author | huffman |
Tue, 01 Jul 2008 20:26:48 +0200 | |
changeset 27430 | 1e25ac05cd87 |
parent 27418 | 564117b58d73 |
child 27611 | 2c01c0bdb385 |
permissions | -rw-r--r-- |
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(* Title: HOL/Finite_Set.thy |
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ID: $Id$ |
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Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel |
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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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Merged theories about wellfoundedness into one: Wellfounded.thy
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imports Divides Transitive_Closure |
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begin |
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|
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subsection {* Definition and basic properties *} |
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|
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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 prems have "A \<noteq> UNIV" by blast |
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thus ?thesis by blast |
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qed |
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lemma finite_induct [case_names empty insert, induct set: 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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||
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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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||
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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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||
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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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||
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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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||
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lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)" |
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-- {* The union of two finite sets is finite. *} |
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by (induct set: 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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show "finite A" |
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proof cases |
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assume x: "x \<in> A" |
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with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff) |
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with r have "finite (A - {x})" . |
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hence "finite (insert x (A - {x}))" .. |
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also have "insert x (A - {x}) = A" using x by (rule insert_Diff) |
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finally show ?thesis . |
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next |
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show "A \<subseteq> F ==> ?thesis" by fact |
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assume "x \<notin> A" |
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with A show "A \<subseteq> F" by (simp add: subset_insert_iff) |
|
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qed |
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qed |
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qed |
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||
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lemma finite_Collect_subset[simp]: "finite A \<Longrightarrow> finite{x \<in> A. P x}" |
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using finite_subset[of "{x \<in> A. P x}" "A"] by blast |
180 |
||
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lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)" |
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by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI) |
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lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)" |
|
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-- {* The converse obviously fails. *} |
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by (blast intro: finite_subset) |
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lemma finite_insert [simp]: "finite (insert a A) = finite A" |
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apply (subst insert_is_Un) |
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apply (simp only: finite_Un, blast) |
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done |
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lemma finite_Union[simp, intro]: |
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"\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)" |
|
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by (induct rule:finite_induct) simp_all |
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lemma finite_empty_induct: |
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assumes "finite A" |
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and "P A" |
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and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})" |
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shows "P {}" |
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proof - |
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have "P (A - A)" |
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proof - |
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{ |
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fix c b :: "'a set" |
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assume c: "finite c" and b: "finite b" |
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and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})" |
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have "c \<subseteq> b ==> P (b - c)" |
|
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using c |
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proof induct |
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case empty |
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from P1 show ?case by simp |
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next |
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case (insert x F) |
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have "P (b - F - {x})" |
|
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proof (rule P2) |
|
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from _ b show "finite (b - F)" by (rule finite_subset) blast |
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from insert show "x \<in> b - F" by simp |
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from insert show "P (b - F)" by simp |
|
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qed |
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also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric]) |
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finally show ?case . |
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qed |
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} |
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then show ?thesis by this (simp_all add: assms) |
|
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qed |
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then show ?thesis by simp |
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qed |
230 |
||
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lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)" |
|
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by (rule Diff_subset [THEN finite_subset]) |
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||
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lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)" |
|
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apply (subst Diff_insert) |
|
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apply (case_tac "a : A - B") |
|
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apply (rule finite_insert [symmetric, THEN trans]) |
|
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apply (subst insert_Diff, simp_all) |
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done |
240 |
||
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lemma finite_Diff_singleton [simp]: "finite (A - {a}) = finite A" |
242 |
by simp |
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243 |
||
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text {* Image and Inverse Image over Finite Sets *} |
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|
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lemma finite_imageI[simp]: "finite F ==> finite (h ` F)" |
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-- {* The image of a finite set is finite. *} |
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by (induct set: finite) simp_all |
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lemma finite_surj: "finite A ==> B <= f ` A ==> finite B" |
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apply (frule finite_imageI) |
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apply (erule finite_subset, assumption) |
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done |
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|
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lemma finite_range_imageI: |
257 |
"finite (range g) ==> finite (range (%x. f (g x)))" |
|
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apply (drule finite_imageI, simp add: range_composition) |
13825 | 259 |
done |
260 |
||
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lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A" |
262 |
proof - |
|
263 |
have aux: "!!A. finite (A - {}) = finite A" by simp |
|
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fix B :: "'a set" |
|
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assume "finite B" |
|
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thus "!!A. f`A = B ==> inj_on f A ==> finite A" |
|
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apply induct |
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apply simp |
|
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apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})") |
|
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apply clarify |
|
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apply (simp (no_asm_use) add: inj_on_def) |
|
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apply (blast dest!: aux [THEN iffD1], atomize) |
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apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl) |
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apply (frule subsetD [OF equalityD2 insertI1], clarify) |
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apply (rule_tac x = xa in bexI) |
276 |
apply (simp_all add: inj_on_image_set_diff) |
|
277 |
done |
|
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qed (rule refl) |
|
279 |
||
280 |
||
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lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}" |
282 |
-- {* The inverse image of a singleton under an injective function |
|
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is included in a singleton. *} |
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apply (auto simp add: inj_on_def) |
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apply (blast intro: the_equality [symmetric]) |
13825 | 286 |
done |
287 |
||
288 |
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)" |
|
289 |
-- {* The inverse image of a finite set under an injective function |
|
290 |
is finite. *} |
|
22262 | 291 |
apply (induct set: finite) |
21575 | 292 |
apply simp_all |
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apply (subst vimage_insert) |
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apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton]) |
13825 | 295 |
done |
296 |
||
297 |
||
15392 | 298 |
text {* The finite UNION of finite sets *} |
12396 | 299 |
|
300 |
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)" |
|
22262 | 301 |
by (induct set: finite) simp_all |
12396 | 302 |
|
303 |
text {* |
|
304 |
Strengthen RHS to |
|
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@{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}? |
12396 | 306 |
|
307 |
We'd need to prove |
|
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@{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"} |
12396 | 309 |
by induction. *} |
310 |
||
311 |
lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))" |
|
312 |
by (blast intro: finite_UN_I finite_subset) |
|
313 |
||
314 |
||
17022 | 315 |
lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)" |
316 |
by (simp add: Plus_def) |
|
317 |
||
15392 | 318 |
text {* Sigma of finite sets *} |
12396 | 319 |
|
320 |
lemma finite_SigmaI [simp]: |
|
321 |
"finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)" |
|
322 |
by (unfold Sigma_def) (blast intro!: finite_UN_I) |
|
323 |
||
15402 | 324 |
lemma finite_cartesian_product: "[| finite A; finite B |] ==> |
325 |
finite (A <*> B)" |
|
326 |
by (rule finite_SigmaI) |
|
327 |
||
12396 | 328 |
lemma finite_Prod_UNIV: |
329 |
"finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)" |
|
330 |
apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)") |
|
331 |
apply (erule ssubst) |
|
14208 | 332 |
apply (erule finite_SigmaI, auto) |
12396 | 333 |
done |
334 |
||
15409
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335 |
lemma finite_cartesian_productD1: |
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336 |
"[| finite (A <*> B); B \<noteq> {} |] ==> finite A" |
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apply (auto simp add: finite_conv_nat_seg_image) |
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apply (drule_tac x=n in spec) |
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apply (drule_tac x="fst o f" in spec) |
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apply (auto simp add: o_def) |
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prefer 2 apply (force dest!: equalityD2) |
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apply (drule equalityD1) |
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apply (rename_tac y x) |
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344 |
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
|
345 |
prefer 2 apply force |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
346 |
apply clarify |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
347 |
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
|
348 |
done |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
349 |
|
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
350 |
lemma finite_cartesian_productD2: |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
351 |
"[| finite (A <*> B); A \<noteq> {} |] ==> finite B" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
352 |
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
|
353 |
apply (drule_tac x=n in spec) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
354 |
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
|
355 |
apply (auto simp add: o_def) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
356 |
prefer 2 apply (force dest!: equalityD2) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
357 |
apply (drule equalityD1) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
358 |
apply (rename_tac x y) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
359 |
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
|
360 |
prefer 2 apply force |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
361 |
apply clarify |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
362 |
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
|
363 |
done |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
364 |
|
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
365 |
|
15392 | 366 |
text {* The powerset of a finite set *} |
12396 | 367 |
|
368 |
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A" |
|
369 |
proof |
|
370 |
assume "finite (Pow A)" |
|
371 |
with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast |
|
372 |
thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp |
|
373 |
next |
|
374 |
assume "finite A" |
|
375 |
thus "finite (Pow A)" |
|
376 |
by induct (simp_all add: finite_UnI finite_imageI Pow_insert) |
|
377 |
qed |
|
378 |
||
15392 | 379 |
|
380 |
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A" |
|
381 |
by(blast intro: finite_subset[OF subset_Pow_Union]) |
|
382 |
||
383 |
||
12396 | 384 |
lemma finite_converse [iff]: "finite (r^-1) = finite r" |
385 |
apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r") |
|
386 |
apply simp |
|
387 |
apply (rule iffI) |
|
388 |
apply (erule finite_imageD [unfolded inj_on_def]) |
|
389 |
apply (simp split add: split_split) |
|
390 |
apply (erule finite_imageI) |
|
14208 | 391 |
apply (simp add: converse_def image_def, auto) |
12396 | 392 |
apply (rule bexI) |
393 |
prefer 2 apply assumption |
|
394 |
apply simp |
|
395 |
done |
|
396 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset
|
397 |
|
15392 | 398 |
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi |
399 |
Ehmety) *} |
|
12396 | 400 |
|
401 |
lemma finite_Field: "finite r ==> finite (Field r)" |
|
402 |
-- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *} |
|
22262 | 403 |
apply (induct set: finite) |
12396 | 404 |
apply (auto simp add: Field_def Domain_insert Range_insert) |
405 |
done |
|
406 |
||
407 |
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r" |
|
408 |
apply clarify |
|
409 |
apply (erule trancl_induct) |
|
410 |
apply (auto simp add: Field_def) |
|
411 |
done |
|
412 |
||
413 |
lemma finite_trancl: "finite (r^+) = finite r" |
|
414 |
apply auto |
|
415 |
prefer 2 |
|
416 |
apply (rule trancl_subset_Field2 [THEN finite_subset]) |
|
417 |
apply (rule finite_SigmaI) |
|
418 |
prefer 3 |
|
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
13595
diff
changeset
|
419 |
apply (blast intro: r_into_trancl' finite_subset) |
12396 | 420 |
apply (auto simp add: finite_Field) |
421 |
done |
|
422 |
||
423 |
||
26441 | 424 |
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
|
425 |
|
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 |
setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*} |
26441 | 427 |
class finite = itself + |
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
|
428 |
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
|
429 |
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
|
430 |
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
|
431 |
|
27430 | 432 |
context finite |
433 |
begin |
|
434 |
||
435 |
lemma finite [simp]: "finite (A \<Colon> 'a set)" |
|
26441 | 436 |
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
|
437 |
|
27430 | 438 |
end |
439 |
||
26146 | 440 |
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
|
441 |
"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
|
442 |
|
26146 | 443 |
instance unit :: finite |
444 |
by default (simp add: UNIV_unit) |
|
445 |
||
446 |
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
|
447 |
"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
|
448 |
|
26146 | 449 |
instance bool :: finite |
450 |
by default (simp add: UNIV_bool) |
|
451 |
||
452 |
instance * :: (finite, finite) finite |
|
453 |
by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite) |
|
454 |
||
455 |
instance "+" :: (finite, finite) finite |
|
456 |
by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite) |
|
457 |
||
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
|
458 |
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
|
459 |
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
|
460 |
|
26146 | 461 |
instance "fun" :: (finite, finite) finite |
462 |
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
|
463 |
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
|
464 |
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
|
465 |
let ?graph = "%f::'a => 'b. {(x, y). y = f x}" |
26792 | 466 |
have "range ?graph \<subseteq> Pow UNIV" by simp |
467 |
moreover have "finite (Pow (UNIV :: ('a * 'b) set))" |
|
468 |
by (simp only: finite_Pow_iff finite) |
|
469 |
ultimately show "finite (range ?graph)" |
|
470 |
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
|
471 |
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
|
472 |
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
|
473 |
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
|
474 |
|
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
|
475 |
|
15392 | 476 |
subsection {* A fold functional for finite sets *} |
477 |
||
478 |
text {* The intended behaviour is |
|
15480 | 479 |
@{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<dots>)"} |
15392 | 480 |
if @{text f} is associative-commutative. For an application of @{text fold} |
481 |
se the definitions of sums and products over finite sets. |
|
482 |
*} |
|
483 |
||
23736 | 484 |
inductive |
22262 | 485 |
foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a => bool" |
486 |
for f :: "'a => 'a => 'a" |
|
487 |
and g :: "'b => 'a" |
|
488 |
and z :: 'a |
|
489 |
where |
|
490 |
emptyI [intro]: "foldSet f g z {} z" |
|
491 |
| insertI [intro]: |
|
492 |
"\<lbrakk> x \<notin> A; foldSet f g z A y \<rbrakk> |
|
493 |
\<Longrightarrow> foldSet f g z (insert x A) (f (g x) y)" |
|
494 |
||
23736 | 495 |
inductive_cases empty_foldSetE [elim!]: "foldSet f g z {} x" |
15392 | 496 |
|
497 |
constdefs |
|
21733 | 498 |
fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a" |
22262 | 499 |
"fold f g z A == THE x. foldSet f g z A x" |
15392 | 500 |
|
15498 | 501 |
text{*A tempting alternative for the definiens is |
22262 | 502 |
@{term "if finite A then THE x. foldSet f g e A x else e"}. |
15498 | 503 |
It allows the removal of finiteness assumptions from the theorems |
504 |
@{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}. |
|
505 |
The proofs become ugly, with @{text rule_format}. It is not worth the effort.*} |
|
506 |
||
507 |
||
15392 | 508 |
lemma Diff1_foldSet: |
22262 | 509 |
"foldSet f g z (A - {x}) y ==> x: A ==> foldSet f g z A (f (g x) y)" |
15392 | 510 |
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto) |
511 |
||
22262 | 512 |
lemma foldSet_imp_finite: "foldSet f g z A x==> finite A" |
15392 | 513 |
by (induct set: foldSet) auto |
514 |
||
22262 | 515 |
lemma finite_imp_foldSet: "finite A ==> EX x. foldSet f g z A x" |
516 |
by (induct set: finite) auto |
|
15392 | 517 |
|
518 |
||
519 |
subsubsection{*From @{term foldSet} to @{term fold}*} |
|
520 |
||
15510 | 521 |
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})" |
19868 | 522 |
by (auto simp add: less_Suc_eq) |
15510 | 523 |
|
524 |
lemma insert_image_inj_on_eq: |
|
525 |
"[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; |
|
526 |
inj_on h {i. i < Suc m}|] |
|
527 |
==> A = h ` {i. i < m}" |
|
528 |
apply (auto simp add: image_less_Suc inj_on_def) |
|
529 |
apply (blast intro: less_trans) |
|
530 |
done |
|
531 |
||
532 |
lemma insert_inj_onE: |
|
533 |
assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" |
|
534 |
and inj_on: "inj_on h {i::nat. i<n}" |
|
535 |
shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n" |
|
536 |
proof (cases n) |
|
537 |
case 0 thus ?thesis using aA by auto |
|
538 |
next |
|
539 |
case (Suc m) |
|
23389 | 540 |
have nSuc: "n = Suc m" by fact |
15510 | 541 |
have mlessn: "m<n" by (simp add: nSuc) |
15532 | 542 |
from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE) |
27165 | 543 |
let ?hm = "Fun.swap k m h" |
15520 | 544 |
have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn |
545 |
by (simp add: inj_on_swap_iff inj_on) |
|
15510 | 546 |
show ?thesis |
15520 | 547 |
proof (intro exI conjI) |
548 |
show "inj_on ?hm {i. i < m}" using inj_hm |
|
15510 | 549 |
by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on) |
15520 | 550 |
show "m<n" by (rule mlessn) |
551 |
show "A = ?hm ` {i. i < m}" |
|
552 |
proof (rule insert_image_inj_on_eq) |
|
27165 | 553 |
show "inj_on (Fun.swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp |
15520 | 554 |
show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) |
555 |
show "insert (?hm m) A = ?hm ` {i. i < Suc m}" |
|
556 |
using aA hkeq nSuc klessn |
|
557 |
by (auto simp add: swap_def image_less_Suc fun_upd_image |
|
558 |
less_Suc_eq inj_on_image_set_diff [OF inj_on]) |
|
15479 | 559 |
qed |
560 |
qed |
|
561 |
qed |
|
562 |
||
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
|
563 |
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
|
564 |
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
|
565 |
|
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
|
566 |
lemma foldSet_determ_aux: |
15510 | 567 |
"!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n}; |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
568 |
foldSet times g z A x; foldSet times g z A x' \<rbrakk> |
15392 | 569 |
\<Longrightarrow> x' = x" |
15510 | 570 |
proof (induct n rule: less_induct) |
571 |
case (less n) |
|
572 |
have IH: "!!m h A x x'. |
|
573 |
\<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m}; |
|
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
|
574 |
foldSet times g z A x; foldSet times g z A x'\<rbrakk> \<Longrightarrow> x' = x" by fact |
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
|
575 |
have Afoldx: "foldSet times g z A x" and Afoldx': "foldSet times g z A x'" |
23389 | 576 |
and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" by fact+ |
15510 | 577 |
show ?case |
578 |
proof (rule foldSet.cases [OF Afoldx]) |
|
22262 | 579 |
assume "A = {}" and "x = z" |
15510 | 580 |
with Afoldx' show "x' = x" by blast |
15392 | 581 |
next |
15510 | 582 |
fix B b u |
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
|
583 |
assume AbB: "A = insert b B" and x: "x = g b * u" |
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
|
584 |
and notinB: "b \<notin> B" and Bu: "foldSet times g z B u" |
15510 | 585 |
show "x'=x" |
586 |
proof (rule foldSet.cases [OF Afoldx']) |
|
22262 | 587 |
assume "A = {}" and "x' = z" |
15510 | 588 |
with AbB show "x' = x" by blast |
15392 | 589 |
next |
15510 | 590 |
fix C c v |
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
|
591 |
assume AcC: "A = insert c C" and x': "x' = g c * v" |
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
|
592 |
and notinC: "c \<notin> C" and Cv: "foldSet times g z C v" |
15510 | 593 |
from A AbB have Beq: "insert b B = h`{i. i<n}" by simp |
594 |
from insert_inj_onE [OF Beq notinB injh] |
|
595 |
obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" |
|
596 |
and Beq: "B = hB ` {i. i < mB}" |
|
597 |
and lessB: "mB < n" by auto |
|
598 |
from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp |
|
599 |
from insert_inj_onE [OF Ceq notinC injh] |
|
600 |
obtain hC mC where inj_onC: "inj_on hC {i. i < mC}" |
|
601 |
and Ceq: "C = hC ` {i. i < mC}" |
|
602 |
and lessC: "mC < n" by auto |
|
603 |
show "x'=x" |
|
15392 | 604 |
proof cases |
15510 | 605 |
assume "b=c" |
606 |
then moreover have "B = C" using AbB AcC notinB notinC by auto |
|
607 |
ultimately show ?thesis using Bu Cv x x' IH[OF lessC Ceq inj_onC] |
|
608 |
by auto |
|
15392 | 609 |
next |
610 |
assume diff: "b \<noteq> c" |
|
611 |
let ?D = "B - {c}" |
|
612 |
have B: "B = insert c ?D" and C: "C = insert b ?D" |
|
15510 | 613 |
using AbB AcC notinB notinC diff by(blast elim!:equalityE)+ |
15402 | 614 |
have "finite A" by(rule foldSet_imp_finite[OF Afoldx]) |
15510 | 615 |
with AbB have "finite ?D" by 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
|
616 |
then obtain d where Dfoldd: "foldSet times g z ?D d" |
17589 | 617 |
using finite_imp_foldSet by iprover |
15506 | 618 |
moreover have cinB: "c \<in> B" using B by auto |
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
|
619 |
ultimately have "foldSet times g z B (g c * d)" |
15392 | 620 |
by(rule Diff1_foldSet) |
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
|
621 |
then have "g c * d = u" by (rule IH [OF lessB Beq inj_onB Bu]) |
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
|
622 |
then have "u = g c * d" .. |
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
|
623 |
moreover have "v = g b * d" |
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
|
624 |
proof (rule sym, rule IH [OF lessC Ceq inj_onC Cv]) |
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
|
625 |
show "foldSet times g z C (g b * d)" using C notinB Dfoldd |
15392 | 626 |
by fastsimp |
627 |
qed |
|
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
|
628 |
ultimately show ?thesis using x 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
|
629 |
by (simp add: mult_left_commute) |
15392 | 630 |
qed |
631 |
qed |
|
632 |
qed |
|
633 |
qed |
|
634 |
||
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
|
635 |
lemma foldSet_determ: |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
636 |
"foldSet times g z A x ==> foldSet times g z A y ==> y = x" |
15510 | 637 |
apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) |
638 |
apply (blast intro: foldSet_determ_aux [rule_format]) |
|
15392 | 639 |
done |
640 |
||
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
641 |
lemma fold_equality: "foldSet times g z A y ==> fold times g z A = y" |
15392 | 642 |
by (unfold fold_def) (blast intro: foldSet_determ) |
643 |
||
644 |
text{* The base case for @{text fold}: *} |
|
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 (in -) fold_empty [simp]: "fold f g z {} = z" |
15392 | 647 |
by (unfold fold_def) blast |
648 |
||
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
|
649 |
lemma fold_insert_aux: "x \<notin> 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
|
650 |
(foldSet times g z (insert x A) v) = |
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
|
651 |
(EX y. foldSet times g z A y & v = g x * y)" |
15392 | 652 |
apply auto |
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
|
653 |
apply (rule_tac A1 = A and f1 = times in finite_imp_foldSet [THEN exE]) |
15392 | 654 |
apply (fastsimp dest: foldSet_imp_finite) |
655 |
apply (blast intro: foldSet_determ) |
|
656 |
done |
|
657 |
||
658 |
text{* The recursion equation for @{text fold}: *} |
|
659 |
||
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
|
660 |
lemma fold_insert [simp]: |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
661 |
"finite A ==> x \<notin> A ==> fold times g z (insert x A) = g x * fold times g z A" |
15392 | 662 |
apply (unfold fold_def) |
663 |
apply (simp add: fold_insert_aux) |
|
664 |
apply (rule the_equality) |
|
665 |
apply (auto intro: finite_imp_foldSet |
|
666 |
cong add: conj_cong simp add: fold_def [symmetric] fold_equality) |
|
667 |
done |
|
668 |
||
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
|
669 |
lemma fold_rec: |
15535 | 670 |
assumes fin: "finite A" and a: "a:A" |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
671 |
shows "fold times g z A = g a * fold times g z (A - {a})" |
15535 | 672 |
proof- |
673 |
have A: "A = insert a (A - {a})" using a by blast |
|
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
|
674 |
hence "fold times g z A = fold times g z (insert a (A - {a}))" by simp |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
675 |
also have "\<dots> = g a * fold times g z (A - {a})" |
15535 | 676 |
by(rule fold_insert) (simp add:fin)+ |
677 |
finally show ?thesis . |
|
678 |
qed |
|
679 |
||
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
|
680 |
end |
15392 | 681 |
|
15480 | 682 |
text{* A simplified version for idempotent functions: *} |
683 |
||
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
|
684 |
context ab_semigroup_idem_mult |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
685 |
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
|
686 |
|
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
|
687 |
lemma fold_insert_idem: |
15480 | 688 |
assumes finA: "finite A" |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
689 |
shows "fold times g z (insert a A) = g a * fold times g z A" |
15480 | 690 |
proof cases |
691 |
assume "a \<in> A" |
|
692 |
then obtain B where A: "A = insert a B" and disj: "a \<notin> B" |
|
693 |
by(blast dest: mk_disjoint_insert) |
|
694 |
show ?thesis |
|
695 |
proof - |
|
696 |
from finA A have finB: "finite B" by(blast intro: 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
|
697 |
have "fold times g z (insert a A) = fold times g z (insert a B)" using A by simp |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
698 |
also have "\<dots> = g a * fold times g z B" |
15506 | 699 |
using finB disj by 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
|
700 |
also have "\<dots> = g a * fold times g z 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
|
701 |
using A finB disj |
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
|
702 |
by (simp add: mult_idem mult_assoc [symmetric]) |
15480 | 703 |
finally show ?thesis . |
704 |
qed |
|
705 |
next |
|
706 |
assume "a \<notin> A" |
|
707 |
with finA show ?thesis by simp |
|
708 |
qed |
|
709 |
||
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
|
710 |
lemma foldI_conv_id: |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
711 |
"finite A \<Longrightarrow> fold times g z A = fold times id z (g ` A)" |
15509 | 712 |
by(erule finite_induct)(simp_all add: fold_insert_idem del: fold_insert) |
15484 | 713 |
|
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
|
714 |
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
|
715 |
|
15392 | 716 |
subsubsection{*Lemmas about @{text fold}*} |
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 |
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
|
719 |
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
|
720 |
|
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
|
721 |
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
|
722 |
"finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)" |
22262 | 723 |
apply (induct set: finite) |
21575 | 724 |
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
|
725 |
apply (simp add: mult_left_commute [of x]) |
15392 | 726 |
done |
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 |
lemma fold_nest_Un_Int: |
15392 | 729 |
"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
|
730 |
==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)" |
22262 | 731 |
apply (induct set: finite) |
21575 | 732 |
apply simp |
15392 | 733 |
apply (simp add: fold_commute Int_insert_left insert_absorb) |
734 |
done |
|
735 |
||
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
736 |
lemma fold_nest_Un_disjoint: |
15392 | 737 |
"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
|
738 |
==> fold times g z (A Un B) = fold times g (fold times g z B) A" |
15392 | 739 |
by (simp add: fold_nest_Un_Int) |
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_reindex: |
15487 | 742 |
assumes fin: "finite A" |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
743 |
shows "inj_on h A \<Longrightarrow> fold times g z (h ` A) = fold times (g \<circ> h) z A" |
15506 | 744 |
using fin apply induct |
15392 | 745 |
apply simp |
746 |
apply simp |
|
747 |
done |
|
748 |
||
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
|
749 |
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
|
750 |
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
|
751 |
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
|
752 |
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
|
753 |
*} |
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
|
754 |
|
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
|
755 |
lemma fold_fusion: |
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 |
includes ab_semigroup_mult g |
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
|
757 |
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
|
758 |
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
|
759 |
shows "h (fold g j w A) = fold times j (h w) 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
|
760 |
using fin hyp by (induct set: finite) simp_all |
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
|
761 |
|
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
|
762 |
lemma fold_cong: |
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
|
763 |
"finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold times g z A = fold times h z 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
|
764 |
apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold times g z C = fold times h z C") |
15392 | 765 |
apply simp |
766 |
apply (erule finite_induct, simp) |
|
767 |
apply (simp add: subset_insert_iff, clarify) |
|
768 |
apply (subgoal_tac "finite C") |
|
769 |
prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl]) |
|
770 |
apply (subgoal_tac "C = insert x (C - {x})") |
|
771 |
prefer 2 apply blast |
|
772 |
apply (erule ssubst) |
|
773 |
apply (drule spec) |
|
774 |
apply (erule (1) notE impE) |
|
775 |
apply (simp add: Ball_def del: insert_Diff_single) |
|
776 |
done |
|
777 |
||
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
|
778 |
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
|
779 |
|
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 |
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
|
781 |
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
|
782 |
|
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
|
783 |
lemma fold_Un_Int: |
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
|
784 |
"finite A ==> finite B ==> |
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
|
785 |
fold times g 1 A * fold times g 1 B = |
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
|
786 |
fold times g 1 (A Un B) * fold times g 1 (A Int B)" |
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
|
787 |
by (induct set: 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
|
788 |
(auto simp add: mult_ac insert_absorb Int_insert_left) |
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
|
789 |
|
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
|
790 |
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
|
791 |
"finite A ==> finite B ==> A Int B = {} ==> |
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
|
792 |
fold times g 1 (A Un B) = fold times g 1 A * fold times g 1 B" |
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
|
793 |
by (simp add: fold_Un_Int) |
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
|
794 |
|
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
|
795 |
lemma 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
|
796 |
"\<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
|
797 |
ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk> |
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
|
798 |
\<Longrightarrow> fold times g 1 (UNION I 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
|
799 |
fold times (%i. fold times g 1 (A i)) 1 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
|
800 |
apply (induct set: finite, simp, atomize) |
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
|
801 |
apply (subgoal_tac "ALL i:F. x \<noteq> 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
|
802 |
prefer 2 apply blast |
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
|
803 |
apply (subgoal_tac "A x Int UNION F 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
|
804 |
prefer 2 apply blast |
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 |
apply (simp add: 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
|
806 |
done |
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 |
|
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 |
lemma fold_Sigma: "finite A ==> ALL x:A. finite (B 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
|
809 |
fold times (%x. fold times (g x) 1 (B x)) 1 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
|
810 |
fold times (split g) 1 (SIGMA x:A. B x)" |
15392 | 811 |
apply (subst Sigma_def) |
15506 | 812 |
apply (subst fold_UN_disjoint, assumption, simp) |
15392 | 813 |
apply blast |
814 |
apply (erule fold_cong) |
|
15506 | 815 |
apply (subst fold_UN_disjoint, simp, simp) |
15392 | 816 |
apply blast |
15506 | 817 |
apply simp |
15392 | 818 |
done |
819 |
||
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
|
820 |
lemma fold_distrib: "finite A \<Longrightarrow> |
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
|
821 |
fold times (%x. g x * h x) 1 A = fold times g 1 A * fold times h 1 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
|
822 |
by (erule finite_induct) (simp_all add: mult_ac) |
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
|
823 |
|
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 |
end |
22917 | 825 |
|
826 |
||
15402 | 827 |
subsection {* Generalized summation over a set *} |
828 |
||
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
|
829 |
interpretation comm_monoid_add: comm_monoid_mult ["0::'a::comm_monoid_add" "op +"] |
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
|
830 |
by unfold_locales (auto intro: add_assoc add_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
|
831 |
|
15402 | 832 |
constdefs |
833 |
setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add" |
|
834 |
"setsum f A == if finite A then fold (op +) f 0 A else 0" |
|
835 |
||
19535 | 836 |
abbreviation |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21249
diff
changeset
|
837 |
Setsum ("\<Sum>_" [1000] 999) where |
19535 | 838 |
"\<Sum>A == setsum (%x. x) A" |
839 |
||
15402 | 840 |
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is |
841 |
written @{text"\<Sum>x\<in>A. e"}. *} |
|
842 |
||
843 |
syntax |
|
17189 | 844 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3SUM _:_. _)" [0, 51, 10] 10) |
15402 | 845 |
syntax (xsymbols) |
17189 | 846 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10) |
15402 | 847 |
syntax (HTML output) |
17189 | 848 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10) |
15402 | 849 |
|
850 |
translations -- {* Beware of argument permutation! *} |
|
851 |
"SUM i:A. b" == "setsum (%i. b) A" |
|
852 |
"\<Sum>i\<in>A. b" == "setsum (%i. b) A" |
|
853 |
||
854 |
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter |
|
855 |
@{text"\<Sum>x|P. e"}. *} |
|
856 |
||
857 |
syntax |
|
17189 | 858 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10) |
15402 | 859 |
syntax (xsymbols) |
17189 | 860 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10) |
15402 | 861 |
syntax (HTML output) |
17189 | 862 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10) |
15402 | 863 |
|
864 |
translations |
|
865 |
"SUM x|P. t" => "setsum (%x. t) {x. P}" |
|
866 |
"\<Sum>x|P. t" => "setsum (%x. t) {x. P}" |
|
867 |
||
868 |
print_translation {* |
|
869 |
let |
|
19535 | 870 |
fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = |
871 |
if x<>y then raise Match |
|
872 |
else let val x' = Syntax.mark_bound x |
|
873 |
val t' = subst_bound(x',t) |
|
874 |
val P' = subst_bound(x',P) |
|
875 |
in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end |
|
876 |
in [("setsum", setsum_tr')] end |
|
15402 | 877 |
*} |
878 |
||
19535 | 879 |
|
15402 | 880 |
lemma setsum_empty [simp]: "setsum f {} = 0" |
881 |
by (simp add: setsum_def) |
|
882 |
||
883 |
lemma setsum_insert [simp]: |
|
884 |
"finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F" |
|
15765 | 885 |
by (simp add: setsum_def) |
15402 | 886 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
887 |
lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
888 |
by (simp add: setsum_def) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
889 |
|
15402 | 890 |
lemma setsum_reindex: |
891 |
"inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) 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
|
892 |
by(auto simp add: setsum_def comm_monoid_add.fold_reindex dest!:finite_imageD) |
15402 | 893 |
|
894 |
lemma setsum_reindex_id: |
|
895 |
"inj_on f B ==> setsum f B = setsum id (f ` B)" |
|
896 |
by (auto simp add: setsum_reindex) |
|
897 |
||
898 |
lemma setsum_cong: |
|
899 |
"A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g 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
|
900 |
by(fastsimp simp: setsum_def intro: comm_monoid_add.fold_cong) |
15402 | 901 |
|
16733
236dfafbeb63
linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
16632
diff
changeset
|
902 |
lemma strong_setsum_cong[cong]: |
236dfafbeb63
linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
16632
diff
changeset
|
903 |
"A = B ==> (!!x. x:B =simp=> f x = g x) |
236dfafbeb63
linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
16632
diff
changeset
|
904 |
==> setsum (%x. f x) A = setsum (%x. g x) 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
|
905 |
by(fastsimp simp: simp_implies_def setsum_def intro: comm_monoid_add.fold_cong) |
16632
ad2895beef79
Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents:
16550
diff
changeset
|
906 |
|
15554 | 907 |
lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A"; |
908 |
by (rule setsum_cong[OF refl], auto); |
|
909 |
||
15402 | 910 |
lemma setsum_reindex_cong: |
15554 | 911 |
"[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] |
15402 | 912 |
==> setsum h B = setsum g A" |
913 |
by (simp add: setsum_reindex cong: setsum_cong) |
|
914 |
||
15542 | 915 |
lemma setsum_0[simp]: "setsum (%i. 0) A = 0" |
15402 | 916 |
apply (clarsimp simp: setsum_def) |
15765 | 917 |
apply (erule finite_induct, auto) |
15402 | 918 |
done |
919 |
||
15543 | 920 |
lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0" |
921 |
by(simp add:setsum_cong) |
|
15402 | 922 |
|
923 |
lemma setsum_Un_Int: "finite A ==> finite B ==> |
|
924 |
setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B" |
|
925 |
-- {* The reversed orientation looks more natural, but LOOPS as a simprule! *} |
|
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
|
926 |
by(simp add: setsum_def comm_monoid_add.fold_Un_Int [symmetric]) |
15402 | 927 |
|
928 |
lemma setsum_Un_disjoint: "finite A ==> finite B |
|
929 |
==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B" |
|
930 |
by (subst setsum_Un_Int [symmetric], auto) |
|
931 |
||
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
932 |
(*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
|
933 |
the lhs need not be, since UNION I A could still be finite.*) |
15402 | 934 |
lemma setsum_UN_disjoint: |
935 |
"finite I ==> (ALL i:I. finite (A i)) ==> |
|
936 |
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==> |
|
937 |
setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))" |
|
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
|
938 |
by(simp add: setsum_def comm_monoid_add.fold_UN_disjoint cong: setsum_cong) |
15402 | 939 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
940 |
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
|
941 |
directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*} |
15402 | 942 |
lemma setsum_Union_disjoint: |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
943 |
"[| (ALL A:C. finite A); |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
944 |
(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
|
945 |
==> setsum f (Union C) = setsum (setsum f) C" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
946 |
apply (cases "finite C") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
947 |
prefer 2 apply (force dest: finite_UnionD simp add: setsum_def) |
15402 | 948 |
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
|
949 |
apply (unfold Union_def id_def, assumption+) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
950 |
done |
15402 | 951 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
952 |
(*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
|
953 |
the rhs need not be, since SIGMA A B could still be finite.*) |
15402 | 954 |
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==> |
17189 | 955 |
(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)" |
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
|
956 |
by(simp add:setsum_def comm_monoid_add.fold_Sigma split_def cong:setsum_cong) |
15402 | 957 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
958 |
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
|
959 |
lemma setsum_cartesian_product: |
17189 | 960 |
"(\<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
|
961 |
apply (cases "finite A") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
962 |
apply (cases "finite B") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
963 |
apply (simp add: setsum_Sigma) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
964 |
apply (cases "A={}", simp) |
15543 | 965 |
apply (simp) |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
966 |
apply (auto simp add: setsum_def |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
967 |
dest: finite_cartesian_productD1 finite_cartesian_productD2) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
968 |
done |
15402 | 969 |
|
970 |
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)" |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
971 |
by(simp add:setsum_def comm_monoid_add.fold_distrib) |
15402 | 972 |
|
973 |
||
974 |
subsubsection {* Properties in more restricted classes of structures *} |
|
975 |
||
976 |
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a" |
|
977 |
apply (case_tac "finite A") |
|
978 |
prefer 2 apply (simp add: setsum_def) |
|
979 |
apply (erule rev_mp) |
|
980 |
apply (erule finite_induct, auto) |
|
981 |
done |
|
982 |
||
983 |
lemma setsum_eq_0_iff [simp]: |
|
984 |
"finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))" |
|
22262 | 985 |
by (induct set: finite) auto |
15402 | 986 |
|
987 |
lemma setsum_Un_nat: "finite A ==> finite B ==> |
|
988 |
(setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)" |
|
989 |
-- {* For the natural numbers, we have subtraction. *} |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
990 |
by (subst setsum_Un_Int [symmetric], auto simp add: ring_simps) |
15402 | 991 |
|
992 |
lemma setsum_Un: "finite A ==> finite B ==> |
|
993 |
(setsum f (A Un B) :: 'a :: ab_group_add) = |
|
994 |
setsum f A + setsum f B - setsum f (A Int B)" |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset
|
995 |
by (subst setsum_Un_Int [symmetric], auto simp add: ring_simps) |
15402 | 996 |
|
997 |
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) = |
|
998 |
(if a:A then setsum f A - f a else setsum f A)" |
|
999 |
apply (case_tac "finite A") |
|
1000 |
prefer 2 apply (simp add: setsum_def) |
|
1001 |
apply (erule finite_induct) |
|
1002 |
apply (auto simp add: insert_Diff_if) |
|
1003 |
apply (drule_tac a = a in mk_disjoint_insert, auto) |
|
1004 |
done |
|
1005 |
||
1006 |
lemma setsum_diff1: "finite A \<Longrightarrow> |
|
1007 |
(setsum f (A - {a}) :: ('a::ab_group_add)) = |
|
1008 |
(if a:A then setsum f A - f a else setsum f A)" |
|
1009 |
by (erule finite_induct) (auto simp add: insert_Diff_if) |
|
1010 |
||
15552
8ab8e425410b
added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents:
15543
diff
changeset
|
1011 |
lemma setsum_diff1'[rule_format]: "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)" |
8ab8e425410b
added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents:
15543
diff
changeset
|
1012 |
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))"]) |
8ab8e425410b
added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents:
15543
diff
changeset
|
1013 |
apply (auto simp add: insert_Diff_if add_ac) |
8ab8e425410b
added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents:
15543
diff
changeset
|
1014 |
done |
8ab8e425410b
added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents:
15543
diff
changeset
|
1015 |
|
15402 | 1016 |
(* By Jeremy Siek: *) |
1017 |
||
1018 |
lemma setsum_diff_nat: |
|
19535 | 1019 |
assumes "finite B" |
1020 |
and "B \<subseteq> A" |
|
1021 |
shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)" |
|
1022 |
using prems |
|
1023 |
proof induct |
|
15402 | 1024 |
show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp |
1025 |
next |
|
1026 |
fix F x assume finF: "finite F" and xnotinF: "x \<notin> F" |
|
1027 |
and xFinA: "insert x F \<subseteq> A" |
|
1028 |
and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F" |
|
1029 |
from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp |
|
1030 |
from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x" |
|
1031 |
by (simp add: setsum_diff1_nat) |
|
1032 |
from xFinA have "F \<subseteq> A" by simp |
|
1033 |
with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp |
|
1034 |
with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x" |
|
1035 |
by simp |
|
1036 |
from xnotinF have "A - insert x F = (A - F) - {x}" by auto |
|
1037 |
with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x" |
|
1038 |
by simp |
|
1039 |
from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp |
|
1040 |
with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" |
|
1041 |
by simp |
|
1042 |
thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp |
|
1043 |
qed |
|
1044 |
||
1045 |
lemma setsum_diff: |
|
1046 |
assumes le: "finite A" "B \<subseteq> A" |
|
1047 |
shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))" |
|
1048 |
proof - |
|
1049 |
from le have finiteB: "finite B" using finite_subset by auto |
|
1050 |
show ?thesis using finiteB le |
|
21575 | 1051 |
proof induct |
19535 | 1052 |
case empty |
1053 |
thus ?case by auto |
|
1054 |
next |
|
1055 |
case (insert x F) |
|
1056 |
thus ?case using le finiteB |
|
1057 |
by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb) |
|
15402 | 1058 |
qed |
19535 | 1059 |
qed |
15402 | 1060 |
|
1061 |
lemma setsum_mono: |
|
1062 |
assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))" |
|
1063 |
shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)" |
|
1064 |
proof (cases "finite K") |
|
1065 |
case True |
|
1066 |
thus ?thesis using le |
|
19535 | 1067 |
proof induct |
15402 | 1068 |
case empty |
1069 |
thus ?case by simp |
|
1070 |
next |
|
1071 |
case insert |
|
19535 | 1072 |
thus ?case using add_mono by fastsimp |
15402 | 1073 |
qed |
1074 |
next |
|
1075 |
case False |
|
1076 |
thus ?thesis |
|
1077 |
by (simp add: setsum_def) |
|
1078 |
qed |
|
1079 |
||
15554 | 1080 |
lemma setsum_strict_mono: |
19535 | 1081 |
fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}" |
1082 |
assumes "finite A" "A \<noteq> {}" |
|
1083 |
and "!!x. x:A \<Longrightarrow> f x < g x" |
|
1084 |
shows "setsum f A < setsum g A" |
|
1085 |
using prems |
|
15554 | 1086 |
proof (induct rule: finite_ne_induct) |
1087 |
case singleton thus ?case by simp |
|
1088 |
next |
|
1089 |
case insert thus ?case by (auto simp: add_strict_mono) |
|
1090 |
qed |
|
1091 |
||
15535 | 1092 |
lemma setsum_negf: |
19535 | 1093 |
"setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A" |
15535 | 1094 |
proof (cases "finite A") |
22262 | 1095 |
case True thus ?thesis by (induct set: finite) auto |
15535 | 1096 |
next |
1097 |
case False thus ?thesis by (simp add: setsum_def) |
|
1098 |
qed |
|
15402 | 1099 |
|
15535 | 1100 |
lemma setsum_subtractf: |
19535 | 1101 |
"setsum (%x. ((f x)::'a::ab_group_add) - g x) A = |
1102 |
setsum f A - setsum g A" |
|
15535 | 1103 |
proof (cases "finite A") |
1104 |
case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf) |
|
1105 |
next |
|
1106 |
case False thus ?thesis by (simp add: setsum_def) |
|
1107 |
qed |
|
15402 | 1108 |
|
15535 | 1109 |
lemma setsum_nonneg: |
19535 | 1110 |
assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x" |
1111 |
shows "0 \<le> setsum f A" |
|
15535 | 1112 |
proof (cases "finite A") |
1113 |
case True thus ?thesis using nn |
|
21575 | 1114 |
proof induct |
19535 | 1115 |
case empty then show ?case by simp |
1116 |
next |
|
1117 |
case (insert x F) |
|
1118 |
then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono) |
|
1119 |
with insert show ?case by simp |
|
1120 |
qed |
|
15535 | 1121 |
next |
1122 |
case False thus ?thesis by (simp add: setsum_def) |
|
1123 |
qed |
|
15402 | 1124 |
|
15535 | 1125 |
lemma setsum_nonpos: |
19535 | 1126 |
assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})" |
1127 |
shows "setsum f A \<le> 0" |
|
15535 | 1128 |
proof (cases "finite A") |
1129 |
case True thus ?thesis using np |
|
21575 | 1130 |
proof induct |
19535 | 1131 |
case empty then show ?case by simp |
1132 |
next |
|
1133 |
case (insert x F) |
|
1134 |
then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono) |
|
1135 |
with insert show ?case by simp |
|
1136 |
qed |
|
15535 | 1137 |
next |
1138 |
case False thus ?thesis by (simp add: setsum_def) |
|
1139 |
qed |
|
15402 | 1140 |
|
15539 | 1141 |
lemma setsum_mono2: |
1142 |
fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}" |
|
1143 |
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b" |
|
1144 |
shows "setsum f A \<le> setsum f B" |
|
1145 |
proof - |
|
1146 |
have "setsum f A \<le> setsum f A + setsum f (B-A)" |
|
1147 |
by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def) |
|
1148 |
also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin] |
|
1149 |
by (simp add:setsum_Un_disjoint del:Un_Diff_cancel) |
|
1150 |
also have "A \<union> (B-A) = B" using sub by blast |
|
1151 |
finally show ?thesis . |
|
1152 |
qed |
|
15542 | 1153 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1154 |
lemma setsum_mono3: "finite B ==> A <= B ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1155 |
ALL x: B - A. |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1156 |
0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1157 |
setsum f A <= setsum f B" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1158 |
apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1159 |
apply (erule ssubst) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1160 |
apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1161 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1162 |
apply (rule add_left_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1163 |
apply (erule setsum_nonneg) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1164 |
apply (subst setsum_Un_disjoint [THEN sym]) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1165 |
apply (erule finite_subset, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1166 |
apply (rule finite_subset) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1167 |
prefer 2 |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1168 |
apply assumption |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1169 |
apply auto |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1170 |
apply (rule setsum_cong) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1171 |
apply auto |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1172 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1173 |
|
19279 | 1174 |
lemma setsum_right_distrib: |
22934
64ecb3d6790a
generalize setsum lemmas from semiring_0_cancel to semiring_0
huffman
parents:
22917
diff
changeset
|
1175 |
fixes f :: "'a => ('b::semiring_0)" |
15402 | 1176 |
shows "r * setsum f A = setsum (%n. r * f n) A" |
1177 |
proof (cases "finite A") |
|
1178 |
case True |
|
1179 |
thus ?thesis |
|
21575 | 1180 |
proof induct |
15402 | 1181 |
case empty thus ?case by simp |
1182 |
next |
|
1183 |
case (insert x A) thus ?case by (simp add: right_distrib) |
|
1184 |
qed |
|
1185 |
next |
|
1186 |
case False thus ?thesis by (simp add: setsum_def) |
|
1187 |
qed |
|
1188 |
||
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1189 |
lemma setsum_left_distrib: |
22934
64ecb3d6790a
generalize setsum lemmas from semiring_0_cancel to semiring_0
huffman
parents:
22917
diff
changeset
|
1190 |
"setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)" |
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1191 |
proof (cases "finite A") |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1192 |
case True |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1193 |
then show ?thesis |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1194 |
proof induct |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1195 |
case empty thus ?case by simp |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1196 |
next |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1197 |
case (insert x A) thus ?case by (simp add: left_distrib) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1198 |
qed |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1199 |
next |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1200 |
case False thus ?thesis by (simp add: setsum_def) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1201 |
qed |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1202 |
|
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1203 |
lemma setsum_divide_distrib: |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1204 |
"setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)" |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1205 |
proof (cases "finite A") |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1206 |
case True |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1207 |
then show ?thesis |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1208 |
proof induct |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1209 |
case empty thus ?case by simp |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1210 |
next |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1211 |
case (insert x A) thus ?case by (simp add: add_divide_distrib) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1212 |
qed |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1213 |
next |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1214 |
case False thus ?thesis by (simp add: setsum_def) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1215 |
qed |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1216 |
|
15535 | 1217 |
lemma setsum_abs[iff]: |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25205
diff
changeset
|
1218 |
fixes f :: "'a => ('b::pordered_ab_group_add_abs)" |
15402 | 1219 |
shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A" |
15535 | 1220 |
proof (cases "finite A") |
1221 |
case True |
|
1222 |
thus ?thesis |
|
21575 | 1223 |
proof induct |
15535 | 1224 |
case empty thus ?case by simp |
1225 |
next |
|
1226 |
case (insert x A) |
|
1227 |
thus ?case by (auto intro: abs_triangle_ineq order_trans) |
|
1228 |
qed |
|
15402 | 1229 |
next |
15535 | 1230 |
case False thus ?thesis by (simp add: setsum_def) |
15402 | 1231 |
qed |
1232 |
||
15535 | 1233 |
lemma setsum_abs_ge_zero[iff]: |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25205
diff
changeset
|
1234 |
fixes f :: "'a => ('b::pordered_ab_group_add_abs)" |
15402 | 1235 |
shows "0 \<le> setsum (%i. abs(f i)) A" |
15535 | 1236 |
proof (cases "finite A") |
1237 |
case True |
|
1238 |
thus ?thesis |
|
21575 | 1239 |
proof induct |
15535 | 1240 |
case empty thus ?case by simp |
1241 |
next |
|
21733 | 1242 |
case (insert x A) thus ?case by (auto simp: add_nonneg_nonneg) |
15535 | 1243 |
qed |
15402 | 1244 |
next |
15535 | 1245 |
case False thus ?thesis by (simp add: setsum_def) |
15402 | 1246 |
qed |
1247 |
||
15539 | 1248 |
lemma abs_setsum_abs[simp]: |
25303
0699e20feabd
renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents:
25205
diff
changeset
|
1249 |
fixes f :: "'a => ('b::pordered_ab_group_add_abs)" |
15539 | 1250 |
shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))" |
1251 |
proof (cases "finite A") |
|
1252 |
case True |
|
1253 |
thus ?thesis |
|
21575 | 1254 |
proof induct |
15539 | 1255 |
case empty thus ?case by simp |
1256 |
next |
|
1257 |
case (insert a A) |
|
1258 |
hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp |
|
1259 |
also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>" using insert by simp |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1260 |
also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1261 |
by (simp del: abs_of_nonneg) |
15539 | 1262 |
also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp |
1263 |
finally show ?case . |
|
1264 |
qed |
|
1265 |
next |
|
1266 |
case False thus ?thesis by (simp add: setsum_def) |
|
1267 |
qed |
|
1268 |
||
15402 | 1269 |
|
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1270 |
text {* Commuting outer and inner summation *} |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1271 |
|
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1272 |
lemma swap_inj_on: |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1273 |
"inj_on (%(i, j). (j, i)) (A \<times> B)" |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1274 |
by (unfold inj_on_def) fast |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1275 |
|
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1276 |
lemma swap_product: |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1277 |
"(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A" |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1278 |
by (simp add: split_def image_def) blast |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1279 |
|
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1280 |
lemma setsum_commute: |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1281 |
"(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)" |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1282 |
proof (simp add: setsum_cartesian_product) |
17189 | 1283 |
have "(\<Sum>(x,y) \<in> A <*> B. f x y) = |
1284 |
(\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)" |
|
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1285 |
(is "?s = _") |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1286 |
apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1287 |
apply (simp add: split_def) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1288 |
done |
17189 | 1289 |
also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)" |
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1290 |
(is "_ = ?t") |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1291 |
apply (simp add: swap_product) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1292 |
done |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1293 |
finally show "?s = ?t" . |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1294 |
qed |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1295 |
|
19279 | 1296 |
lemma setsum_product: |
22934
64ecb3d6790a
generalize setsum lemmas from semiring_0_cancel to semiring_0
huffman
parents:
22917
diff
changeset
|
1297 |
fixes f :: "'a => ('b::semiring_0)" |
19279 | 1298 |
shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)" |
1299 |
by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute) |
|