src/HOL/Finite_Set.thy
author nipkow
Sun, 15 Feb 2009 07:54:16 +0100
changeset 29918 214755b03df3
parent 29916 f24137b42d9b
child 29920 b95f5b8b93dd
permissions -rw-r--r--
more finiteness
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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 Nat Product_Type Power
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begin
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subsection {* Definition and basic properties *}
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inductive finite :: "'a set => bool"
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  where
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    emptyI [simp, intro!]: "finite {}"
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  | insertI [simp, intro!]: "finite A ==> finite (insert a A)"
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lemma ex_new_if_finite: -- "does not depend on def of finite at all"
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  assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
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  shows "\<exists>a::'a. a \<notin> A"
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proof -
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  from assms have "A \<noteq> UNIV" by blast
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  thus ?thesis by blast
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qed
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lemma finite_induct [case_names empty insert, induct set: finite]:
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  "finite F ==>
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    P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
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  -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
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proof -
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  assume "P {}" and
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    insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
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  assume "finite F"
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  thus "P F"
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  proof induct
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    show "P {}" by fact
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    fix x F assume F: "finite F" and P: "P F"
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    show "P (insert x F)"
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    proof cases
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      assume "x \<in> F"
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      hence "insert x F = F" by (rule insert_absorb)
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      with P show ?thesis by (simp only:)
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    next
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      assume "x \<notin> F"
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      from F this P show ?thesis by (rule insert)
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    qed
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  qed
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qed
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lemma finite_ne_induct[case_names singleton insert, consumes 2]:
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assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>
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 \<lbrakk> \<And>x. P{x};
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   \<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk>
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 \<Longrightarrow> P F"
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using fin
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proof induct
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  case empty thus ?case by simp
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next
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  case (insert x F)
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  show ?case
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  proof cases
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    assume "F = {}"
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    thus ?thesis using `P {x}` by simp
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  next
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    assume "F \<noteq> {}"
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    thus ?thesis using insert by blast
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  qed
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qed
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lemma finite_subset_induct [consumes 2, case_names empty insert]:
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  assumes "finite F" and "F \<subseteq> A"
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    and empty: "P {}"
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    and insert: "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
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  shows "P F"
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proof -
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  from `finite F` and `F \<subseteq> A`
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  show ?thesis
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  proof induct
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    show "P {}" by fact
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  next
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    fix x F
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    assume "finite F" and "x \<notin> F" and
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      P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
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    show "P (insert x F)"
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    proof (rule insert)
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      from i show "x \<in> A" by blast
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      from i have "F \<subseteq> A" by blast
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      with P show "P F" .
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      show "finite F" by fact
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      show "x \<notin> F" by fact
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    qed
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  qed
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qed
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text{* Finite sets are the images of initial segments of natural numbers: *}
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lemma finite_imp_nat_seg_image_inj_on:
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  assumes fin: "finite A" 
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  shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}"
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using fin
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proof induct
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  case empty
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  show ?case  
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  proof show "\<exists>f. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp 
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  qed
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next
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  case (insert a A)
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  have notinA: "a \<notin> A" by fact
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  from insert.hyps obtain n f
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    where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
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  hence "insert a A = f(n:=a) ` {i. i < Suc n}"
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        "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
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    by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
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  thus ?case by blast
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qed
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lemma nat_seg_image_imp_finite:
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  "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
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proof (induct n)
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  case 0 thus ?case by simp
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next
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  case (Suc n)
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  let ?B = "f ` {i. i < n}"
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  have finB: "finite ?B" by(rule Suc.hyps[OF refl])
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  show ?case
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  proof cases
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    assume "\<exists>k<n. f n = f k"
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    hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  next
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    assume "\<not>(\<exists> k<n. f n = f k)"
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    hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
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    thus ?thesis using finB by simp
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  qed
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qed
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lemma finite_conv_nat_seg_image:
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  "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
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by(blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
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subsubsection{* Finiteness and set theoretic constructions *}
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lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
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by (induct set: finite) simp_all
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lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
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  -- {* Every subset of a finite set is finite. *}
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proof -
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  assume "finite B"
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  thus "!!A. A \<subseteq> B ==> finite A"
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  proof induct
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    case empty
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    thus ?case by simp
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  next
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    case (insert x F A)
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    have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" by fact+
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   160
    show "finite A"
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wenzelm
parents:
diff changeset
   161
    proof cases
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   162
      assume x: "x \<in> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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diff changeset
   163
      with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   164
      with r have "finite (A - {x})" .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   165
      hence "finite (insert x (A - {x}))" ..
23389
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   166
      also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
12396
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   167
      finally show ?thesis .
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
diff changeset
   168
    next
23389
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   169
      show "A \<subseteq> F ==> ?thesis" by fact
12396
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   170
      assume "x \<notin> A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   171
      with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
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   172
    qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   173
  qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   174
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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parents:
diff changeset
   175
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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   176
lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
29901
f4b3f8fbf599 finiteness lemmas
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   177
by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
f4b3f8fbf599 finiteness lemmas
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diff changeset
   178
29916
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diff changeset
   179
lemma finite_Collect_disjI[simp]:
29901
f4b3f8fbf599 finiteness lemmas
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   180
  "finite{x. P x | Q x} = (finite{x. P x} & finite{x. Q x})"
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   181
by(simp add:Collect_disj_eq)
12396
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   182
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
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   183
lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
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   184
  -- {* The converse obviously fails. *}
29901
f4b3f8fbf599 finiteness lemmas
nipkow
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diff changeset
   185
by (blast intro: finite_subset)
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   186
29916
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diff changeset
   187
lemma finite_Collect_conjI [simp, intro]:
29901
f4b3f8fbf599 finiteness lemmas
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diff changeset
   188
  "finite{x. P x} | finite{x. Q x} ==> finite{x. P x & Q x}"
f4b3f8fbf599 finiteness lemmas
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diff changeset
   189
  -- {* The converse obviously fails. *}
f4b3f8fbf599 finiteness lemmas
nipkow
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diff changeset
   190
by(simp add:Collect_conj_eq)
12396
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wenzelm
parents:
diff changeset
   191
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   192
lemma finite_insert [simp]: "finite (insert a A) = finite A"
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wenzelm
parents:
diff changeset
   193
  apply (subst insert_is_Un)
14208
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paulson
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diff changeset
   194
  apply (simp only: finite_Un, blast)
12396
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wenzelm
parents:
diff changeset
   195
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   196
15281
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diff changeset
   197
lemma finite_Union[simp, intro]:
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diff changeset
   198
 "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
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nipkow
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diff changeset
   199
by (induct rule:finite_induct) simp_all
bd4611956c7b More lemmas
nipkow
parents: 15234
diff changeset
   200
12396
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   201
lemma finite_empty_induct:
23389
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   202
  assumes "finite A"
aaca6a8e5414 tuned proofs: avoid implicit prems;
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diff changeset
   203
    and "P A"
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diff changeset
   204
    and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
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diff changeset
   205
  shows "P {}"
12396
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wenzelm
parents:
diff changeset
   206
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   207
  have "P (A - A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   208
  proof -
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   209
    {
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   210
      fix c b :: "'a set"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   211
      assume c: "finite c" and b: "finite b"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   212
	and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   213
      have "c \<subseteq> b ==> P (b - c)"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   214
	using c
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   215
      proof induct
aaca6a8e5414 tuned proofs: avoid implicit prems;
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parents: 23277
diff changeset
   216
	case empty
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   217
	from P1 show ?case by simp
aaca6a8e5414 tuned proofs: avoid implicit prems;
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parents: 23277
diff changeset
   218
      next
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   219
	case (insert x F)
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   220
	have "P (b - F - {x})"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   221
	proof (rule P2)
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   222
          from _ b show "finite (b - F)" by (rule finite_subset) blast
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   223
          from insert show "x \<in> b - F" by simp
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   224
          from insert show "P (b - F)" by simp
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   225
	qed
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   226
	also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
aaca6a8e5414 tuned proofs: avoid implicit prems;
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parents: 23277
diff changeset
   227
	finally show ?case .
12396
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parents:
diff changeset
   228
      qed
23389
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parents: 23277
diff changeset
   229
    }
aaca6a8e5414 tuned proofs: avoid implicit prems;
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diff changeset
   230
    then show ?thesis by this (simp_all add: assms)
12396
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parents:
diff changeset
   231
  qed
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
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parents: 23277
diff changeset
   232
  then show ?thesis by simp
12396
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wenzelm
parents:
diff changeset
   233
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   234
29901
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parents: 29879
diff changeset
   235
lemma finite_Diff [simp]: "finite A ==> finite (A - B)"
f4b3f8fbf599 finiteness lemmas
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parents: 29879
diff changeset
   236
by (rule Diff_subset [THEN finite_subset])
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   237
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   238
lemma finite_Diff2 [simp]:
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   239
  assumes "finite B" shows "finite (A - B) = finite A"
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   240
proof -
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   241
  have "finite A \<longleftrightarrow> finite((A-B) Un (A Int B))" by(simp add: Un_Diff_Int)
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   242
  also have "\<dots> \<longleftrightarrow> finite(A-B)" using `finite B` by(simp)
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   243
  finally show ?thesis ..
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   244
qed
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   245
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   246
lemma finite_compl[simp]:
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   247
  "finite(A::'a set) \<Longrightarrow> finite(-A) = finite(UNIV::'a set)"
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29879
diff changeset
   248
by(simp add:Compl_eq_Diff_UNIV)
12396
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parents:
diff changeset
   249
29916
f24137b42d9b more finiteness
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parents: 29903
diff changeset
   250
lemma finite_Collect_not[simp]:
29903
2c0046b26f80 more finiteness changes
nipkow
parents: 29901
diff changeset
   251
  "finite{x::'a. P x} \<Longrightarrow> finite{x. ~P x} = finite(UNIV::'a set)"
2c0046b26f80 more finiteness changes
nipkow
parents: 29901
diff changeset
   252
by(simp add:Collect_neg_eq)
2c0046b26f80 more finiteness changes
nipkow
parents: 29901
diff changeset
   253
12396
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wenzelm
parents:
diff changeset
   254
lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   255
  apply (subst Diff_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   256
  apply (case_tac "a : A - B")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   257
   apply (rule finite_insert [symmetric, THEN trans])
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   258
   apply (subst insert_Diff, simp_all)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   259
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   260
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   261
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   262
text {* Image and Inverse Image over Finite Sets *}
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   263
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   264
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   265
  -- {* The image of a finite set is finite. *}
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   266
  by (induct set: finite) simp_all
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   267
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   268
lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   269
  apply (frule finite_imageI)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   270
  apply (erule finite_subset, assumption)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   271
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   272
13825
ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   273
lemma finite_range_imageI:
ef4c41e7956a new inverse image lemmas
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parents: 13737
diff changeset
   274
    "finite (range g) ==> finite (range (%x. f (g x)))"
27418
564117b58d73 remove simp attribute from range_composition
huffman
parents: 27165
diff changeset
   275
  apply (drule finite_imageI, simp add: range_composition)
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   276
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   277
12396
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wenzelm
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diff changeset
   278
lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   279
proof -
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   280
  have aux: "!!A. finite (A - {}) = finite A" by simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   281
  fix B :: "'a set"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   282
  assume "finite B"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   283
  thus "!!A. f`A = B ==> inj_on f A ==> finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   284
    apply induct
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   285
     apply simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   286
    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
   287
     apply clarify
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   288
     apply (simp (no_asm_use) add: inj_on_def)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   289
     apply (blast dest!: aux [THEN iffD1], atomize)
12396
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wenzelm
parents:
diff changeset
   290
    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   291
    apply (frule subsetD [OF equalityD2 insertI1], clarify)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   292
    apply (rule_tac x = xa in bexI)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   293
     apply (simp_all add: inj_on_image_set_diff)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   294
    done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   295
qed (rule refl)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   296
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   297
13825
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paulson
parents: 13737
diff changeset
   298
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   299
  -- {* The inverse image of a singleton under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   300
         is included in a singleton. *}
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   301
  apply (auto simp add: inj_on_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   302
  apply (blast intro: the_equality [symmetric])
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   303
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   304
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   305
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   306
  -- {* The inverse image of a finite set under an injective function
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   307
         is finite. *}
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   308
  apply (induct set: finite)
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
   309
   apply simp_all
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   310
  apply (subst vimage_insert)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   311
  apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
13825
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   312
  done
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   313
ef4c41e7956a new inverse image lemmas
paulson
parents: 13737
diff changeset
   314
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   315
text {* The finite UNION of finite sets *}
12396
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wenzelm
parents:
diff changeset
   316
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   317
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
   318
  by (induct set: finite) simp_all
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   319
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   320
text {*
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   321
  Strengthen RHS to
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   322
  @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   323
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   324
  We'd need to prove
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14331
diff changeset
   325
  @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   326
  by induction. *}
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   327
29918
214755b03df3 more finiteness
nipkow
parents: 29916
diff changeset
   328
lemma finite_UN [simp]:
214755b03df3 more finiteness
nipkow
parents: 29916
diff changeset
   329
  "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
214755b03df3 more finiteness
nipkow
parents: 29916
diff changeset
   330
by (blast intro: finite_UN_I finite_subset)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   331
17022
b257300c3a9c added Brian Hufmann's finite instances
nipkow
parents: 16775
diff changeset
   332
lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"
b257300c3a9c added Brian Hufmann's finite instances
nipkow
parents: 16775
diff changeset
   333
by (simp add: Plus_def)
b257300c3a9c added Brian Hufmann's finite instances
nipkow
parents: 16775
diff changeset
   334
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   335
text {* Sigma of finite sets *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   336
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   337
lemma finite_SigmaI [simp]:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   338
    "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   339
  by (unfold Sigma_def) (blast intro!: finite_UN_I)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   340
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   341
lemma finite_cartesian_product: "[| finite A; finite B |] ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   342
    finite (A <*> B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   343
  by (rule finite_SigmaI)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   344
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   345
lemma finite_Prod_UNIV:
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   346
    "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   347
  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   348
   apply (erule ssubst)
14208
144f45277d5a misc tidying
paulson
parents: 13825
diff changeset
   349
   apply (erule finite_SigmaI, auto)
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   350
  done
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   351
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   352
lemma finite_cartesian_productD1:
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   353
     "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   354
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
   355
apply (drule_tac x=n in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   356
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
   357
apply (auto simp add: o_def) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   358
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   359
apply (drule equalityD1) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   360
apply (rename_tac y x)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   361
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
   362
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   363
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   364
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
   365
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   366
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   367
lemma finite_cartesian_productD2:
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   368
     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   369
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
   370
apply (drule_tac x=n in spec) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   371
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
   372
apply (auto simp add: o_def) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   373
 prefer 2 apply (force dest!: equalityD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   374
apply (drule equalityD1)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   375
apply (rename_tac x y)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   376
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
   377
 prefer 2 apply force
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   378
apply clarify
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   379
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
   380
done
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   381
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   382
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   383
text {* The powerset of a finite set *}
12396
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   384
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   385
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   386
proof
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   387
  assume "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   388
  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   389
  thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   390
next
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   391
  assume "finite A"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   392
  thus "finite (Pow A)"
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   393
    by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   394
qed
2298d5b8e530 renamed theory Finite to Finite_Set and converted;
wenzelm
parents:
diff changeset
   395
29916
f24137b42d9b more finiteness
nipkow
parents: 29903
diff changeset
   396
lemma finite_Collect_subsets[simp,intro]: "finite A \<Longrightarrow> finite{B. B \<subseteq> A}"
f24137b42d9b more finiteness
nipkow
parents: 29903
diff changeset
   397
by(simp add: Pow_def[symmetric])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   398
29918
214755b03df3 more finiteness
nipkow
parents: 29916
diff changeset
   399
lemma finite_bex_subset[simp]:
214755b03df3 more finiteness
nipkow
parents: 29916
diff changeset
   400
  "finite B \<Longrightarrow> finite{x. EX A<=B. P x A} = (ALL A<=B. finite{x. P x A})"
214755b03df3 more finiteness
nipkow
parents: 29916
diff changeset
   401
apply(subgoal_tac "{x. EX A<=B. P x A} = (UN A:Pow B. {x. P x A})")
214755b03df3 more finiteness
nipkow
parents: 29916
diff changeset
   402
 apply auto
214755b03df3 more finiteness
nipkow
parents: 29916
diff changeset
   403
done
214755b03df3 more finiteness
nipkow
parents: 29916
diff changeset
   404
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   405
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   406
by(blast intro: finite_subset[OF subset_Pow_Union])
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   407
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   408
26441
7914697ff104 no "attach UNIV" any more
haftmann
parents: 26146
diff changeset
   409
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
   410
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   411
setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*}
29797
08ef36ed2f8a handling type classes without parameters
haftmann
parents: 29675
diff changeset
   412
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
   413
  assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   414
setup {* Sign.parent_path *}
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   415
hide const finite
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   416
27430
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   417
context finite
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   418
begin
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   419
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   420
lemma finite [simp]: "finite (A \<Colon> 'a set)"
26441
7914697ff104 no "attach UNIV" any more
haftmann
parents: 26146
diff changeset
   421
  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
   422
27430
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   423
end
1e25ac05cd87 prove lemma finite in context of finite class
huffman
parents: 27418
diff changeset
   424
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   425
lemma UNIV_unit [noatp]:
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   426
  "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
   427
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   428
instance unit :: finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   429
  by default (simp add: UNIV_unit)
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   430
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   431
lemma UNIV_bool [noatp]:
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   432
  "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
   433
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   434
instance bool :: finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   435
  by default (simp add: UNIV_bool)
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   436
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   437
instance * :: (finite, finite) finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   438
  by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   439
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
   440
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   441
  by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   442
26146
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   443
instance "fun" :: (finite, finite) finite
61cb176d0385 tuned proofs
haftmann
parents: 26041
diff changeset
   444
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
   445
  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
   446
  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
   447
    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
26792
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   448
    have "range ?graph \<subseteq> Pow UNIV" by simp
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   449
    moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   450
      by (simp only: finite_Pow_iff finite)
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   451
    ultimately show "finite (range ?graph)"
f2d75fd23124 - Deleted code setup for finite and card
berghofe
parents: 26757
diff changeset
   452
      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
   453
    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
   454
  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
   455
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
   456
27981
feb0c01cf0fb tuned import order
haftmann
parents: 27611
diff changeset
   457
instance "+" :: (finite, finite) finite
feb0c01cf0fb tuned import order
haftmann
parents: 27611
diff changeset
   458
  by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
feb0c01cf0fb tuned import order
haftmann
parents: 27611
diff changeset
   459
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
   460
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   461
subsection {* A fold functional for finite sets *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   462
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   463
text {* The intended behaviour is
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   464
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   465
if @{text f} is ``left-commutative'':
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   466
*}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   467
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   468
locale fun_left_comm =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   469
  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
   470
  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
   471
begin
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   472
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   473
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
   474
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   475
lemma fun_comp_comm:  "f x \<circ> f y = f y \<circ> f x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   476
by (simp add: fun_left_comm expand_fun_eq)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   477
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   478
end
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   479
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   480
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
   481
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
   482
  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
   483
  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
   484
      \<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
   485
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   486
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
   487
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   488
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   489
[code del]: "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
   490
15498
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   491
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
   492
@{term "if finite A then THE y. fold_graph f z A y else e"}.
15498
3988e90613d4 comment
paulson
parents: 15497
diff changeset
   493
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
   494
@{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
   495
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
   496
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   497
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   498
lemma Diff1_fold_graph:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   499
  "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
   500
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
   501
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   502
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
   503
by (induct set: fold_graph) auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   504
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   505
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   506
by (induct set: finite) auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   507
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   508
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   509
subsubsection{*From @{const fold_graph} to @{term fold}*}
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   510
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   511
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
19868
wenzelm
parents: 19793
diff changeset
   512
  by (auto simp add: less_Suc_eq) 
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   513
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   514
lemma insert_image_inj_on_eq:
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   515
     "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   516
        inj_on h {i. i < Suc m}|] 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   517
      ==> A = h ` {i. i < m}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   518
apply (auto simp add: image_less_Suc inj_on_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   519
apply (blast intro: less_trans) 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   520
done
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   521
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   522
lemma insert_inj_onE:
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   523
  assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" 
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   524
      and inj_on: "inj_on h {i::nat. i<n}"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   525
  shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   526
proof (cases n)
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   527
  case 0 thus ?thesis using aA by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   528
next
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   529
  case (Suc m)
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23277
diff changeset
   530
  have nSuc: "n = Suc m" by fact
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   531
  have mlessn: "m<n" by (simp add: nSuc)
15532
9712d41db5b8 simplified a proof
paulson
parents: 15526
diff changeset
   532
  from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
27165
e1c49eb8cee6 Hid swap
nipkow
parents: 26792
diff changeset
   533
  let ?hm = "Fun.swap k m h"
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   534
  have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   535
    by (simp add: inj_on_swap_iff inj_on)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   536
  show ?thesis
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   537
  proof (intro exI conjI)
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   538
    show "inj_on ?hm {i. i < m}" using inj_hm
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   539
      by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   540
    show "m<n" by (rule mlessn)
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   541
    show "A = ?hm ` {i. i < m}" 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   542
    proof (rule insert_image_inj_on_eq)
27165
e1c49eb8cee6 Hid swap
nipkow
parents: 26792
diff changeset
   543
      show "inj_on (Fun.swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
15520
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   544
      show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   545
      show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   546
	using aA hkeq nSuc klessn
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   547
	by (auto simp add: swap_def image_less_Suc fun_upd_image 
0ed33cd8f238 simplified a key lemma for foldSet
paulson
parents: 15517
diff changeset
   548
			   less_Suc_eq inj_on_image_set_diff [OF inj_on])
15479
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   549
    qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   550
  qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   551
qed
fbc473ea9d3c proof simpification
nipkow
parents: 15447
diff changeset
   552
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   553
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
   554
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
   555
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   556
lemma fold_graph_determ_aux:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   557
  "A = h`{i::nat. i<n} \<Longrightarrow> inj_on h {i. i<n}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   558
   \<Longrightarrow> fold_graph f z A x \<Longrightarrow> fold_graph f z A x'
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   559
   \<Longrightarrow> x' = x"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   560
proof (induct n arbitrary: A x x' h rule: less_induct)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15509
diff changeset
   561
  case (less n)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   562
  have IH: "\<And>m h A x x'. m < n \<Longrightarrow> A = h ` {i. i<m}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   563
      \<Longrightarrow> inj_on h {i. i<m} \<Longrightarrow> fold_graph f z A x
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   564
      \<Longrightarrow> fold_graph f z A x' \<Longrightarrow> x' = x" by fact
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   565
  have Afoldx: "fold_graph f z A x" and Afoldx': "fold_graph f z A x'"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   566
    and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" by fact+
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   567
  show ?case
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   568
  proof (rule fold_graph.cases [OF Afoldx])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   569
    assume "A = {}" and "x = z"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   570
    with Afoldx' show "x' = x" by auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   571
  next
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   572
    fix B b u
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   573
    assume AbB: "A = insert b B" and x: "x = f b u"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   574
      and notinB: "b \<notin> B" and Bu: "fold_graph f z B u"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   575
    show "x'=x" 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   576
    proof (rule fold_graph.cases [OF Afoldx'])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   577
      assume "A = {}" and "x' = z"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   578
      with AbB show "x' = x" by blast
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   579
    next
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   580
      fix C c v
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   581
      assume AcC: "A = insert c C" and x': "x' = f c v"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   582
        and notinC: "c \<notin> C" and Cv: "fold_graph f z C v"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   583
      from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   584
      from insert_inj_onE [OF Beq notinB injh]
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   585
      obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   586
        and Beq: "B = hB ` {i. i < mB}" and lessB: "mB < n" by auto 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   587
      from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   588
      from insert_inj_onE [OF Ceq notinC injh]
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   589
      obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   590
        and Ceq: "C = hC ` {i. i < mC}" and lessC: "mC < n" by auto 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   591
      show "x'=x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   592
      proof cases
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   593
        assume "b=c"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   594
	then moreover have "B = C" using AbB AcC notinB notinC by auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   595
	ultimately show ?thesis  using Bu Cv x x' IH [OF lessC Ceq inj_onC]
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   596
          by auto
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   597
      next
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   598
	assume diff: "b \<noteq> c"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   599
	let ?D = "B - {c}"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   600
	have B: "B = insert c ?D" and C: "C = insert b ?D"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   601
	  using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   602
	have "finite A" by(rule fold_graph_imp_finite [OF Afoldx])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   603
	with AbB have "finite ?D" by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   604
	then obtain d where Dfoldd: "fold_graph f z ?D d"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   605
	  using finite_imp_fold_graph by iprover
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   606
	moreover have cinB: "c \<in> B" using B by auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   607
	ultimately have "fold_graph f z B (f c d)" by(rule Diff1_fold_graph)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   608
	hence "f c d = u" by (rule IH [OF lessB Beq inj_onB Bu]) 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   609
        moreover have "f b d = v"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   610
	proof (rule IH[OF lessC Ceq inj_onC Cv])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   611
	  show "fold_graph f z C (f b d)" using C notinB Dfoldd by fastsimp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   612
	qed
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   613
	ultimately show ?thesis
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   614
          using fun_left_comm [of c b] x x' by (auto simp add: o_def)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   615
      qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   616
    qed
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   617
  qed
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   618
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   619
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   620
lemma fold_graph_determ:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   621
  "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   622
apply (frule fold_graph_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   623
apply (blast intro: fold_graph_determ_aux [rule_format])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   624
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   625
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   626
lemma fold_equality:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   627
  "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
   628
by (unfold fold_def) (blast intro: fold_graph_determ)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   629
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   630
text{* The base case for @{text fold}: *}
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   631
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   632
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
   633
by (unfold fold_def) blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   634
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   635
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
   636
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   637
lemma fold_insert_aux: "x \<notin> A
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   638
  \<Longrightarrow> fold_graph f z (insert x A) v \<longleftrightarrow>
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   639
      (\<exists>y. fold_graph f z A y \<and> v = f x y)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   640
apply auto
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   641
apply (rule_tac A1 = A and f1 = f in finite_imp_fold_graph [THEN exE])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   642
 apply (fastsimp dest: fold_graph_imp_finite)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   643
apply (blast intro: fold_graph_determ)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   644
done
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   645
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   646
lemma fold_insert [simp]:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   647
  "finite A ==> x \<notin> A ==> fold f z (insert x A) = f x (fold f z A)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   648
apply (simp add: fold_def fold_insert_aux)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   649
apply (rule the_equality)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   650
 apply (auto intro: finite_imp_fold_graph
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   651
        cong add: conj_cong simp add: fold_def[symmetric] fold_equality)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   652
done
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   653
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   654
lemma fold_fun_comm:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   655
  "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
   656
proof (induct rule: finite_induct)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   657
  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
   658
next
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   659
  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
   660
    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
   661
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   662
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   663
lemma fold_insert2:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   664
  "finite A \<Longrightarrow> x \<notin> 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
   665
by (simp add: fold_insert fold_fun_comm)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   666
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
   667
lemma fold_rec:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   668
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
   669
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
   670
proof -
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   671
  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
   672
  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
   673
  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
   674
    by (rule fold_insert) (simp add: `finite A`)+
15535
nipkow
parents: 15532
diff changeset
   675
  finally show ?thesis .
nipkow
parents: 15532
diff changeset
   676
qed
nipkow
parents: 15532
diff changeset
   677
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   678
lemma fold_insert_remove:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   679
  assumes "finite A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   680
  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
   681
proof -
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   682
  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
   683
  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
   684
  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
   685
    by (rule fold_rec)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   686
  then show ?thesis by simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   687
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   688
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
end
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   690
15480
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   691
text{* A simplified version for idempotent functions: *}
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   692
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   693
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
   694
  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
   695
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
   696
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   697
text{* The nice version: *}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   698
lemma fun_comp_idem : "f x o f x = f x"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   699
by (simp add: fun_left_idem expand_fun_eq)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   700
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
   701
lemma fold_insert_idem:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   702
  assumes fin: "finite A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   703
  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
   704
proof cases
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   705
  assume "x \<in> A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   706
  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
   707
  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
   708
next
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   709
  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
   710
qed
cb3612cc41a3 renamed a few vars, added a lemma
nipkow
parents: 15479
diff changeset
   711
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   712
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
   713
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   714
lemma fold_insert_idem2:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   715
  "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
   716
by(simp add:fold_fun_comm)
15484
2636ec211ec8 fold and fol1 changes
nipkow
parents: 15483
diff changeset
   717
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   718
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
   719
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   720
subsubsection{* The derived combinator @{text fold_image} *}
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   721
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   722
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
   723
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
   724
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   725
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
   726
by(simp add:fold_image_def)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   727
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   728
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
   729
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
   730
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   731
lemma fold_image_insert[simp]:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   732
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
   733
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
   734
proof -
29223
e09c53289830 Conversion of HOL-Main and ZF to new locales.
ballarin
parents: 29025
diff changeset
   735
  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
   736
    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
   737
  show ?thesis using assms by(simp add:fold_image_def I.fold_insert)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   738
qed
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   739
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   740
(*
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   741
lemma fold_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
   742
  "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
   743
  apply (induct set: finite)
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
   744
   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
   745
  apply (simp add: mult_left_commute [of x])
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   746
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   747
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
   748
lemma fold_nest_Un_Int:
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   749
  "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
   750
    ==> 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
   751
  apply (induct set: finite)
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
   752
   apply simp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   753
  apply (simp add: fold_commute Int_insert_left insert_absorb)
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   754
  done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   755
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
   756
lemma fold_nest_Un_disjoint:
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   757
  "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
   758
    ==> 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
   759
  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
   760
*)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   761
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   762
lemma fold_image_reindex:
15487
55497029b255 generalization and tidying
paulson
parents: 15484
diff changeset
   763
assumes fin: "finite A"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   764
shows "inj_on h A \<Longrightarrow> fold_image times g z (h`A) = fold_image times (g\<circ>h) z A"
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   765
using fin apply induct
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   766
 apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   767
apply simp
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   768
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   769
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   770
(*
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
   771
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
   772
  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
   773
  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
   774
  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
   775
*}
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   776
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   777
lemma fold_fusion:
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 27430
diff changeset
   778
  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
   779
  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
   780
    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
   781
  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
   782
proof -
29223
e09c53289830 Conversion of HOL-Main and ZF to new locales.
ballarin
parents: 29025
diff changeset
   783
  class_interpret ab_semigroup_mult [g] by fact
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 27430
diff changeset
   784
  show ?thesis using fin hyp by (induct set: finite) simp_all
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 27430
diff changeset
   785
qed
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   786
*)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   787
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   788
lemma fold_image_cong:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   789
  "finite A \<Longrightarrow>
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   790
  (!!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
   791
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
   792
 apply simp
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   793
apply (erule finite_induct, simp)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   794
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
   795
apply (subgoal_tac "finite C")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   796
 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
   797
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
   798
 prefer 2 apply blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   799
apply (erule ssubst)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   800
apply (drule spec)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   801
apply (erule (1) notE impE)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   802
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
   803
done
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   804
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
   805
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
   806
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   807
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
   808
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
   809
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   810
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
   811
  "finite A ==> finite B ==>
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   812
    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
   813
    fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   814
by (induct set: finite) 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   815
   (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
   816
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   817
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
   818
  "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
   819
   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
   820
   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
   821
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
   822
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   823
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
   824
  "\<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
   825
     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
   826
   \<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
   827
       fold_image times (%i. fold_image times g 1 (A i)) 1 I"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   828
apply (induct set: finite, simp, atomize)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   829
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
   830
 prefer 2 apply blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   831
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
   832
 prefer 2 apply blast
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   833
apply (simp add: fold_Un_disjoint)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   834
done
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   835
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   836
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
   837
  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
   838
  fold_image times (split g) 1 (SIGMA x:A. B x)"
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   839
apply (subst Sigma_def)
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   840
apply (subst fold_image_UN_disjoint, assumption, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   841
 apply blast
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   842
apply (erule fold_image_cong)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   843
apply (subst fold_image_UN_disjoint, simp, simp)
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   844
 apply blast
15506
864238c95b56 new treatment of fold1
paulson
parents: 15505
diff changeset
   845
apply simp
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   846
done
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15376
diff changeset
   847
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   848
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
   849
   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
   850
   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
   851
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
   852
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   853
end
22917
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
   854
3c56b12fd946 localized Min/Max
haftmann
parents: 22616
diff changeset
   855
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   856
subsection {* Generalized summation over a set *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   857
29509
1ff0f3f08a7b migrated class package to new locale implementation
haftmann
parents: 29223
diff changeset
   858
interpretation comm_monoid_add!: comm_monoid_mult "0::'a::comm_monoid_add" "op +"
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 27981
diff changeset
   859
  proof qed (auto intro: add_assoc add_commute)
26041
c2e15e65165f locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents: 25571
diff changeset
   860
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   861
definition setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   862
where "setsum f A == if finite A then fold_image (op +) f 0 A else 0"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   863
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   864
abbreviation
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21249
diff changeset
   865
  Setsum  ("\<Sum>_" [1000] 999) where
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   866
  "\<Sum>A == setsum (%x. x) A"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   867
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   868
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   869
written @{text"\<Sum>x\<in>A. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   870
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   871
syntax
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   872
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   873
syntax (xsymbols)
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   874
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   875
syntax (HTML output)
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   876
  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   877
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   878
translations -- {* Beware of argument permutation! *}
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   879
  "SUM i:A. b" == "CONST setsum (%i. b) A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   880
  "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   881
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   882
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   883
 @{text"\<Sum>x|P. e"}. *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   884
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   885
syntax
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   886
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   887
syntax (xsymbols)
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   888
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   889
syntax (HTML output)
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
   890
  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   891
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   892
translations
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   893
  "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   894
  "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   895
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   896
print_translation {*
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   897
let
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   898
  fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   899
    if x<>y then raise Match
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   900
    else let val x' = Syntax.mark_bound x
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   901
             val t' = subst_bound(x',t)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   902
             val P' = subst_bound(x',P)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   903
         in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   904
in [("setsum", setsum_tr')] end
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   905
*}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   906
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
   907
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   908
lemma setsum_empty [simp]: "setsum f {} = 0"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   909
by (simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   910
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   911
lemma setsum_insert [simp]:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   912
  "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   913
by (simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   914
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   915
lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   916
by (simp add: setsum_def)
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
   917
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   918
lemma setsum_reindex:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   919
     "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   920
by(auto simp add: setsum_def comm_monoid_add.fold_image_reindex dest!:finite_imageD)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   921
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   922
lemma setsum_reindex_id:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   923
     "inj_on f B ==> setsum f B = setsum id (f ` B)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   924
by (auto simp add: setsum_reindex)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   925
29674
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   926
lemma setsum_reindex_nonzero: 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   927
  assumes fS: "finite S"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   928
  and nz: "\<And> x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   929
  shows "setsum h (f ` S) = setsum (h o f) S"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   930
using nz
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   931
proof(induct rule: finite_induct[OF fS])
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   932
  case 1 thus ?case by simp
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   933
next
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   934
  case (2 x F) 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   935
  {assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   936
    then obtain y where y: "y \<in> F" "f x = f y" by auto 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   937
    from "2.hyps" y have xy: "x \<noteq> y" by auto
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   938
    
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   939
    from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   940
    have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   941
    also have "\<dots> = setsum (h o f) (insert x F)" 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   942
      unfolding setsum_insert[OF `finite F` `x\<notin>F`]
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   943
      using h0 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   944
      apply simp
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   945
      apply (rule "2.hyps"(3))
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   946
      apply (rule_tac y="y" in  "2.prems")
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   947
      apply simp_all
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   948
      done
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   949
    finally have ?case .}
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   950
  moreover
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   951
  {assume fxF: "f x \<notin> f ` F"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   952
    have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   953
      using fxF "2.hyps" by simp 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   954
    also have "\<dots> = setsum (h o f) (insert x F)"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   955
      unfolding setsum_insert[OF `finite F` `x\<notin>F`]
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   956
      apply simp
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   957
      apply (rule cong[OF refl[of "op + (h (f x))"]])
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   958
      apply (rule "2.hyps"(3))
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   959
      apply (rule_tac y="y" in  "2.prems")
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   960
      apply simp_all
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   961
      done
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   962
    finally have ?case .}
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   963
  ultimately show ?case by blast
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   964
qed
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   965
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   966
lemma setsum_cong:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   967
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   968
by(fastsimp simp: setsum_def intro: comm_monoid_add.fold_image_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   969
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16632
diff changeset
   970
lemma strong_setsum_cong[cong]:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16632
diff changeset
   971
  "A = B ==> (!!x. x:B =simp=> f x = g x)
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16632
diff changeset
   972
   ==> setsum (%x. f x) A = setsum (%x. g x) B"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   973
by(fastsimp simp: simp_implies_def setsum_def intro: comm_monoid_add.fold_image_cong)
16632
ad2895beef79 Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents: 16550
diff changeset
   974
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
   975
lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A";
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   976
by (rule setsum_cong[OF refl], auto);
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
   977
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   978
lemma setsum_reindex_cong:
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   979
   "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] 
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   980
    ==> setsum h B = setsum g A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   981
by (simp add: setsum_reindex cong: setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   982
29674
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
   983
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   984
lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   985
apply (clarsimp simp: setsum_def)
15765
6472d4942992 Cleaned up, now uses interpretation.
ballarin
parents: 15554
diff changeset
   986
apply (erule finite_induct, auto)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   987
done
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   988
15543
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   989
lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   990
by(simp add:setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   991
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   992
lemma setsum_Un_Int: "finite A ==> finite B ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   993
  setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   994
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
   995
by(simp add: setsum_def comm_monoid_add.fold_image_Un_Int [symmetric])
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   996
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   997
lemma setsum_Un_disjoint: "finite A ==> finite B
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   998
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
   999
by (subst setsum_Un_Int [symmetric], auto)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1000
29674
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1001
lemma setsum_mono_zero_left: 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1002
  assumes fT: "finite T" and ST: "S \<subseteq> T"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1003
  and z: "\<forall>i \<in> T - S. f i = 0"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1004
  shows "setsum f S = setsum f T"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1005
proof-
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1006
  have eq: "T = S \<union> (T - S)" using ST by blast
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1007
  have d: "S \<inter> (T - S) = {}" using ST by blast
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1008
  from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1009
  show ?thesis 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1010
  by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z])
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1011
qed
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1012
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1013
lemma setsum_mono_zero_right: 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1014
  assumes fT: "finite T" and ST: "S \<subseteq> T"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1015
  and z: "\<forall>i \<in> T - S. f i = 0"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1016
  shows "setsum f T = setsum f S"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1017
using setsum_mono_zero_left[OF fT ST z] by simp
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1018
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1019
lemma setsum_mono_zero_cong_left: 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1020
  assumes fT: "finite T" and ST: "S \<subseteq> T"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1021
  and z: "\<forall>i \<in> T - S. g i = 0"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1022
  and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1023
  shows "setsum f S = setsum g T"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1024
proof-
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1025
  have eq: "T = S \<union> (T - S)" using ST by blast
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1026
  have d: "S \<inter> (T - S) = {}" using ST by blast
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1027
  from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1028
  show ?thesis 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1029
    using fg by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z])
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1030
qed
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1031
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1032
lemma setsum_mono_zero_cong_right: 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1033
  assumes fT: "finite T" and ST: "S \<subseteq> T"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1034
  and z: "\<forall>i \<in> T - S. f i = 0"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1035
  and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1036
  shows "setsum f T = setsum g S"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1037
using setsum_mono_zero_cong_left[OF fT ST z] fg[symmetric] by auto 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1038
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1039
lemma setsum_delta: 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1040
  assumes fS: "finite S"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1041
  shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1042
proof-
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1043
  let ?f = "(\<lambda>k. if k=a then b k else 0)"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1044
  {assume a: "a \<notin> S"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1045
    hence "\<forall> k\<in> S. ?f k = 0" by simp
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1046
    hence ?thesis  using a by simp}
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1047
  moreover 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1048
  {assume a: "a \<in> S"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1049
    let ?A = "S - {a}"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1050
    let ?B = "{a}"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1051
    have eq: "S = ?A \<union> ?B" using a by blast 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1052
    have dj: "?A \<inter> ?B = {}" by simp
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1053
    from fS have fAB: "finite ?A" "finite ?B" by auto  
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1054
    have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1055
      using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1056
      by simp
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1057
    then have ?thesis  using a by simp}
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1058
  ultimately show ?thesis by blast
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1059
qed
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1060
lemma setsum_delta': 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1061
  assumes fS: "finite S" shows 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1062
  "setsum (\<lambda>k. if a = k then b k else 0) S = 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1063
     (if a\<in> S then b a else 0)"
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1064
  using setsum_delta[OF fS, of a b, symmetric] 
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1065
  by (auto intro: setsum_cong)
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1066
3857d7eba390 Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents: 29609
diff changeset
  1067
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1068
(*But we can't get rid of finite I. If infinite, although the rhs is 0, 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1069
  the lhs need not be, since UNION I A could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1070
lemma setsum_UN_disjoint:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1071
    "finite I ==> (ALL i:I. finite (A i)) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1072
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1073
      setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1074
by(simp add: setsum_def comm_monoid_add.fold_image_UN_disjoint cong: setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1075
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1076
text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1077
directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1078
lemma setsum_Union_disjoint:
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1079
  "[| (ALL A:C. finite A);
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1080
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1081
   ==> setsum f (Union C) = setsum (setsum f) C"
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1082
apply (cases "finite C") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1083
 prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1084
  apply (frule setsum_UN_disjoint [of C id f])
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1085
 apply (unfold Union_def id_def, assumption+)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1086
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1087
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1088
(*But we can't get rid of finite A. If infinite, although the lhs is 0, 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1089
  the rhs need not be, since SIGMA A B could still be finite.*)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1090
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
  1091
    (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1092
by(simp add:setsum_def comm_monoid_add.fold_image_Sigma split_def cong:setsum_cong)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1093
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1094
text{*Here we can eliminate the finiteness assumptions, by cases.*}
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1095
lemma setsum_cartesian_product: 
17189
b15f8e094874 patterns in setsum and setprod
paulson
parents: 17149
diff changeset
  1096
   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1097
apply (cases "finite A") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1098
 apply (cases "finite B") 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1099
  apply (simp add: setsum_Sigma)
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1100
 apply (cases "A={}", simp)
15543
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
  1101
 apply (simp) 
15409
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1102
apply (auto simp add: setsum_def
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1103
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
a063687d24eb new and stronger lemmas and improved simplification for finite sets
paulson
parents: 15402
diff changeset
  1104
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1105
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1106
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1107
by(simp add:setsum_def comm_monoid_add.fold_image_distrib)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1108
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1109
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1110
subsubsection {* Properties in more restricted classes of structures *}
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1111
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1112
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1113
apply (case_tac "finite A")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1114
 prefer 2 apply (simp add: setsum_def)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1115
apply (erule rev_mp)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1116
apply (erule finite_induct, auto)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1117
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1118
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1119
lemma setsum_eq_0_iff [simp]:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1120
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1121
by (induct set: finite) auto
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1122
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1123
lemma setsum_Un_nat: "finite A ==> finite B ==>
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1124
  (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1125
  -- {* For the natural numbers, we have subtraction. *}
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
  1126
by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1127
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1128
lemma setsum_Un: "finite A ==> finite B ==>
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1129
  (setsum f (A Un B) :: 'a :: ab_group_add) =
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1130
   setsum f A + setsum f B - setsum f (A Int B)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
  1131
by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1132
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1133
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1134
  (if a:A then setsum f A - f a else setsum f A)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1135
apply (case_tac "finite A")
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1136
 prefer 2 apply (simp add: setsum_def)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1137
apply (erule finite_induct)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1138
 apply (auto simp add: insert_Diff_if)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1139
apply (drule_tac a = a in mk_disjoint_insert, auto)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1140
done
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1141
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1142
lemma setsum_diff1: "finite A \<Longrightarrow>
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1143
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1144
  (if a:A then setsum f A - f a else setsum f A)"
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1145
by (erule finite_induct) (auto simp add: insert_Diff_if)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1146
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1147
lemma setsum_diff1'[rule_format]:
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1148
  "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1149
apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1150
apply (auto simp add: insert_Diff_if add_ac)
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1151
done
15552
8ab8e425410b added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents: 15543
diff changeset
  1152
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1153
(* By Jeremy Siek: *)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1154
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1155
lemma setsum_diff_nat: 
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1156
assumes "finite B" and "B \<subseteq> A"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1157
shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28823
diff changeset
  1158
using assms
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1159
proof induct
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1160
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1161
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1162
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1163
    and xFinA: "insert x F \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1164
    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1165
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1166
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1167
    by (simp add: setsum_diff1_nat)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1168
  from xFinA have "F \<subseteq> A" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1169
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1170
  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1171
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1172
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1173
  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1174
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1175
  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1176
  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1177
    by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1178
  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1179
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1180
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1181
lemma setsum_diff:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1182
  assumes le: "finite A" "B \<subseteq> A"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1183
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1184
proof -
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1185
  from le have finiteB: "finite B" using finite_subset by auto
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1186
  show ?thesis using finiteB le
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1187
  proof induct
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1188
    case empty
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1189
    thus ?case by auto
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1190
  next
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1191
    case (insert x F)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1192
    thus ?case using le finiteB 
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1193
      by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1194
  qed
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1195
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1196
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1197
lemma setsum_mono:
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1198
  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1199
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1200
proof (cases "finite K")
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1201
  case True
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1202
  thus ?thesis using le
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1203
  proof induct
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1204
    case empty
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1205
    thus ?case by simp
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1206
  next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1207
    case insert
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1208
    thus ?case using add_mono by fastsimp
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1209
  qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1210
next
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1211
  case False
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1212
  thus ?thesis
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1213
    by (simp add: setsum_def)
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1214
qed
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1215
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1216
lemma setsum_strict_mono:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1217
  fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1218
  assumes "finite A"  "A \<noteq> {}"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1219
    and "!!x. x:A \<Longrightarrow> f x < g x"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1220
  shows "setsum f A < setsum g A"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1221
  using prems
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1222
proof (induct rule: finite_ne_induct)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1223
  case singleton thus ?case by simp
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1224
next
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1225
  case insert thus ?case by (auto simp: add_strict_mono)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1226
qed
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15552
diff changeset
  1227
15535
nipkow
parents: 15532
diff changeset
  1228
lemma setsum_negf:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1229
  "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
15535
nipkow
parents: 15532
diff changeset
  1230
proof (cases "finite A")
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 21733
diff changeset
  1231
  case True thus ?thesis by (induct set: finite) auto
15535
nipkow
parents: 15532
diff changeset
  1232
next
nipkow
parents: 15532
diff changeset
  1233
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1234
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1235
15535
nipkow
parents: 15532
diff changeset
  1236
lemma setsum_subtractf:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1237
  "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1238
    setsum f A - setsum g A"
15535
nipkow
parents: 15532
diff changeset
  1239
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1240
  case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
nipkow
parents: 15532
diff changeset
  1241
next
nipkow
parents: 15532
diff changeset
  1242
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1243
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1244
15535
nipkow
parents: 15532
diff changeset
  1245
lemma setsum_nonneg:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1246
  assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1247
  shows "0 \<le> setsum f A"
15535
nipkow
parents: 15532
diff changeset
  1248
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1249
  case True thus ?thesis using nn
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1250
  proof induct
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1251
    case empty then show ?case by simp
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1252
  next
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1253
    case (insert x F)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1254
    then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1255
    with insert show ?case by simp
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1256
  qed
15535
nipkow
parents: 15532
diff changeset
  1257
next
nipkow
parents: 15532
diff changeset
  1258
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1259
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1260
15535
nipkow
parents: 15532
diff changeset
  1261
lemma setsum_nonpos:
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1262
  assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1263
  shows "setsum f A \<le> 0"
15535
nipkow
parents: 15532
diff changeset
  1264
proof (cases "finite A")
nipkow
parents: 15532
diff changeset
  1265
  case True thus ?thesis using np
21575
89463ae2612d tuned proofs;
wenzelm
parents: 21409
diff changeset
  1266
  proof induct
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1267
    case empty then show ?case by simp
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1268
  next
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1269
    case (insert x F)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1270
    then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1271
    with insert show ?case by simp
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
  1272
  qed
15535
nipkow
parents: 15532
diff changeset
  1273
next
nipkow
parents: 15532
diff changeset
  1274
  case False thus ?thesis by (simp add: setsum_def)
nipkow
parents: 15532
diff changeset
  1275
qed
15402
97204f3b4705 REorganized Finite_Set
nipkow
parents: 15392
diff changeset
  1276
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1277
lemma setsum_mono2:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1278
fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1279
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1280
shows "setsum f A \<le> setsum f B"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1281
proof -
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1282
  have "setsum f A \<le> setsum f A + setsum f (B-A)"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1283
    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1284
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1285
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1286
  also have "A \<union> (B-A) = B" using sub by blast
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15535
diff changeset
  1287
  finally show ?thesis .