src/HOL/Finite_Set.thy
author nipkow
Thu, 17 Mar 2011 09:58:13 +0100
changeset 41988 c2583bbb92f5
parent 41987 4ad8f1dc2e0b
child 42206 0920f709610f
permissions -rw-r--r--
tuned lemma
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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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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` 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 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 (fastsimp 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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  case (insert x F A)
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  have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
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  show "finite A"
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  proof cases
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    assume x: "x \<in> A"
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    with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
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    with r have "finite (A - {x})" .
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    hence "finite (insert x (A - {x}))" ..
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    also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
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    finally show ?thesis .
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  next
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    show "A \<subseteq> F ==> ?thesis" by fact
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    assume "x \<notin> A"
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    with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
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  qed
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qed
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lemma finite_subset:
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  "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
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  by (rule rev_finite_subset)
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   162
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   163
lemma finite_UnI:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   164
  assumes "finite F" and "finite G"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   165
  shows "finite (F \<union> G)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   166
  using assms by induct simp_all
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   167
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   168
lemma finite_Un [iff]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   169
  "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   170
  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
   171
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   172
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
   173
proof -
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   174
  have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   175
  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
   176
  then show ?thesis by simp
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   177
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   178
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   179
lemma finite_Int [simp, intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   180
  "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   181
  by (blast intro: finite_subset)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   182
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   183
lemma finite_Collect_conjI [simp, intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   184
  "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
   185
  by (simp add: Collect_conj_eq)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   186
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   187
lemma finite_Collect_disjI [simp]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   188
  "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
   189
  by (simp add: Collect_disj_eq)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   190
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   191
lemma finite_Diff [simp, intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   192
  "finite A \<Longrightarrow> finite (A - B)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   193
  by (rule finite_subset, rule Diff_subset)
29901
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   194
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   195
lemma finite_Diff2 [simp]:
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   196
  assumes "finite B"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   197
  shows "finite (A - B) \<longleftrightarrow> finite A"
29901
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   198
proof -
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   199
  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
   200
  also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp
29901
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   201
  finally show ?thesis ..
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   202
qed
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   203
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   204
lemma finite_Diff_insert [iff]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   205
  "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   206
proof -
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   207
  have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   208
  moreover have "A - insert a B = A - B - {a}" by auto
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   209
  ultimately show ?thesis by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   210
qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   211
29901
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   212
lemma finite_compl[simp]:
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   213
  "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   214
  by (simp add: Compl_eq_Diff_UNIV)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   215
29916
f24137b42d9b more finiteness
nipkow
parents: 29903
diff changeset
   216
lemma finite_Collect_not[simp]:
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   217
  "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   218
  by (simp add: Collect_neg_eq)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   219
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   220
lemma finite_Union [simp, intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   221
  "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   222
  by (induct rule: finite_induct) simp_all
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   223
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   224
lemma finite_UN_I [intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   225
  "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
   226
  by (induct rule: finite_induct) simp_all
29903
2c0046b26f80 more finiteness changes
nipkow
parents: 29901
diff changeset
   227
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   228
lemma finite_UN [simp]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   229
  "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   230
  by (blast intro: finite_subset)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   231
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   232
lemma finite_Inter [intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   233
  "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   234
  by (blast intro: Inter_lower finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   235
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   236
lemma finite_INT [intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   237
  "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   238
  by (blast intro: INT_lower finite_subset)
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   239
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   240
lemma finite_imageI [simp, intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   241
  "finite F \<Longrightarrow> finite (h ` F)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   242
  by (induct rule: finite_induct) simp_all
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   243
31768
159cd6b5e5d4 lemma finite_image_set by Jeremy Avigad
haftmann
parents: 31465
diff changeset
   244
lemma finite_image_set [simp]:
159cd6b5e5d4 lemma finite_image_set by Jeremy Avigad
haftmann
parents: 31465
diff changeset
   245
  "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
159cd6b5e5d4 lemma finite_image_set by Jeremy Avigad
haftmann
parents: 31465
diff changeset
   246
  by (simp add: image_Collect [symmetric])
159cd6b5e5d4 lemma finite_image_set by Jeremy Avigad
haftmann
parents: 31465
diff changeset
   247
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   248
lemma finite_imageD:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   249
  "finite (f ` A) \<Longrightarrow> inj_on f A \<Longrightarrow> finite A"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   250
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   251
  have aux: "!!A. finite (A - {}) = finite A" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   252
  fix B :: "'a set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   253
  assume "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   254
  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   255
    apply induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   256
     apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   257
    apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   258
     apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   259
     apply (simp (no_asm_use) add: inj_on_def)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   260
     apply (blast dest!: aux [THEN iffD1], atomize)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   261
    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   262
    apply (frule subsetD [OF equalityD2 insertI1], clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   263
    apply (rule_tac x = xa in bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   264
     apply (simp_all add: inj_on_image_set_diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   265
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   266
qed (rule refl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   267
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   268
lemma finite_surj:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   269
  "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   270
  by (erule finite_subset) (rule finite_imageI)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   271
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   272
lemma finite_range_imageI:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   273
  "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   274
  by (drule finite_imageI) (simp add: range_composition)
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   275
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   276
lemma finite_subset_image:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   277
  assumes "finite B"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   278
  shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   279
using assms proof induct
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   280
  case empty then show ?case by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   281
next
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   282
  case insert then show ?case
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   283
    by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   284
       blast
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   285
qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   286
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   287
lemma finite_vimageI:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   288
  "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   289
  apply (induct rule: finite_induct)
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
   290
   apply simp_all
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   291
  apply (subst vimage_insert)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
   292
  apply (simp add: finite_subset [OF inj_vimage_singleton])
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   293
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   294
34111
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   295
lemma finite_vimageD:
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   296
  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
   297
  shows "finite F"
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   298
proof -
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   299
  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
   300
  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
   301
  finally show "finite F" .
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   302
qed
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   303
1b015caba46c add lemmas rev_finite_subset, finite_vimageD, finite_vimage_iff
huffman
parents: 34007
diff changeset
   304
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
   305
  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
   306
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   307
lemma finite_Collect_bex [simp]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   308
  assumes "finite A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   309
  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
   310
proof -
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   311
  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
   312
  with assms show ?thesis by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   313
qed
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   314
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   315
lemma finite_Collect_bounded_ex [simp]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   316
  assumes "finite {y. P y}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   317
  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
   318
proof -
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   319
  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
   320
  with assms show ?thesis by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   321
qed
29920
b95f5b8b93dd more finiteness
nipkow
parents: 29918
diff changeset
   322
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   323
lemma finite_Plus:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   324
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   325
  by (simp add: Plus_def)
17022
b257300c3a9c added Brian Hufmann's finite instances
nipkow
parents: 16775
diff changeset
   326
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   327
lemma finite_PlusD: 
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   328
  fixes A :: "'a set" and B :: "'b set"
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   329
  assumes fin: "finite (A <+> B)"
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   330
  shows "finite A" "finite B"
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   331
proof -
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   332
  have "Inl ` A \<subseteq> A <+> B" by auto
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   333
  then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   334
  then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   335
next
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   336
  have "Inr ` B \<subseteq> A <+> B" by auto
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   337
  then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   338
  then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   339
qed
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   340
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   341
lemma finite_Plus_iff [simp]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   342
  "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   343
  by (auto intro: finite_PlusD finite_Plus)
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 31017
diff changeset
   344
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   345
lemma finite_Plus_UNIV_iff [simp]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   346
  "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   347
  by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   348
40786
0a54cfc9add3 gave more standard finite set rules simp and intro attribute
nipkow
parents: 40716
diff changeset
   349
lemma finite_SigmaI [simp, intro]:
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   350
  "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
   351
  by (unfold Sigma_def) blast
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   352
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   353
lemma finite_cartesian_product:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   354
  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   355
  by (rule finite_SigmaI)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   356
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   357
lemma finite_Prod_UNIV:
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   358
  "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   359
  by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   360
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   361
lemma finite_cartesian_productD1:
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   362
  "finite (A \<times> B) \<Longrightarrow> B \<noteq> {} \<Longrightarrow> finite A"
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   363
apply (auto simp add: finite_conv_nat_seg_image) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   364
apply (drule_tac x=n in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   365
apply (drule_tac x="fst o f" in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   366
apply (auto simp add: o_def) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   367
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   368
apply (drule equalityD1) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   369
apply (rename_tac y x)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   370
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   371
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   372
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   373
apply (rule_tac x=k in image_eqI, auto)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   374
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   375
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   376
lemma finite_cartesian_productD2:
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   377
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   378
apply (auto simp add: finite_conv_nat_seg_image) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   379
apply (drule_tac x=n in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   380
apply (drule_tac x="snd o f" in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   381
apply (auto simp add: o_def) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   382
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   383
apply (drule equalityD1)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   384
apply (rename_tac x y)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   385
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   386
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   387
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   388
apply (rule_tac x=k in image_eqI, auto)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   389
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   390
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   391
lemma finite_Pow_iff [iff]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   392
  "finite (Pow A) \<longleftrightarrow> finite A"
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   393
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   394
  assume "finite (Pow A)"
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   395
  then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   396
  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
   397
next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   398
  assume "finite A"
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   399
  then show "finite (Pow A)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
   400
    by induct (simp_all add: Pow_insert)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   401
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   402
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   403
corollary finite_Collect_subsets [simp, intro]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   404
  "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   405
  by (simp add: Pow_def [symmetric])
29918
214755b03df3 more finiteness
nipkow
parents: 29916
diff changeset
   406
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   407
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   408
  by (blast intro: finite_subset [OF subset_Pow_Union])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   409
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   410
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   411
subsubsection {* Further induction rules on finite sets *}
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   412
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   413
lemma finite_ne_induct [case_names singleton insert, consumes 2]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   414
  assumes "finite F" and "F \<noteq> {}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   415
  assumes "\<And>x. P {x}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   416
    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
   417
  shows "P F"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   418
using assms proof induct
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   419
  case empty then show ?case by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   420
next
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   421
  case (insert x F) then show ?case by cases auto
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   422
qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   423
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   424
lemma finite_subset_induct [consumes 2, case_names empty insert]:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   425
  assumes "finite F" and "F \<subseteq> A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   426
  assumes empty: "P {}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   427
    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
   428
  shows "P F"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   429
using `finite F` `F \<subseteq> A` proof induct
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   430
  show "P {}" by fact
31441
428e4caf2299 finite lemmas
nipkow
parents: 31438
diff changeset
   431
next
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   432
  fix x F
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   433
  assume "finite F" and "x \<notin> F" and
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   434
    P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   435
  show "P (insert x F)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   436
  proof (rule insert)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   437
    from i show "x \<in> A" by blast
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   438
    from i have "F \<subseteq> A" by blast
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   439
    with P show "P F" .
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   440
    show "finite F" by fact
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   441
    show "x \<notin> F" by fact
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   442
  qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   443
qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   444
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   445
lemma finite_empty_induct:
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   446
  assumes "finite A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   447
  assumes "P A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   448
    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
   449
  shows "P {}"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   450
proof -
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   451
  have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   452
  proof -
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   453
    fix B :: "'a set"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   454
    assume "B \<subseteq> A"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   455
    with `finite A` have "finite B" by (rule rev_finite_subset)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   456
    from this `B \<subseteq> A` show "P (A - B)"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   457
    proof induct
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   458
      case empty
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   459
      from `P A` show ?case by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   460
    next
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   461
      case (insert b B)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   462
      have "P (A - B - {b})"
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   463
      proof (rule remove)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   464
        from `finite A` show "finite (A - B)" by induct auto
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   465
        from insert show "b \<in> A - B" by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   466
        from insert show "P (A - B)" by simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   467
      qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   468
      also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   469
      finally show ?case .
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   470
    qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   471
  qed
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   472
  then have "P (A - A)" by blast
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   473
  then show ?thesis by simp
31441
428e4caf2299 finite lemmas
nipkow
parents: 31438
diff changeset
   474
qed
428e4caf2299 finite lemmas
nipkow
parents: 31438
diff changeset
   475
428e4caf2299 finite lemmas
nipkow
parents: 31438
diff changeset
   476
26441
7914697ff104 no "attach UNIV" any more
haftmann
parents: 26146
diff changeset
   477
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
   478
29797
08ef36ed2f8a handling type classes without parameters
haftmann
parents: 29675
diff changeset
   479
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
   480
  assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
27430
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   481
begin
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   482
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   483
lemma finite [simp]: "finite (A \<Colon> 'a set)"
26441
7914697ff104 no "attach UNIV" any more
haftmann
parents: 26146
diff changeset
   484
  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
   485
40922
4d0f96a54e76 adding code equation for finiteness of finite types
bulwahn
parents: 40786
diff changeset
   486
lemma finite_code [code]: "finite (A \<Colon> 'a set) = True"
4d0f96a54e76 adding code equation for finiteness of finite types
bulwahn
parents: 40786
diff changeset
   487
  by simp
4d0f96a54e76 adding code equation for finiteness of finite types
bulwahn
parents: 40786
diff changeset
   488
27430
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   489
end
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   490
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35796
diff changeset
   491
lemma UNIV_unit [no_atp]:
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
   492
  "UNIV = {()}" by auto
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   493
35719
99b6152aedf5 split off theory Big_Operators from theory Finite_Set
haftmann
parents: 35577
diff changeset
   494
instance unit :: finite proof
99b6152aedf5 split off theory Big_Operators from theory Finite_Set
haftmann
parents: 35577
diff changeset
   495
qed (simp add: UNIV_unit)
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   496
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35796
diff changeset
   497
lemma UNIV_bool [no_atp]:
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
   498
  "UNIV = {False, True}" by auto
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   499
35719
99b6152aedf5 split off theory Big_Operators from theory Finite_Set
haftmann
parents: 35577
diff changeset
   500
instance bool :: finite proof
99b6152aedf5 split off theory Big_Operators from theory Finite_Set
haftmann
parents: 35577
diff changeset
   501
qed (simp add: UNIV_bool)
99b6152aedf5 split off theory Big_Operators from theory Finite_Set
haftmann
parents: 35577
diff changeset
   502
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37466
diff changeset
   503
instance prod :: (finite, finite) finite proof
35719
99b6152aedf5 split off theory Big_Operators from theory Finite_Set
haftmann
parents: 35577
diff changeset
   504
qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   505
35719
99b6152aedf5 split off theory Big_Operators from theory Finite_Set
haftmann
parents: 35577
diff changeset
   506
lemma finite_option_UNIV [simp]:
99b6152aedf5 split off theory Big_Operators from theory Finite_Set
haftmann
parents: 35577
diff changeset
   507
  "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
99b6152aedf5 split off theory Big_Operators from theory Finite_Set
haftmann
parents: 35577
diff changeset
   508
  by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
99b6152aedf5 split off theory Big_Operators from theory Finite_Set
haftmann
parents: 35577
diff changeset
   509
99b6152aedf5 split off theory Big_Operators from theory Finite_Set
haftmann
parents: 35577
diff changeset
   510
instance option :: (finite) finite proof
99b6152aedf5 split off theory Big_Operators from theory Finite_Set
haftmann
parents: 35577
diff changeset
   511
qed (simp add: UNIV_option_conv)
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   512
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
   513
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
   514
  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
   515
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   516
instance "fun" :: (finite, finite) finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   517
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
   518
  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
   519
  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
   520
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
26792
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   521
    have "range ?graph \<subseteq> Pow UNIV" by simp
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   522
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   523
      by (simp only: finite_Pow_iff finite)
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   524
    ultimately show "finite (range ?graph)"
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   525
      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
   526
    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
   527
  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
   528
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
   529
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37466
diff changeset
   530
instance sum :: (finite, finite) finite proof
35719
99b6152aedf5 split off theory Big_Operators from theory Finite_Set
haftmann
parents: 35577
diff changeset
   531
qed (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
27981
feb0c01cf0fb tuned import order
haftmann
parents: 27611
diff changeset
   532
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
   533
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   534
subsection {* A basic fold functional for finite sets *}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   535
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   536
text {* The intended behaviour is
31916
f3227bb306a4 recovered subscripts, which were lost in b41d61c768e2 (due to Emacs accident?);
wenzelm
parents: 31907
diff changeset
   537
@{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
   538
if @{text f} is ``left-commutative'':
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   539
*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   540
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   541
locale fun_left_comm =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   542
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   543
  assumes fun_left_comm: "f x (f y z) = f y (f x z)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   544
begin
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   545
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   546
text{* On a functional level it looks much nicer: *}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   547
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   548
lemma fun_comp_comm:  "f x \<circ> f y = f y \<circ> 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
   549
by (simp add: fun_left_comm fun_eq_iff)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   550
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   551
end
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   552
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   553
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
   554
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
   555
  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
   556
  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
   557
      \<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
   558
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   559
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
   560
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   561
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
   562
  "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
   563
15498
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   564
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
   565
@{term "if finite A then THE y. fold_graph f z A y else e"}.
15498
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   566
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
   567
@{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
   568
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
   569
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
lemma Diff1_fold_graph:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   572
  "fold_graph f z (A - {x}) y \<Longrightarrow> x \<in> A \<Longrightarrow> fold_graph f z A (f x y)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   573
by (erule insert_Diff [THEN subst], rule fold_graph.intros, auto)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   574
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   575
lemma fold_graph_imp_finite: "fold_graph f z A x \<Longrightarrow> finite A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   576
by (induct set: fold_graph) auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   577
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   578
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
   579
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
   580
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   581
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   582
subsubsection{*From @{const fold_graph} to @{term fold}*}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   583
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   584
context fun_left_comm
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
   585
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
   586
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   587
lemma fold_graph_insertE_aux:
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   588
  "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
   589
proof (induct set: fold_graph)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   590
  case (insertI x A y) show ?case
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   591
  proof (cases "x = a")
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   592
    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
   593
  next
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   594
    assume "x \<noteq> a"
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   595
    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
   596
      using insertI by auto
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   597
    have 1: "f x y = f a (f x y')"
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   598
      unfolding y by (rule fun_left_comm)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   599
    have 2: "fold_graph f z (insert x A - {a}) (f x y')"
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   600
      using y' and `x \<noteq> a` and `x \<notin> A`
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   601
      by (simp add: insert_Diff_if fold_graph.insertI)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   602
    from 1 2 show ?case by fast
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   603
  qed
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   604
qed simp
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   605
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   606
lemma fold_graph_insertE:
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   607
  assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   608
  obtains y where "v = f x y" and "fold_graph f z A y"
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   609
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
   610
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   611
lemma fold_graph_determ:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   612
  "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
   613
proof (induct arbitrary: y set: fold_graph)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   614
  case (insertI x A y v)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   615
  from `fold_graph f z (insert x A) v` and `x \<notin> A`
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   616
  obtain y' where "v = f x y'" and "fold_graph f z A y'"
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   617
    by (rule fold_graph_insertE)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   618
  from `fold_graph f z A y'` have "y' = y" by (rule insertI)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   619
  with `v = f x y'` show "v = f x y" by simp
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   620
qed fast
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   621
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   622
lemma fold_equality:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   623
  "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
   624
by (unfold fold_def) (blast intro: fold_graph_determ)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   625
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   626
lemma fold_graph_fold: "finite A \<Longrightarrow> fold_graph f z A (fold f z A)"
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   627
unfolding fold_def
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   628
apply (rule theI')
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   629
apply (rule ex_ex1I)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   630
apply (erule finite_imp_fold_graph)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   631
apply (erule (1) fold_graph_determ)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   632
done
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   633
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   634
text{* The base case for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   635
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   636
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
   637
by (unfold fold_def) blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   638
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   639
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
   640
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   641
lemma fold_insert [simp]:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   642
  "finite A ==> x \<notin> A ==> fold f z (insert x A) = f x (fold f z A)"
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   643
apply (rule fold_equality)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   644
apply (erule fold_graph.insertI)
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
   645
apply (erule fold_graph_fold)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   646
done
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   647
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   648
lemma fold_fun_comm:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   649
  "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
   650
proof (induct rule: finite_induct)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   651
  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
   652
next
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   653
  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
   654
    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
   655
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   656
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   657
lemma fold_insert2:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   658
  "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
   659
by (simp add: fold_fun_comm)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   660
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
   661
lemma fold_rec:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   662
assumes "finite A" and "x \<in> A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   663
shows "fold f z 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
   664
proof -
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   665
  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
   666
  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
   667
  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
   668
    by (rule fold_insert) (simp add: `finite A`)+
15535
nipkow
parents: 15532
diff changeset
   669
  finally show ?thesis .
nipkow
parents: 15532
diff changeset
   670
qed
nipkow
parents: 15532
diff changeset
   671
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   672
lemma fold_insert_remove:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   673
  assumes "finite A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   674
  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
   675
proof -
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   676
  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
   677
  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
   678
  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
   679
    by (rule fold_rec)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   680
  then show ?thesis by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   681
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   682
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
   683
end
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   684
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   685
text{* A simplified version for idempotent functions: *}
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   686
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   687
locale fun_left_comm_idem = fun_left_comm +
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   688
  assumes fun_left_idem: "f x (f x z) = f x z"
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   689
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
   690
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   691
text{* The nice version: *}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   692
lemma fun_comp_idem : "f x o f x = 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
   693
by (simp add: fun_left_idem fun_eq_iff)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   694
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
   695
lemma fold_insert_idem:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   696
  assumes fin: "finite A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   697
  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
   698
proof cases
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   699
  assume "x \<in> A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   700
  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
   701
  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
   702
next
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   703
  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
   704
qed
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   705
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   706
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
   707
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   708
lemma fold_insert_idem2:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   709
  "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
   710
by(simp add:fold_fun_comm)
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   711
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
   712
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
   713
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   714
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   715
subsubsection {* Expressing set operations via @{const fold} *}
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   716
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   717
lemma (in fun_left_comm) fun_left_comm_apply:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   718
  "fun_left_comm (\<lambda>x. f (g x))"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   719
proof
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   720
qed (simp_all add: fun_left_comm)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   721
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   722
lemma (in fun_left_comm_idem) fun_left_comm_idem_apply:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   723
  "fun_left_comm_idem (\<lambda>x. f (g x))"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   724
  by (rule fun_left_comm_idem.intro, rule fun_left_comm_apply, unfold_locales)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   725
    (simp_all add: fun_left_idem)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   726
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   727
lemma fun_left_comm_idem_insert:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   728
  "fun_left_comm_idem insert"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   729
proof
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   730
qed auto
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   731
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   732
lemma fun_left_comm_idem_remove:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   733
  "fun_left_comm_idem (\<lambda>x A. A - {x})"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   734
proof
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   735
qed auto
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   736
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   737
lemma (in semilattice_inf) fun_left_comm_idem_inf:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   738
  "fun_left_comm_idem inf"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   739
proof
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   740
qed (auto simp add: inf_left_commute)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   741
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   742
lemma (in semilattice_sup) fun_left_comm_idem_sup:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   743
  "fun_left_comm_idem sup"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   744
proof
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   745
qed (auto simp add: sup_left_commute)
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   746
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   747
lemma union_fold_insert:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   748
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   749
  shows "A \<union> B = fold insert B A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   750
proof -
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   751
  interpret fun_left_comm_idem insert by (fact fun_left_comm_idem_insert)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   752
  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
   753
qed
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   754
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   755
lemma minus_fold_remove:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   756
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   757
  shows "B - A = fold (\<lambda>x A. A - {x}) B A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   758
proof -
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   759
  interpret fun_left_comm_idem "\<lambda>x A. A - {x}" by (fact fun_left_comm_idem_remove)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   760
  from `finite A` show ?thesis by (induct A arbitrary: B) auto
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   761
qed
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   762
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   763
context complete_lattice
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   764
begin
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   765
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   766
lemma inf_Inf_fold_inf:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   767
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   768
  shows "inf B (Inf A) = fold inf B A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   769
proof -
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   770
  interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   771
  from `finite A` show ?thesis by (induct A arbitrary: B)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 40945
diff changeset
   772
    (simp_all add: Inf_insert inf_commute fold_fun_comm)
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   773
qed
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   774
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   775
lemma sup_Sup_fold_sup:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   776
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   777
  shows "sup B (Sup A) = fold sup B A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   778
proof -
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   779
  interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   780
  from `finite A` show ?thesis by (induct A arbitrary: B)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 40945
diff changeset
   781
    (simp_all add: Sup_insert sup_commute fold_fun_comm)
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   782
qed
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   783
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   784
lemma Inf_fold_inf:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   785
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   786
  shows "Inf A = fold inf top A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   787
  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
   788
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   789
lemma Sup_fold_sup:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   790
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   791
  shows "Sup A = fold sup bot A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   792
  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
   793
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   794
lemma inf_INFI_fold_inf:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   795
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   796
  shows "inf B (INFI A f) = fold (\<lambda>A. inf (f A)) B A" (is "?inf = ?fold") 
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   797
proof (rule sym)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   798
  interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   799
  interpret fun_left_comm_idem "\<lambda>A. inf (f A)" by (fact fun_left_comm_idem_apply)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   800
  from `finite A` show "?fold = ?inf"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   801
  by (induct A arbitrary: B)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 40945
diff changeset
   802
    (simp_all add: INFI_def Inf_insert inf_left_commute)
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   803
qed
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   804
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   805
lemma sup_SUPR_fold_sup:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   806
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   807
  shows "sup B (SUPR A f) = fold (\<lambda>A. sup (f A)) B A" (is "?sup = ?fold") 
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   808
proof (rule sym)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   809
  interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   810
  interpret fun_left_comm_idem "\<lambda>A. sup (f A)" by (fact fun_left_comm_idem_apply)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   811
  from `finite A` show "?fold = ?sup"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   812
  by (induct A arbitrary: B)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 40945
diff changeset
   813
    (simp_all add: SUPR_def Sup_insert sup_left_commute)
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   814
qed
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   815
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   816
lemma INFI_fold_inf:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   817
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   818
  shows "INFI A f = fold (\<lambda>A. inf (f A)) top A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   819
  using assms inf_INFI_fold_inf [of A top] by simp
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   820
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   821
lemma SUPR_fold_sup:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   822
  assumes "finite A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   823
  shows "SUPR A f = fold (\<lambda>A. sup (f A)) bot A"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   824
  using assms sup_SUPR_fold_sup [of A bot] by simp
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   825
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   826
end
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   827
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   828
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   829
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
   830
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   831
definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   832
where "fold_image f g = fold (%x y. f (g x) y)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   833
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   834
lemma fold_image_empty[simp]: "fold_image f g z {} = z"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   835
by(simp add:fold_image_def)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   836
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
   837
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
   838
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
   839
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   840
lemma fold_image_insert[simp]:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   841
assumes "finite A" and "a \<notin> A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   842
shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   843
proof -
29223
e09c53289830 Conversion of HOL-Main and ZF to new locales.
ballarin
parents: 29025
diff changeset
   844
  interpret I: fun_left_comm "%x y. (g x) * y"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   845
    by unfold_locales (simp add: mult_ac)
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   846
  show ?thesis using assms 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
   847
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   848
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   849
(*
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
   850
lemma fold_commute:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   851
  "finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   852
  apply (induct set: finite)
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
   853
   apply simp
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   854
  apply (simp add: mult_left_commute [of x])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   855
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   856
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
   857
lemma fold_nest_Un_Int:
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   858
  "finite A ==> finite B
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   859
    ==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   860
  apply (induct set: finite)
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
   861
   apply simp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   862
  apply (simp add: fold_commute Int_insert_left insert_absorb)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   863
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   864
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
   865
lemma fold_nest_Un_disjoint:
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   866
  "finite A ==> finite B ==> A Int B = {}
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   867
    ==> fold times g z (A Un B) = fold times g (fold times g z B) A"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   868
  by (simp add: fold_nest_Un_Int)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   869
*)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   870
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   871
lemma fold_image_reindex:
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   872
assumes fin: "finite A"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   873
shows "inj_on h A \<Longrightarrow> fold_image times g z (h`A) = fold_image times (g\<circ>h) z A"
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31916
diff changeset
   874
using fin by induct auto
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   875
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   876
(*
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
   877
text{*
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   878
  Fusion theorem, as described in Graham Hutton's paper,
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   879
  A Tutorial on the Universality and Expressiveness of Fold,
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   880
  JFP 9:4 (355-372), 1999.
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   881
*}
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
   882
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
   883
lemma fold_fusion:
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 27430
diff changeset
   884
  assumes "ab_semigroup_mult g"
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
   885
  assumes fin: "finite A"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   886
    and hyp: "\<And>x y. h (g x y) = times x (h y)"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   887
  shows "h (fold g j w A) = fold times j (h w) A"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 27430
diff changeset
   888
proof -
29223
e09c53289830 Conversion of HOL-Main and ZF to new locales.
ballarin
parents: 29025
diff changeset
   889
  class_interpret ab_semigroup_mult [g] by fact
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 27430
diff changeset
   890
  show ?thesis using fin hyp by (induct set: finite) simp_all
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 27430
diff changeset
   891
qed
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   892
*)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   893
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   894
lemma fold_image_cong:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   895
  "finite A \<Longrightarrow>
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   896
  (!!x. x:A ==> g x = h x) ==> fold_image times g z A = fold_image times h z A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   897
apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   898
 apply simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   899
apply (erule finite_induct, simp)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   900
apply (simp add: subset_insert_iff, clarify)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   901
apply (subgoal_tac "finite C")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   902
 prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   903
apply (subgoal_tac "C = insert x (C - {x})")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   904
 prefer 2 apply blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   905
apply (erule ssubst)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   906
apply (drule spec)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   907
apply (erule (1) notE impE)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   908
apply (simp add: Ball_def del: insert_Diff_single)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   909
done
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   910
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
   911
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
   912
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
   913
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
   914
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
   915
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   916
lemma fold_image_1:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   917
  "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
   918
  apply (induct rule: finite_induct)
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   919
  apply simp by auto
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   920
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   921
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
   922
  "finite A ==> finite B ==>
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   923
    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
   924
    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
   925
  apply (induct rule: finite_induct)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   926
by (induct set: finite) 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   927
   (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
   928
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   929
lemma fold_image_Un_one:
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   930
  assumes fS: "finite S" and fT: "finite T"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   931
  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
   932
  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
   933
proof-
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   934
  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
   935
    apply (rule fold_image_1)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   936
    using fS fT I0 by auto 
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   937
  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
   938
qed
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
   939
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
   940
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
   941
  "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
   942
   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
   943
   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
   944
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
   945
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   946
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
   947
  "\<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
   948
     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
   949
   \<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
   950
       fold_image times (%i. fold_image times g 1 (A i)) 1 I"
41656
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   951
apply (induct rule: finite_induct)
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   952
apply simp
011fcb70e32f restructured theory;
haftmann
parents: 41550
diff changeset
   953
apply atomize
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   954
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
   955
 prefer 2 apply blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   956
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
   957
 prefer 2 apply blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   958
apply (simp add: fold_Un_disjoint)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   959
done
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   960
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   961
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
   962
  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
   963
  fold_image times (split g) 1 (SIGMA x:A. B x)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   964
apply (subst Sigma_def)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   965
apply (subst fold_image_UN_disjoint, assumption, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   966
 apply blast
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   967
apply (erule fold_image_cong)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   968
apply (subst fold_image_UN_disjoint, simp, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   969
 apply blast
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   970
apply simp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   971
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   972
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   973
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
   974
   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
   975
   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
   976
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
   977
30260
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
   978
lemma fold_image_related: 
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
   979
  assumes Re: "R e e" 
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
   980
  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
   981
  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
   982
  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
   983
  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
   984
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
   985
lemma  fold_image_eq_general:
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
   986
  assumes fS: "finite S"
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
   987
  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
   988
  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
   989
  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
   990
proof-
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
   991
  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
   992
  {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
   993
    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
   994
  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
   995
  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
   996
  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
   997
  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
   998
    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
   999
  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
  1000
    by blast
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1001
  finally show ?thesis ..
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1002
qed
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1003
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1004
lemma fold_image_eq_general_inverses:
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1005
  assumes fS: "finite S" 
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1006
  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
  1007
  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
  1008
  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
  1009
  (* metis solves it, but not yet available here *)
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1010
  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
  1011
  apply (rule ballI)
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1012
  apply (frule kh)
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1013
  apply (rule ex1I[])
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1014
  apply blast
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1015
  apply clarsimp
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1016
  apply (drule hk) apply simp
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1017
  apply (rule sym)
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1018
  apply (erule conjunct1[OF conjunct2[OF hk]])
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1019
  apply (rule ballI)
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1020
  apply (drule  hk)
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1021
  apply blast
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1022
  done
be39acd3ac85 Added general theorems for fold_image, setsum and set_prod
chaieb
parents: 29966
diff changeset
  1023
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
  1024
end
22917
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1025
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25062
diff changeset
  1026
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1027
subsection {* A fold functional for non-empty sets *}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1028
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1029
text{* Does not require start value. *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1030
23736
bf8d4a46452d Renamed inductive2 to inductive.
berghofe
parents: 23706
diff changeset
  1031
inductive
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1032
  fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1033
  for f :: "'a => 'a => 'a"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1034
where
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1035
  fold1Set_insertI [intro]:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1036
   "\<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
  1037
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35267
diff changeset
  1038
definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1039
  "fold1 f A == THE x. fold1Set f A x"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1040
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1041
lemma fold1Set_nonempty:
22917
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1042
  "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
  1043
by(erule fold1Set.cases, simp_all)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1044
23736
bf8d4a46452d Renamed inductive2 to inductive.
berghofe
parents: 23706
diff changeset
  1045
inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
bf8d4a46452d Renamed inductive2 to inductive.
berghofe
parents: 23706
diff changeset
  1046
bf8d4a46452d Renamed inductive2 to inductive.
berghofe
parents: 23706
diff changeset
  1047
inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1048
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1049
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1050
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
  1051
by (blast elim: fold_graph.cases)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1052
22917
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1053
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
  1054
by (unfold fold1_def) blast
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1055
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1056
lemma finite_nonempty_imp_fold1Set:
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1057
  "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1058
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
  1059
apply (auto dest: finite_imp_fold_graph [of _ f])
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1060
done
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1061
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1062
text{*First, some lemmas about @{const fold_graph}.*}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1063
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
  1064
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
  1065
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
  1066
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1067
lemma fun_left_comm: "fun_left_comm(op *)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1068
by unfold_locales (simp add: mult_ac)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1069
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1070
lemma fold_graph_insert_swap:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1071
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
  1072
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
  1073
proof -
29223
e09c53289830 Conversion of HOL-Main and ZF to new locales.
ballarin
parents: 29025
diff changeset
  1074
  interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1075
from assms show ?thesis
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1076
proof (induct rule: fold_graph.induct)
36045
b846881928ea simplify fold_graph proofs
huffman
parents: 35831
diff changeset
  1077
  case emptyI show ?case by (subst mult_commute [of z b], fast)
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1078
next
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1079
  case (insertI x A y)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1080
    have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1081
      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
  1082
    thus ?case by (simp add: insert_commute mult_ac)
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1083
qed
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1084
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1085
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1086
lemma fold_graph_permute_diff:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1087
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
  1088
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
  1089
using fold
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1090
proof (induct rule: fold_graph.induct)
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1091
  case emptyI thus ?case by simp
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1092
next
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1093
  case (insertI x A y)
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1094
  have "a = x \<or> a \<in> A" using insertI by simp
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1095
  thus ?case
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1096
  proof
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1097
    assume "a = x"
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1098
    with insertI show ?thesis
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1099
      by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1100
  next
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1101
    assume ainA: "a \<in> A"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1102
    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
  1103
      using insertI by force
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1104
    moreover
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1105
    have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1106
      using ainA insertI by blast
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1107
    ultimately show ?thesis by simp
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1108
  qed
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1109
qed
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1110
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
  1111
lemma fold1_eq_fold:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1112
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
  1113
proof -
29223
e09c53289830 Conversion of HOL-Main and ZF to new locales.
ballarin
parents: 29025
diff changeset
  1114
  interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1115
  from assms show ?thesis
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1116
apply (simp add: fold1_def fold_def)
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1117
apply (rule the_equality)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1118
apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1119
apply (rule sym, clarify)
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1120
apply (case_tac "Aa=A")
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
  1121
 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
  1122
apply (subgoal_tac "fold_graph times a A x")
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
  1123
 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
  1124
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
  1125
 prefer 2 apply (blast elim: equalityE)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1126
apply (auto dest: fold_graph_permute_diff [where a=a])
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1127
done
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1128
qed
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1129
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1130
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
  1131
apply safe
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1132
 apply simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1133
 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
  1134
 apply (drule_tac x="A-{x}" in spec, auto)
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1135
done
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1136
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
  1137
lemma fold1_insert:
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1138
  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
  1139
  shows "fold1 times (insert x A) = x * fold1 times A"
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1140
proof -
29223
e09c53289830 Conversion of HOL-Main and ZF to new locales.
ballarin
parents: 29025
diff changeset
  1141
  interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1142
  from nonempty obtain a A' where "A = insert a A' & a ~: A'"
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1143
    by (auto simp add: nonempty_iff)
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1144
  with A show ?thesis
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1145
    by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1146
qed
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1147
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
  1148
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
  1149
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
  1150
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
  1151
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
  1152
35817
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1153
lemma fun_left_comm_idem: "fun_left_comm_idem(op *)"
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1154
apply unfold_locales
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1155
 apply (rule mult_left_commute)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1156
apply (rule mult_left_idem)
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1157
done
d8b8527102f5 added locales folding_one_(idem); various streamlining and tuning
haftmann
parents: 35796
diff changeset
  1158
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
  1159
lemma fold1_insert_idem [simp]:
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1160
  assumes nonempty: "A \<noteq> {}" and A: "finite A" 
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1161
  shows "fold1 times (insert x A) = x * fold1 times A"
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1162
proof -
29223
e09c53289830 Conversion of HOL-Main and ZF to new locales.
ballarin
parents: 29025
diff changeset
  1163
  interpret fun_left_comm_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1164
    by (rule fun_left_comm_idem)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1165
  from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1166
    by (auto simp add: nonempty_iff)
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1167
  show ?thesis
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1168
  proof cases
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 40945
diff changeset
  1169
    assume a: "a = x"
efa734d9b221 eliminated global prems;
wenzelm
parents: 40945
diff changeset
  1170
    show ?thesis
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1171
    proof cases
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1172
      assume "A' = {}"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 40945
diff changeset
  1173
      with A' a show ?thesis by simp
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1174
    next
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1175
      assume "A' \<noteq> {}"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 40945
diff changeset
  1176
      with A A' a show ?thesis
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
  1177
        by (simp add: fold1_insert mult_assoc [symmetric])
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1178
    qed
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1179
  next
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1180
    assume "a \<noteq> x"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 40945
diff changeset
  1181
    with A A' show ?thesis
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
  1182
      by (simp add: insert_commute fold1_eq_fold)
15521
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1183
  qed
1ffd04343ac9 non-inductive fold1Set proofs
paulson
parents: 15520
diff changeset
  1184
qed
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1185
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
  1186
lemma hom_fold1_commute:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1187
assumes hom: "!!x y. h (x * y) = h x * h y"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1188
and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
22917
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1189
using N proof (induct rule: finite_ne_induct)
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1190
  case singleton thus ?case by simp
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1191
next
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1192
  case (insert n N)
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1193
  then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1194
  also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
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
  1195
  also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
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
  1196
  also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
22917
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1197
    using insert by(simp)
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1198
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1199
  finally show ?case .
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1200
qed
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1201
32679
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1202
lemma fold1_eq_fold_idem:
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1203
  assumes "finite A"
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1204
  shows "fold1 times (insert a A) = fold times a A"
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1205
proof (cases "a \<in> A")
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1206
  case False
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1207
  with assms show ?thesis by (simp add: fold1_eq_fold)
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1208
next
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1209
  interpret fun_left_comm_idem times by (fact fun_left_comm_idem)
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1210
  case True then obtain b B
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1211
    where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1212
  with assms have "finite B" by auto
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1213
  then have "fold times a (insert a B) = fold times (a * a) B"
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1214
    using `a \<notin> B` by (rule fold_insert2)
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1215
  then show ?thesis
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1216
    using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1217
qed
096306d7391d idempotency case for fold1
haftmann
parents: 32642
diff changeset
  1218
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
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
  1220
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1221
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1222
text{* Now the recursion rules for definitions: *}
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1223
22917
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
  1224
lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
  1225
by simp
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1226
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
  1227
lemma (in ab_semigroup_mult) fold1_insert_def:
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
  1228
  "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1229
by (simp add:fold1_insert)
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
  1230
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
  1231
lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
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
  1232
  "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1233
by simp
15508
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1234
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1235
subsubsection{* Determinacy for @{term fold1Set} *}
c09defa4c956 revised fold1 proofs
paulson
parents: 15507
diff changeset
  1236
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1237
(*Not actually used!!*)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1238
(*
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
  1239
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
  1240
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
  1241
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1242
lemma fold_graph_permute:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1243
  "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1244
   ==> fold_graph times id a (insert b A) 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
  1245
apply (cases "a=b") 
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1246
apply (auto dest: fold_graph_permute_diff) 
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1247
done
15376
302ef111b621 Started to clean up and generalize FiniteSet
nipkow
parents: 15327
diff changeset
  1248
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
  1249
lemma fold1Set_determ:
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
  1250
  "fold1Set times A x ==> fold1Set times A y ==> y = x"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1251
proof (clarify elim!: fold1Set.cases)
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1252
  fix A x B y a b
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1253
  assume Ax: "fold_graph times id a A x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1254
  assume By: "fold_graph times id b B y"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1255
  assume anotA:  "a \<notin> A"
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1256
  assume bnotB:  "b \<notin> B"
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1257
  assume eq: "insert a A = insert b B"
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1258
  show "y=x"
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1259
  proof cases
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1260
    assume same: "a=b"
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1261
    hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1262
    thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1263
  next
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1264
    assume diff: "a\<noteq>b"
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1265
    let ?D = "B - {a}"
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1266
    have B: "B = insert a ?D" and A: "A = insert b ?D"
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1267
     and aB: "a \<in> B" and bA: "b \<in> A"
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1268
      using eq anotA bnotB diff by (blast elim!:equalityE)+
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1269
    with aB bnotB By
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1270
    have "fold_graph times id a (insert b ?D) y" 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1271
      by (auto intro: fold_graph_permute simp add: insert_absorb)
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1272
    moreover
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1273
    have "fold_graph times id a (insert b ?D) x"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1274
      by (simp add: A [symmetric] Ax) 
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1275
    ultimately show ?thesis by (blast intro: fold_graph_determ) 
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
  1276
  qed
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1277
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
  1278
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
  1279
lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1280
  by (unfold fold1_def) (blast intro: fold1Set_determ)
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
  1281
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
  1282
end
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1283
*)
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
  1284
15506
864238c95b56 new treatment of fold1
paulson
parents: