ?f ` A = {i. i<n}"
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    by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
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  thus ?thesis by blast
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qed
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lemma finite_Collect_less_nat [iff]:
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  "finite {n::nat. n < k}"
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  by (fastforce simp: finite_conv_nat_seg_image)
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lemma finite_Collect_le_nat [iff]:
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  "finite {n::nat. n \<le> k}"
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  by (simp add: le_eq_less_or_eq Collect_disj_eq)
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subsubsection {* Finiteness and common set operations *}
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lemma rev_finite_subset:
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  "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
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proof (induct arbitrary: A rule: finite_induct)
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  case empty
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  then show ?case by simp
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next
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   157
  case (insert x F A)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   158
  have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   159
  show "finite A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   160
  proof cases
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   161
    assume x: "x \<in> A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   162
    with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   163
    with r have "finite (A - {x})" .
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   164
    hence "finite (insert x (A - {x}))" ..
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   165
    also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   166
    finally show ?thesis .
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   167
  next
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   168
    show "A \<subseteq> F ==> ?thesis" by fact
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   169
    assume "x \<notin> A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   170
    with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   171
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   172
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   173
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   174
lemma finite_subset:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   175
  "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   176
  by (rule rev_finite_subset)
29901
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   177
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   178
lemma finite_UnI:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   179
  assumes "finite F" and "finite G"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   180
  shows "finite (F \<union> G)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   181
  using assms by induct simp_all
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   182
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   183
lemma finite_Un [iff]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   184
  "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   185
  by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   186
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   187
lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   188
proof -
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   189
  have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   190
  then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   191
  then show ?thesis by simp
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   192
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   193
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   194
lemma finite_Int [simp, intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   195
  "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   196
  by (blast intro: finite_subset)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   197
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   198
lemma finite_Collect_conjI [simp, intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   199
  "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   200
  by (simp add: Collect_conj_eq)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   201
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   202
lemma finite_Collect_disjI [simp]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   203
  "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   204
  by (simp add: Collect_disj_eq)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   205
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   206
lemma finite_Diff [simp, intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   207
  "finite A \<Longrightarrow> finite (A - B)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   208
  by (rule finite_subset, rule Diff_subset)
29901
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   209
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   210
lemma finite_Diff2 [simp]:
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   211
  assumes "finite B"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   212
  shows "finite (A - B) \<longleftrightarrow> finite A"
29901
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   213
proof -
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   214
  have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   215
  also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
29901
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   216
  finally show ?thesis ..
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   217
qed
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   218
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   219
lemma finite_Diff_insert [iff]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   220
  "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   221
proof -
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   222
  have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   223
  moreover have "A - insert a B = A - B - {a}" by auto
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   224
  ultimately show ?thesis by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   225
qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   226
29901
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   227
lemma finite_compl[simp]:
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   228
  "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   229
  by (simp add: Compl_eq_Diff_UNIV)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   230
29916
f24137b42d9b more finiteness
nipkow
parents: 29903
diff changeset
   231
lemma finite_Collect_not[simp]:
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   232
  "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   233
  by (simp add: Collect_neg_eq)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   234
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   235
lemma finite_Union [simp, intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   236
  "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   237
  by (induct rule: finite_induct) simp_all
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   238
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   239
lemma finite_UN_I [intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   240
  "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   241
  by (induct rule: finite_induct) simp_all
29903
2c0046b26f80 more finiteness changes
nipkow
parents: 29901
diff changeset
   242
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   243
lemma finite_UN [simp]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   244
  "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   245
  by (blast intro: finite_subset)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   246
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   247
lemma finite_Inter [intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   248
  "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   249
  by (blast intro: Inter_lower finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   250
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   251
lemma finite_INT [intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   252
  "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   253
  by (blast intro: INT_lower finite_subset)
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   254
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   255
lemma finite_imageI [simp, intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   256
  "finite F \<Longrightarrow> finite (h ` F)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   257
  by (induct rule: finite_induct) simp_all
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   258
31768
159cd6b5e5d4 lemma finite_image_set by Jeremy Avigad
haftmann
parents: 31465
diff changeset
   259
lemma finite_image_set [simp]:
159cd6b5e5d4 lemma finite_image_set by Jeremy Avigad
haftmann
parents: 31465
diff changeset
   260
  "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
159cd6b5e5d4 lemma finite_image_set by Jeremy Avigad
haftmann
parents: 31465
diff changeset
   261
  by (simp add: image_Collect [symmetric])
159cd6b5e5d4 lemma finite_image_set by Jeremy Avigad
haftmann
parents: 31465
diff changeset
   262
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   263
lemma finite_imageD:
42206
0920f709610f tuned proof
haftmann
parents: 41988
diff changeset
   264
  assumes "finite (f ` A)" and "inj_on f A"
0920f709610f tuned proof
haftmann
parents: 41988
diff changeset
   265
  shows "finite A"
46898
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   266
using assms
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   267
proof (induct "f ` A" arbitrary: A)
42206
0920f709610f tuned proof
haftmann
parents: 41988
diff changeset
   268
  case empty then show ?case by simp
0920f709610f tuned proof
haftmann
parents: 41988
diff changeset
   269
next
0920f709610f tuned proof
haftmann
parents: 41988
diff changeset
   270
  case (insert x B)
0920f709610f tuned proof
haftmann
parents: 41988
diff changeset
   271
  then have B_A: "insert x B = f ` A" by simp
0920f709610f tuned proof
haftmann
parents: 41988
diff changeset
   272
  then obtain y where "x = f y" and "y \<in> A" by blast
0920f709610f tuned proof
haftmann
parents: 41988
diff changeset
   273
  from B_A `x \<notin> B` have "B = f ` A - {x}" by blast
0920f709610f tuned proof
haftmann
parents: 41988
diff changeset
   274
  with B_A `x \<notin> B` `x = f y` `inj_on f A` `y \<in> A` have "B = f ` (A - {y})" by (simp add: inj_on_image_set_diff)
0920f709610f tuned proof
haftmann
parents: 41988
diff changeset
   275
  moreover from `inj_on f A` have "inj_on f (A - {y})" by (rule inj_on_diff)
0920f709610f tuned proof
haftmann
parents: 41988
diff changeset
   276
  ultimately have "finite (A - {y})" by (rule insert.hyps)
0920f709610f tuned proof
haftmann
parents: 41988
diff changeset
   277
  then show "finite A" by simp
0920f709610f tuned proof
haftmann
parents: 41988
diff changeset
   278
qed
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   279
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   280
lemma finite_surj:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   281
  "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   282
  by (erule finite_subset) (rule finite_imageI)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   283
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   284
lemma finite_range_imageI:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   285
  "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   286
  by (drule finite_imageI) (simp add: range_composition)
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   287
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   288
lemma finite_subset_image:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   289
  assumes "finite B"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   290
  shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
46898
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   291
using assms
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   292
proof induct
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   293
  case empty then show ?case by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   294
next
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   295
  case insert then show ?case
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   296
    by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   297
       blast
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   298
qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   299
43991
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43866
diff changeset
   300
lemma finite_vimage_IntI:
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43866
diff changeset
   301
  "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   302
  apply (induct rule: finite_induct)
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
   303
   apply simp_all
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   304
  apply (subst vimage_insert)
43991
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43866
diff changeset
   305
  apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2)
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   306
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   307
43991
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43866
diff changeset
   308
lemma finite_vimageI:
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43866
diff changeset
   309
  "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43866
diff changeset
   310
  using finite_vimage_IntI[of F h UNIV] by auto
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43866
diff changeset
   311
34111
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   312
lemma finite_vimageD:
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   313
  assumes fin: "finite (h -` F)" and surj: "surj h"
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   314
  shows "finite F"
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   315
proof -
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   316
  have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   317
  also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   318
  finally show "finite F" .
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   319
qed
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   320
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   321
lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   322
  unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   323
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   324
lemma finite_Collect_bex [simp]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   325
  assumes "finite A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   326
  shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   327
proof -
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   328
  have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   329
  with assms show ?thesis by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   330
qed
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   331
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   332
lemma finite_Collect_bounded_ex [simp]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   333
  assumes "finite {y. P y}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   334
  shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   335
proof -
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   336
  have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   337
  with assms show ?thesis by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   338
qed
29920
b95f5b8b93dd more finiteness
nipkow
parents: 29918
diff changeset
   339
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   340
lemma finite_Plus:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   341
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   342
  by (simp add: Plus_def)
17022
b257300c3a9c added Brian Hufmann's finite instances
nipkow
parents: 16775
diff changeset
   343
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   344
lemma finite_PlusD: 
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   345
  fixes A :: "'a set" and B :: "'b set"
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   346
  assumes fin: "finite (A <+> B)"
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   347
  shows "finite A" "finite B"
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   348
proof -
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   349
  have "Inl ` A \<subseteq> A <+> B" by auto
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   350
  then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   351
  then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   352
next
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   353
  have "Inr ` B \<subseteq> A <+> B" by auto
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   354
  then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   355
  then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   356
qed
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   357
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   358
lemma finite_Plus_iff [simp]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   359
  "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   360
  by (auto intro: finite_PlusD finite_Plus)
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   361
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   362
lemma finite_Plus_UNIV_iff [simp]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   363
  "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   364
  by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   365
40786
0a54cfc9add3 gave more standard finite set rules simp and intro attribute
nipkow
parents: 40716
diff changeset
   366
lemma finite_SigmaI [simp, intro]:
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   367
  "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
40786
0a54cfc9add3 gave more standard finite set rules simp and intro attribute
nipkow
parents: 40716
diff changeset
   368
  by (unfold Sigma_def) blast
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   369
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   370
lemma finite_cartesian_product:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   371
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   372
  by (rule finite_SigmaI)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   373
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   374
lemma finite_Prod_UNIV:
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   375
  "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   376
  by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   377
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   378
lemma finite_cartesian_productD1:
42207
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   379
  assumes "finite (A \<times> B)" and "B \<noteq> {}"
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   380
  shows "finite A"
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   381
proof -
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   382
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   383
    by (auto simp add: finite_conv_nat_seg_image)
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   384
  then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   385
  with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   386
    by (simp add: image_compose)
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   387
  then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   388
  then show ?thesis
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   389
    by (auto simp add: finite_conv_nat_seg_image)
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   390
qed
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   391
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   392
lemma finite_cartesian_productD2:
42207
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   393
  assumes "finite (A \<times> B)" and "A \<noteq> {}"
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   394
  shows "finite B"
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   395
proof -
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   396
  from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   397
    by (auto simp add: finite_conv_nat_seg_image)
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   398
  then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   399
  with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   400
    by (simp add: image_compose)
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   401
  then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   402
  then show ?thesis
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   403
    by (auto simp add: finite_conv_nat_seg_image)
2bda5eddadf3 tuned proofs
haftmann
parents: 42206
diff changeset
   404
qed
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   405
48175
fea68365c975 add finiteness lemmas for 'a * 'b and 'a set
Andreas Lochbihler
parents: 48128
diff changeset
   406
lemma finite_prod: 
fea68365c975 add finiteness lemmas for 'a * 'b and 'a set
Andreas Lochbihler
parents: 48128
diff changeset
   407
  "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
fea68365c975 add finiteness lemmas for 'a * 'b and 'a set
Andreas Lochbihler
parents: 48128
diff changeset
   408
by(auto simp add: UNIV_Times_UNIV[symmetric] simp del: UNIV_Times_UNIV 
fea68365c975 add finiteness lemmas for 'a * 'b and 'a set
Andreas Lochbihler
parents: 48128
diff changeset
   409
   dest: finite_cartesian_productD1 finite_cartesian_productD2)
fea68365c975 add finiteness lemmas for 'a * 'b and 'a set
Andreas Lochbihler
parents: 48128
diff changeset
   410
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   411
lemma finite_Pow_iff [iff]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   412
  "finite (Pow A) \<longleftrightarrow> finite A"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   413
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   414
  assume "finite (Pow A)"
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   415
  then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   416
  then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   417
next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   418
  assume "finite A"
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   419
  then show "finite (Pow A)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
   420
    by induct (simp_all add: Pow_insert)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   421
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   422
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   423
corollary finite_Collect_subsets [simp, intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   424
  "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   425
  by (simp add: Pow_def [symmetric])
29918
214755b03df3 more finiteness
nipkow
parents: 29916
diff changeset
   426
48175
fea68365c975 add finiteness lemmas for 'a * 'b and 'a set
Andreas Lochbihler
parents: 48128
diff changeset
   427
lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
fea68365c975 add finiteness lemmas for 'a * 'b and 'a set
Andreas Lochbihler
parents: 48128
diff changeset
   428
by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
fea68365c975 add finiteness lemmas for 'a * 'b and 'a set
Andreas Lochbihler
parents: 48128
diff changeset
   429
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   430
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   431
  by (blast intro: finite_subset [OF subset_Pow_Union])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   432
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   433
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   434
subsubsection {* Further induction rules on finite sets *}
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   435
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   436
lemma finite_ne_induct [case_names singleton insert, consumes 2]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   437
  assumes "finite F" and "F \<noteq> {}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   438
  assumes "\<And>x. P {x}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   439
    and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   440
  shows "P F"
46898
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   441
using assms
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   442
proof induct
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   443
  case empty then show ?case by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   444
next
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   445
  case (insert x F) then show ?case by cases auto
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   446
qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   447
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   448
lemma finite_subset_induct [consumes 2, case_names empty insert]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   449
  assumes "finite F" and "F \<subseteq> A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   450
  assumes empty: "P {}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   451
    and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   452
  shows "P F"
46898
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   453
using `finite F` `F \<subseteq> A`
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   454
proof induct
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   455
  show "P {}" by fact
31441
428e4caf2299 finite lemmas
nipkow
parents: 31438
diff changeset
   456
next
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   457
  fix x F
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   458
  assume "finite F" and "x \<notin> F" and
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   459
    P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   460
  show "P (insert x F)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   461
  proof (rule insert)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   462
    from i show "x \<in> A" by blast
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   463
    from i have "F \<subseteq> A" by blast
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   464
    with P show "P F" .
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   465
    show "finite F" by fact
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   466
    show "x \<notin> F" by fact
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   467
  qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   468
qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   469
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   470
lemma finite_empty_induct:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   471
  assumes "finite A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   472
  assumes "P A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   473
    and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   474
  shows "P {}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   475
proof -
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   476
  have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   477
  proof -
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   478
    fix B :: "'a set"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   479
    assume "B \<subseteq> A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   480
    with `finite A` have "finite B" by (rule rev_finite_subset)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   481
    from this `B \<subseteq> A` show "P (A - B)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   482
    proof induct
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   483
      case empty
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   484
      from `P A` show ?case by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   485
    next
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   486
      case (insert b B)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   487
      have "P (A - B - {b})"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   488
      proof (rule remove)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   489
        from `finite A` show "finite (A - B)" by induct auto
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   490
        from insert show "b \<in> A - B" by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   491
        from insert show "P (A - B)" by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   492
      qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   493
      also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   494
      finally show ?case .
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   495
    qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   496
  qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   497
  then have "P (A - A)" by blast
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   498
  then show ?thesis by simp
31441
428e4caf2299 finite lemmas
nipkow
parents: 31438
diff changeset
   499
qed
428e4caf2299 finite lemmas
nipkow
parents: 31438
diff changeset
   500
428e4caf2299 finite lemmas
nipkow
parents: 31438
diff changeset
   501
26441
7914697ff104 no "attach UNIV" any more
haftmann
parents: 26146
diff changeset
   502
subsection {* Class @{text finite}  *}
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   503
29797
08ef36ed2f8a handling type classes without parameters
haftmann
parents: 29675
diff changeset
   504
class finite =
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   505
  assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
27430
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   506
begin
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   507
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   508
lemma finite [simp]: "finite (A \<Colon> 'a set)"
26441
7914697ff104 no "attach UNIV" any more
haftmann
parents: 26146
diff changeset
   509
  by (rule subset_UNIV finite_UNIV finite_subset)+
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   510
43866
8a50dc70cbff moving UNIV = ... equations to their proper theories
haftmann
parents: 42875
diff changeset
   511
lemma finite_code [code]: "finite (A \<Colon> 'a set) \<longleftrightarrow> True"
40922
4d0f96a54e76 adding code equation for finiteness of finite types
bulwahn
parents: 40786
diff changeset
   512
  by simp
4d0f96a54e76 adding code equation for finiteness of finite types
bulwahn
parents: 40786
diff changeset
   513
27430
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   514
end
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   515
46898
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   516
instance prod :: (finite, finite) finite
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   517
  by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   518
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   519
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   520
  by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   521
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   522
instance "fun" :: (finite, finite) finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   523
proof
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   524
  show "finite (UNIV :: ('a => 'b) set)"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   525
  proof (rule finite_imageD)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   526
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
26792
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   527
    have "range ?graph \<subseteq> Pow UNIV" by simp
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   528
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   529
      by (simp only: finite_Pow_iff finite)
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   530
    ultimately show "finite (range ?graph)"
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   531
      by (rule finite_subset)
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   532
    show "inj ?graph" by (rule inj_graph)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   533
  qed
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   534
qed
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   535
46898
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   536
instance bool :: finite
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   537
  by default (simp add: UNIV_bool)
44831
haftmann
parents: 43991
diff changeset
   538
45962
fc77947a7db4 finite type class instance for `set`
haftmann
parents: 45166
diff changeset
   539
instance set :: (finite) finite
fc77947a7db4 finite type class instance for `set`
haftmann
parents: 45166
diff changeset
   540
  by default (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite)
fc77947a7db4 finite type class instance for `set`
haftmann
parents: 45166
diff changeset
   541
46898
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   542
instance unit :: finite
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   543
  by default (simp add: UNIV_unit)
44831
haftmann
parents: 43991
diff changeset
   544
46898
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   545
instance sum :: (finite, finite) finite
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   546
  by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
27981
feb0c01cf0fb tuned import order
haftmann
parents: 27611
diff changeset
   547
44831
haftmann
parents: 43991
diff changeset
   548
lemma finite_option_UNIV [simp]:
haftmann
parents: 43991
diff changeset
   549
  "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
haftmann
parents: 43991
diff changeset
   550
  by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
haftmann
parents: 43991
diff changeset
   551
46898
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   552
instance option :: (finite) finite
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
   553
  by default (simp add: UNIV_option_conv)
44831
haftmann
parents: 43991
diff changeset
   554
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   555
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   556
subsection {* A basic fold functional for finite sets *}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   557
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   558
text {* The intended behaviour is
31916
f3227bb306a4 recovered subscripts, which were lost in b41d61c768e2 (due to Emacs accident?);
wenzelm
parents: 31907
diff changeset
   559
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   560
if @{text f} is ``left-commutative'':
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   561
*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   562
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   563
locale comp_fun_commute =
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   564
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   565
  assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   566
begin
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   567
42809
5b45125b15ba use pointfree characterisation for fold_set locale
haftmann
parents: 42715
diff changeset
   568
lemma fun_left_comm: "f x (f y z) = f y (f x z)"
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   569
  using comp_fun_commute by (simp add: fun_eq_iff)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   570
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   571
end
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   572
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   573
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   574
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   575
  emptyI [intro]: "fold_graph f z {} z" |
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   576
  insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   577
      \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   578
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   579
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   580
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   581
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 37678
diff changeset
   582
  "fold f z A = (THE y. fold_graph f z A y)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   583
15498
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   584
text{*A tempting alternative for the definiens is
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   585
@{term "if finite A then THE y. fold_graph f z A y else e"}.
15498
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   586
It allows the removal of finiteness assumptions from the theorems
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   587
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   588
The proofs become ugly. It is not worth the effort. (???) *}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   589
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   590
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   591
by (induct rule: finite_induct) auto
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   592
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   593
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   594
subsubsection{*From @{const fold_graph} to @{term fold}*}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   595
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   596
context comp_fun_commute
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   597
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   598
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   599
lemma fold_graph_insertE_aux:
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   600
  "fold_graph f z A y \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'"
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   601
proof (induct set: fold_graph)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   602
  case (insertI x A y) show ?case
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   603
  proof (cases "x = a")
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   604
    assume "x = a" with insertI show ?case by auto
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   605
  next
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   606
    assume "x \<noteq> a"
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   607
    then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   608
      using insertI by auto
42875
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
   609
    have "f x y = f a (f x y')"
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   610
      unfolding y by (rule fun_left_comm)
42875
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
   611
    moreover have "fold_graph f z (insert x A - {a}) (f x y')"
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   612
      using y' and `x \<noteq> a` and `x \<notin> A`
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   613
      by (simp add: insert_Diff_if fold_graph.insertI)
42875
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
   614
    ultimately show ?case by fast
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   615
  qed
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   616
qed simp
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   617
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   618
lemma fold_graph_insertE:
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   619
  assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   620
  obtains y where "v = f x y" and "fold_graph f z A y"
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   621
using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   622
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   623
lemma fold_graph_determ:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   624
  "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   625
proof (induct arbitrary: y set: fold_graph)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   626
  case (insertI x A y v)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   627
  from `fold_graph f z (insert x A) v` and `x \<notin> A`
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   628
  obtain y' where "v = f x y'" and "fold_graph f z A y'"
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   629
    by (rule fold_graph_insertE)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   630
  from `fold_graph f z A y'` have "y' = y" by (rule insertI)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   631
  with `v = f x y'` show "v = f x y" by simp
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   632
qed fast
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   633
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   634
lemma fold_equality:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   635
  "fold_graph f z A y \<Longrightarrow> fold f z A = y"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   636
by (unfold fold_def) (blast intro: fold_graph_determ)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   637
42272
a46a13b4be5f dropped unused lemmas; proper Isar proof
haftmann
parents: 42207
diff changeset
   638
lemma fold_graph_fold:
a46a13b4be5f dropped unused lemmas; proper Isar proof
haftmann
parents: 42207
diff changeset
   639
  assumes "finite A"
a46a13b4be5f dropped unused lemmas; proper Isar proof
haftmann
parents: 42207
diff changeset
   640
  shows "fold_graph f z A (fold f z A)"
a46a13b4be5f dropped unused lemmas; proper Isar proof
haftmann
parents: 42207
diff changeset
   641
proof -
a46a13b4be5f dropped unused lemmas; proper Isar proof
haftmann
parents: 42207
diff changeset
   642
  from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
a46a13b4be5f dropped unused lemmas; proper Isar proof
haftmann
parents: 42207
diff changeset
   643
  moreover note fold_graph_determ
a46a13b4be5f dropped unused lemmas; proper Isar proof
haftmann
parents: 42207
diff changeset
   644
  ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
a46a13b4be5f dropped unused lemmas; proper Isar proof
haftmann
parents: 42207
diff changeset
   645
  then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
a46a13b4be5f dropped unused lemmas; proper Isar proof
haftmann
parents: 42207
diff changeset
   646
  then show ?thesis by (unfold fold_def)
a46a13b4be5f dropped unused lemmas; proper Isar proof
haftmann
parents: 42207
diff changeset
   647
qed
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   648
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   649
text{* The base case for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   650
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   651
lemma (in -) fold_empty [simp]: "fold f z {} = z"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   652
by (unfold fold_def) blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   653
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   654
text{* The various recursion equations for @{const fold}: *}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   655
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   656
lemma fold_insert [simp]:
42875
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
   657
  assumes "finite A" and "x \<notin> A"
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
   658
  shows "fold f z (insert x A) = f x (fold f z A)"
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
   659
proof (rule fold_equality)
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
   660
  from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
   661
  with `x \<notin> A`show "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
   662
qed
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   663
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   664
lemma fold_fun_comm:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   665
  "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   666
proof (induct rule: finite_induct)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   667
  case empty then show ?case by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   668
next
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   669
  case (insert y A) then show ?case
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   670
    by (simp add: fun_left_comm[of x])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   671
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   672
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   673
lemma fold_insert2:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   674
  "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
   675
by (simp add: fold_fun_comm)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   676
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   677
lemma fold_rec:
42875
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
   678
  assumes "finite A" and "x \<in> A"
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
   679
  shows "fold f z A = f x (fold f z (A - {x}))"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   680
proof -
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   681
  have A: "A = insert x (A - {x})" using `x \<in> A` by blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   682
  then have "fold f z A = fold f z (insert x (A - {x}))" by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   683
  also have "\<dots> = f x (fold f z (A - {x}))"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   684
    by (rule fold_insert) (simp add: `finite A`)+
15535
nipkow
parents: 15532
diff changeset
   685
  finally show ?thesis .
nipkow
parents: 15532
diff changeset
   686
qed
nipkow
parents: 15532
diff changeset
   687
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   688
lemma fold_insert_remove:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   689
  assumes "finite A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   690
  shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   691
proof -
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   692
  from `finite A` have "finite (insert x A)" by auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   693
  moreover have "x \<in> insert x A" by auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   694
  ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   695
    by (rule fold_rec)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   696
  then show ?thesis by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   697
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   698
48619
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   699
text{* Other properties of @{const fold}: *}
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   700
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   701
lemma fold_image:
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   702
  assumes "finite A" and "inj_on g A"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   703
  shows "fold f x (g ` A) = fold (f \<circ> g) x A"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   704
using assms
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   705
proof induction
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   706
  case (insert a F)
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   707
    interpret comp_fun_commute "\<lambda>x. f (g x)" by default (simp add: comp_fun_commute)
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   708
    from insert show ?case by auto
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   709
qed (simp)
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   710
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   711
end
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   712
49724
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   713
lemma fold_cong:
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   714
  assumes "comp_fun_commute f" "comp_fun_commute g"
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   715
  assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   716
    and "A = B" and "s = t"
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   717
  shows "Finite_Set.fold f s A = Finite_Set.fold g t B"
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   718
proof -
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   719
  have "Finite_Set.fold f s A = Finite_Set.fold g s A"  
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   720
  using `finite A` cong proof (induct A)
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   721
    case empty then show ?case by simp
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   722
  next
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   723
    case (insert x A)
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   724
    interpret f: comp_fun_commute f by (fact `comp_fun_commute f`)
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   725
    interpret g: comp_fun_commute g by (fact `comp_fun_commute g`)
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   726
    from insert show ?case by simp
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   727
  qed
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   728
  with assms show ?thesis by simp
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   729
qed
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   730
a5842f026d4c congruence rule for Finite_Set.fold
haftmann
parents: 49723
diff changeset
   731
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   732
text{* A simplified version for idempotent functions: *}
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   733
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   734
locale comp_fun_idem = comp_fun_commute +
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   735
  assumes comp_fun_idem: "f x o f x = f x"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   736
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   737
42869
43b0f61f56d0 use point-free characterization for locale fun_left_comm_idem
haftmann
parents: 42809
diff changeset
   738
lemma fun_left_idem: "f x (f x z) = f x z"
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   739
  using comp_fun_idem by (simp add: fun_eq_iff)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   740
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   741
lemma fold_insert_idem:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   742
  assumes fin: "finite A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   743
  shows "fold f z (insert x A) = f x (fold f z A)"
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   744
proof cases
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   745
  assume "x \<in> A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   746
  then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   747
  then show ?thesis using assms by (simp add:fun_left_idem)
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   748
next
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   749
  assume "x \<notin> A" then show ?thesis using assms by simp
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   750
qed
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   751
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   752
declare fold_insert[simp del] fold_insert_idem[simp]
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   753
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   754
lemma fold_insert_idem2:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   755
  "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   756
by(simp add:fold_fun_comm)
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   757
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   758
end
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   759
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   760
49723
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   761
subsubsection {* Liftings to @{text comp_fun_commute} etc. *}
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   762
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   763
lemma (in comp_fun_commute) comp_comp_fun_commute:
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   764
  "comp_fun_commute (f \<circ> g)"
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   765
proof
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   766
qed (simp_all add: comp_fun_commute)
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   767
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   768
lemma (in comp_fun_idem) comp_comp_fun_idem:
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   769
  "comp_fun_idem (f \<circ> g)"
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   770
  by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   771
    (simp_all add: comp_fun_idem)
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   772
49723
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   773
lemma (in comp_fun_commute) comp_fun_commute_funpow:
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   774
  "comp_fun_commute (\<lambda>x. f x ^^ g x)"
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   775
proof
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   776
  fix y x
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   777
  show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   778
  proof (cases "x = y")
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   779
    case True then show ?thesis by simp
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   780
  next
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   781
    case False show ?thesis
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   782
    proof (induct "g x" arbitrary: g)
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   783
      case 0 then show ?case by simp
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   784
    next
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   785
      case (Suc n g)
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   786
      have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   787
      proof (induct "g y" arbitrary: g)
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   788
        case 0 then show ?case by simp
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   789
      next
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   790
        case (Suc n g)
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   791
        def h \<equiv> "\<lambda>z. g z - 1"
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   792
        with Suc have "n = h y" by simp
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   793
        with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   794
          by auto
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   795
        from Suc h_def have "g y = Suc (h y)" by simp
49739
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 49724
diff changeset
   796
        then show ?case by (simp add: comp_assoc hyp)
49723
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   797
          (simp add: o_assoc comp_fun_commute)
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   798
      qed
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   799
      def h \<equiv> "\<lambda>z. if z = x then g x - 1 else g z"
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   800
      with Suc have "n = h x" by simp
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   801
      with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   802
        by auto
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   803
      with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y" by simp
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   804
      from Suc h_def have "g x = Suc (h x)" by simp
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   805
      then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
49739
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 49724
diff changeset
   806
        (simp add: comp_assoc hyp1)
49723
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   807
    qed
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   808
  qed
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   809
qed
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   810
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   811
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   812
subsubsection {* Expressing set operations via @{const fold} *}
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 48891
diff changeset
   813
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   814
lemma comp_fun_idem_insert:
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   815
  "comp_fun_idem insert"
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   816
proof
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   817
qed auto
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   818
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   819
lemma comp_fun_idem_remove:
46146
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46033
diff changeset
   820
  "comp_fun_idem Set.remove"
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   821
proof
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   822
qed auto
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   823
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   824
lemma (in semilattice_inf) comp_fun_idem_inf:
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   825
  "comp_fun_idem inf"
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   826
proof
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   827
qed (auto simp add: inf_left_commute)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   828
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   829
lemma (in semilattice_sup) comp_fun_idem_sup:
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   830
  "comp_fun_idem sup"
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   831
proof
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   832
qed (auto simp add: sup_left_commute)
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   833
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   834
lemma union_fold_insert:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   835
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   836
  shows "A \<union> B = fold insert B A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   837
proof -
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   838
  interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   839
  from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   840
qed
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   841
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   842
lemma minus_fold_remove:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   843
  assumes "finite A"
46146
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46033
diff changeset
   844
  shows "B - A = fold Set.remove B A"
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   845
proof -
46146
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46033
diff changeset
   846
  interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46033
diff changeset
   847
  from `finite A` have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46033
diff changeset
   848
  then show ?thesis ..
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   849
qed
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   850
48619
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   851
definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" 
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   852
  where "filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   853
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   854
lemma comp_fun_commute_filter_fold: "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   855
proof - 
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   856
  interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   857
  show ?thesis by default (auto simp: fun_eq_iff)
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   858
qed
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   859
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   860
lemma inter_filter:     
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   861
  assumes "finite B"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   862
  shows "A \<inter> B = filter (\<lambda>x. x \<in> A) B"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   863
using assms 
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   864
by (induct B) (auto simp: filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   865
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   866
lemma project_filter:
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   867
  assumes "finite A"
49757
73ab6d4a9236 rename Set.project to Set.filter - more appropriate name
kuncar
parents: 49739
diff changeset
   868
  shows "Set.filter P A = filter P A"
48619
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   869
using assms
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   870
by (induct A) 
49757
73ab6d4a9236 rename Set.project to Set.filter - more appropriate name
kuncar
parents: 49739
diff changeset
   871
  (auto simp add: filter_def Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
48619
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   872
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   873
lemma image_fold_insert:
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   874
  assumes "finite A"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   875
  shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   876
using assms
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   877
proof -
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   878
  interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by default auto
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   879
  show ?thesis using assms by (induct A) auto
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   880
qed
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   881
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   882
lemma Ball_fold:
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   883
  assumes "finite A"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   884
  shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   885
using assms
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   886
proof -
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   887
  interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by default auto
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   888
  show ?thesis using assms by (induct A) auto
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   889
qed
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   890
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   891
lemma Bex_fold:
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   892
  assumes "finite A"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   893
  shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   894
using assms
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   895
proof -
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   896
  interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by default auto
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   897
  show ?thesis using assms by (induct A) auto
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   898
qed
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   899
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   900
lemma comp_fun_commute_Pow_fold: 
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   901
  "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)" 
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   902
  by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   903
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   904
lemma Pow_fold:
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   905
  assumes "finite A"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   906
  shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   907
using assms
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   908
proof -
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   909
  interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   910
  show ?thesis using assms by (induct A) (auto simp: Pow_insert)
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   911
qed
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   912
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   913
lemma fold_union_pair:
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   914
  assumes "finite B"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   915
  shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   916
proof -
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   917
  interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by default auto
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   918
  show ?thesis using assms  by (induct B arbitrary: A) simp_all
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   919
qed
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   920
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   921
lemma comp_fun_commute_product_fold: 
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   922
  assumes "finite B"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   923
  shows "comp_fun_commute (\<lambda>x A. fold (\<lambda>y. Set.insert (x, y)) A B)" 
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   924
by default (auto simp: fold_union_pair[symmetric] assms)
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   925
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   926
lemma product_fold:
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   927
  assumes "finite A"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   928
  assumes "finite B"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   929
  shows "A \<times> B = fold (\<lambda>x A. fold (\<lambda>y. Set.insert (x, y)) A B) {} A"
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   930
using assms unfolding Sigma_def 
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   931
by (induct A) 
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   932
  (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   933
558e4e77ce69 more set operations expressed by Finite_Set.fold
kuncar
parents: 48175
diff changeset
   934
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   935
context complete_lattice
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   936
begin
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   937
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   938
lemma inf_Inf_fold_inf:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   939
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   940
  shows "inf B (Inf A) = fold inf B A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   941
proof -
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   942
  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   943
  from `finite A` show ?thesis by (induct A arbitrary: B)
44919
482f1807976e tune proofs
noschinl
parents: 44890
diff changeset
   944
    (simp_all add: inf_commute fold_fun_comm)
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   945
qed
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   946
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   947
lemma sup_Sup_fold_sup:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   948
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   949
  shows "sup B (Sup A) = fold sup B A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   950
proof -
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   951
  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   952
  from `finite A` show ?thesis by (induct A arbitrary: B)
44919
482f1807976e tune proofs
noschinl
parents: 44890
diff changeset
   953
    (simp_all add: sup_commute fold_fun_comm)
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   954
qed
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   955
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   956
lemma Inf_fold_inf:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   957
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   958
  shows "Inf A = fold inf top A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   959
  using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   960
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   961
lemma Sup_fold_sup:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   962
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   963
  shows "Sup A = fold sup bot A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   964
  using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   965
46146
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46033
diff changeset
   966
lemma inf_INF_fold_inf:
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   967
  assumes "finite A"
42873
da1253ff1764 point-free characterization of operations on finite sets
haftmann
parents: 42871
diff changeset
   968
  shows "inf B (INFI A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold") 
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   969
proof (rule sym)
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   970
  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   971
  interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
42873
da1253ff1764 point-free characterization of operations on finite sets
haftmann
parents: 42871
diff changeset
   972
  from `finite A` show "?fold = ?inf"
42869
43b0f61f56d0 use point-free characterization for locale fun_left_comm_idem
haftmann
parents: 42809
diff changeset
   973
    by (induct A arbitrary: B)
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44919
diff changeset
   974
      (simp_all add: INF_def inf_left_commute)
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   975
qed
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   976
46146
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46033
diff changeset
   977
lemma sup_SUP_fold_sup:
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   978
  assumes "finite A"
42873
da1253ff1764 point-free characterization of operations on finite sets
haftmann
parents: 42871
diff changeset
   979
  shows "sup B (SUPR A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold") 
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   980
proof (rule sym)
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   981
  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
   982
  interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
42873
da1253ff1764 point-free characterization of operations on finite sets
haftmann
parents: 42871
diff changeset
   983
  from `finite A` show "?fold = ?sup"
42869
43b0f61f56d0 use point-free characterization for locale fun_left_comm_idem
haftmann
parents: 42809
diff changeset
   984
    by (induct A arbitrary: B)
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44919
diff changeset
   985
      (simp_all add: SUP_def sup_left_commute)
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   986
qed
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   987
46146
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46033
diff changeset
   988
lemma INF_fold_inf:
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   989
  assumes "finite A"
42873
da1253ff1764 point-free characterization of operations on finite sets
haftmann
parents: 42871
diff changeset
   990
  shows "INFI A f = fold (inf \<circ> f) top A"
46146
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46033
diff changeset
   991
  using assms inf_INF_fold_inf [of A top] by simp
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   992
46146
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46033
diff changeset
   993
lemma SUP_fold_sup:
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   994
  assumes "finite A"
42873
da1253ff1764 point-free characterization of operations on finite sets
haftmann
parents: 42871
diff changeset
   995
  shows "SUPR A f = fold (sup \<circ> f) bot A"
46146
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46033
diff changeset
   996
  using assms sup_SUP_fold_sup [of A bot] by simp
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   997
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   998
end
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   999
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
  1000
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1001
subsection {* The derived combinator @{text fold_image} *}
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1002
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1003
definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
42875
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1004
  where "fold_image f g = fold (\<lambda>x y. f (g x) y)"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1005
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1006
lemma fold_image_empty[simp]: "fold_image f g z {} = z"
42875
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1007
  by (simp add:fold_image_def)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1008
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1009
context ab_semigroup_mult
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1010
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1011
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1012
lemma fold_image_insert[simp]:
42875
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1013
  assumes "finite A" and "a \<notin> A"
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1014
  shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1015
proof -
46898
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
  1016
  interpret comp_fun_commute "%x y. (g x) * y"
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
  1017
    by default (simp add: fun_eq_iff mult_ac)
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
  1018
  from assms show ?thesis by (simp add: fold_image_def)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1019
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1020
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1021
lemma fold_image_reindex:
42875
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1022
  assumes "finite A"
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1023
  shows "inj_on h A \<Longrightarrow> fold_image times g z (h ` A) = fold_image times (g \<circ> h) z A"
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1024
  using assms by induct auto
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1025
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1026
lemma fold_image_cong:
42875
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1027
  assumes "finite A" and g_h: "\<And>x. x\<in>A \<Longrightarrow> g x = h x"
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1028
  shows "fold_image times g z A = fold_image times h z A"
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1029
proof -
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1030
  from `finite A`
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1031
  have "\<And>C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C"
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1032
  proof (induct arbitrary: C)
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1033
    case empty then show ?case by simp
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1034
  next
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1035
    case (insert x F) then show ?case apply -
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1036
    apply (simp add: subset_insert_iff, clarify)
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1037
    apply (subgoal_tac "finite C")
48125
602dc0215954 tuned proofs -- prefer direct "rotated" instead of old-style COMP;
wenzelm
parents: 48124
diff changeset
  1038
      prefer 2 apply (blast dest: finite_subset [rotated])
42875
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1039
    apply (subgoal_tac "C = insert x (C - {x})")
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1040
      prefer 2 apply blast
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1041
    apply (erule ssubst)
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1042
    apply (simp add: Ball_def del: insert_Diff_single)
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1043
    done
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1044
  qed
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1045
  with g_h show ?thesis by simp
d1aad0957eb2 tuned proofs
haftmann
parents: 42873
diff changeset
  1046
qed
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1047
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1048
end
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1049
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1050
context comm_monoid_mult
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1051
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1052
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1053
lemma fold_image_1:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1054
  "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
  1055
  apply (induct rule: finite_induct)
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1056
  apply simp by auto
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1057
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1058
lemma fold_image_Un_Int:
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1059
  "finite A ==> finite B ==>
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1060
    fold_image times g 1 A * fold_image times g 1 B =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1061
    fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
  1062
  apply (induct rule: finite_induct)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1063
by (induct set: finite) 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1064
   (auto simp add: mult_ac insert_absorb Int_insert_left)
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1065
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1066
lemma fold_image_Un_one:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1067
  assumes fS: "finite S" and fT: "finite T"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1068
  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1069
  shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1070
proof-
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1071
  have "fold_image op * f 1 (S \<inter> T) = 1" 
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1072
    apply (rule fold_image_1)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1073
    using fS fT I0 by auto 
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1074
  with fold_image_Un_Int[OF fS fT] show ?thesis by simp
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1075
qed
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1076
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1077
corollary fold_Un_disjoint:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1078
  "finite A ==> finite B ==> A Int B = {} ==>
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1079
   fold_image times g 1 (A Un B) =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1080
   fold_image times g 1 A * fold_image times g 1 B"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1081
by (simp add: fold_image_Un_Int)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1082
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1083
lemma fold_image_UN_disjoint:
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1084
  "\<lbrakk> finite I; ALL i:I. finite (A i);
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1085
     ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1086
   \<Longrightarrow> fold_image times g 1 (UNION I A) =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1087
       fold_image times (%i. fold_image times g 1 (A i)) 1 I"
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
  1088
apply (induct rule: finite_induct)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
  1089
apply simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
  1090
apply atomize
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1091
apply (subgoal_tac "ALL i:F. x \<noteq> i")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1092
 prefer 2 apply blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1093
apply (subgoal_tac "A x Int UNION F A = {}")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1094
 prefer 2 apply blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1095
apply (simp add: fold_Un_disjoint)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1096
done
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1097
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1098
lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1099
  fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1100
  fold_image times (split g) 1 (SIGMA x:A. B x)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1101
apply (subst Sigma_def)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1102
apply (subst fold_image_UN_disjoint, assumption, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1103
 apply blast
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1104
apply (erule fold_image_cong)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1105
apply (subst fold_image_UN_disjoint, simp, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1106
 apply blast
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1107
apply simp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1108
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1109
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1110
lemma fold_image_distrib: "finite A \<Longrightarrow>
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1111
   fold_image times (%x. g x * h x) 1 A =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1112
   fold_image times g 1 A *  fold_image times h 1 A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1113
by (erule finite_induct) (simp_all add: mult_ac)
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1114
30260
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1115
lemma fold_image_related: 
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1116
  assumes Re: "R e e" 
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1117
  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1118
  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1119
  shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1120
  using fS by (rule finite_subset_induct) (insert assms, auto)
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1121
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1122
lemma  fold_image_eq_general:
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1123
  assumes fS: "finite S"
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1124
  and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1125
  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1126
  shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1127
proof-
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1128
  from h f12 have hS: "h ` S = S'" by auto
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1129
  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1130
    from f12 h H  have "x = y" by auto }
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1131
  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1132
  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1133
  from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1134
  also have "\<dots> = fold_image (op *) (f2 o h) e S" 
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1135
    using fold_image_reindex[OF fS hinj, of f2 e] .
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1136
  also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1137
    by blast
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1138
  finally show ?thesis ..
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1139
qed
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1140
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1141
lemma fold_image_eq_general_inverses:
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1142
  assumes fS: "finite S" 
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1143
  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1144
  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1145
  shows "fold_image (op *) f e S = fold_image (op *) g e T"
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1146
  (* metis solves it, but not yet available here *)
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1147
  apply (rule fold_image_eq_general[OF fS, of T h g f e])
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1148
  apply (rule ballI)
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1149
  apply (frule kh)
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1150
  apply (rule ex1I[])
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1151
  apply blast
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1152
  apply clarsimp
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1153
  apply (drule hk) apply simp
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1154
  apply (rule sym)
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1155
  apply (erule conjunct1[OF conjunct2[OF hk]])
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1156
  apply (rule ballI)
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1157
  apply (drule  hk)
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1158
  apply blast
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1159
  done
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1160
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1161
end
22917
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1162
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25062
diff changeset
  1163
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1164
subsection {* A fold functional for non-empty sets *}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1165
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1166
text{* Does not require start value. *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1167
23736
bf8d4a46452d Renamed inductive2 to inductive.
berghofe
parents: 23706
diff changeset
  1168
inductive
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1169
  fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1170
  for f :: "'a => 'a => 'a"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1171
where
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1172
  fold1Set_insertI [intro]:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1173
   "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1174
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
  1175
definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1176
  "fold1 f A == THE x. fold1Set f A x"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1177
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1178
lemma fold1Set_nonempty:
22917
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1179
  "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1180
by(erule fold1Set.cases, simp_all)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1181
23736
bf8d4a46452d Renamed inductive2 to inductive.
berghofe
parents: 23706
diff changeset
  1182
inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
bf8d4a46452d Renamed inductive2 to inductive.
berghofe
parents: 23706
diff changeset
  1183
bf8d4a46452d Renamed inductive2 to inductive.
berghofe
parents: 23706
diff changeset
  1184
inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1185
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1186
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1187
lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
  1188
by (blast elim: fold_graph.cases)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1189
22917
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1190
lemma fold1_singleton [simp]: "fold1 f {a} = a"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1191
by (unfold fold1_def) blast
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1192
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1193
lemma finite_nonempty_imp_fold1Set:
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1194
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1195
apply (induct A rule: finite_induct)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1196
apply (auto dest: finite_imp_fold_graph [of _ f])
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1197
done
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1198
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1199
text{*First, some lemmas about @{const fold_graph}.*}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1200
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1201
context ab_semigroup_mult
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1202
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1203
46898
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
  1204
lemma comp_fun_commute: "comp_fun_commute (op *)"
1570b30ee040 tuned proofs -- eliminated pointless chaining of facts after 'interpret';
wenzelm
parents: 46146
diff changeset
  1205
  by default (simp add: fun_eq_iff mult_ac)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1206
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1207
lemma fold_graph_insert_swap:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1208
assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1209
shows "fold_graph times z (insert b A) (z * y)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1210
proof -
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
  1211
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1212
from assms show ?thesis
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1213
proof (induct rule: fold_graph.induct)
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
  1214
  case emptyI show ?case by (subst mult_commute [of z b], fast)
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1215
next
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1216
  case (insertI x A y)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1217
    have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1218
      using insertI by force  --{*how does @{term id} get unfolded?*}
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1219
    thus ?case by (simp add: insert_commute mult_ac)
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1220
qed
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1221
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1222
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1223
lemma fold_graph_permute_diff:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1224
assumes fold: "fold_graph times b A x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1225
shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1226
using fold
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1227
proof (induct rule: fold_graph.induct)
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1228
  case emptyI thus ?case by simp
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1229
next
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1230
  case (insertI x A y)
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1231
  have "a = x \<or> a \<in> A" using insertI by simp
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1232
  thus ?case
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1233
  proof
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1234
    assume "a = x"
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1235
    with insertI show ?thesis
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1236
      by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1237
  next
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1238
    assume ainA: "a \<in> A"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1239
    hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1240
      using insertI by force
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1241
    moreover
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1242
    have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1243
      using ainA insertI by blast
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1244
    ultimately show ?thesis by simp
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1245
  qed
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1246
qed
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1247
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1248
lemma fold1_eq_fold:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1249
assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1250
proof -
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
  1251
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1252
  from assms show ?thesis
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1253
apply (simp add: fold1_def fold_def)
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1254
apply (rule the_equality)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1255
apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1256
apply (rule sym, clarify)
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1257
apply (case_tac "Aa=A")
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
  1258
 apply (best intro: fold_graph_determ)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1259
apply (subgoal_tac "fold_graph times a A x")
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
  1260
 apply (best intro: fold_graph_determ)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1261
apply (subgoal_tac "insert aa (Aa - {a}) = A")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1262
 prefer 2 apply (blast elim: equalityE)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1263
apply (auto dest: fold_graph_permute_diff [where a=a])
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1264
done
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1265
qed
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1266
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1267
lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1268
apply safe
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1269
 apply simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1270
 apply (drule_tac x=x in spec)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1271
 apply (drule_tac x="A-{x}" in spec, auto)
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1272
done
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1273
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1274
lemma fold1_insert:
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1275
  assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1276
  shows "fold1 times (insert x A) = x * fold1 times A"
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1277
proof -
42871
1c0b99f950d9 names of fold_set locales resemble name of characteristic property more closely
haftmann
parents: 42869
diff changeset
  1278
  interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1279
  from nonempty obtain a A' where "A = insert a A' & a ~: A'"
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1280
    by (auto simp add: nonempty_iff)
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1281
  with A show ?thesis
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1282
    by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1283
qed
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1284
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1285
end
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1286
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1287
context ab_semigroup_idem_mult
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1288
begin
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of th